Exsanguinated blood volume estimation using fractal analysis of digital images.

PubMed

Sant, Sonia P; Fairgrieve, Scott I

2012-05-01

The estimation of bloodstain volume using fractal analysis of digital images of passive blood stains is presented. Binary digital photos of bloodstains of known volumes (ranging from 1 to 7 mL), dispersed in a defined area, were subjected to image analysis using FracLac V. 2.0 for ImageJ. The box-counting method was used to generate a fractal dimension for each trial. A positive correlation between the generated fractal number and the volume of blood was found (R(2) = 0.99). Regression equations were produced to estimate the volume of blood in blind trials. An error rate ranging from 78% for 1 mL to 7% for 6 mL demonstrated that as the volume increases so does the accuracy of the volume estimation. This method used in the preliminary study proved that bloodstain patterns may be deconstructed into mathematical parameters, thus removing the subjective element inherent in other methods of volume estimation. © 2012 American Academy of Forensic Sciences.

Analysis of fractal dimensions of rat bones from film and digital images

NASA Technical Reports Server (NTRS)

Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Yamauchi, M.

2001-01-01

OBJECTIVES: (1) To compare the effect of two different intra-oral image receptors on estimates of fractal dimension; and (2) to determine the variations in fractal dimensions between the femur, tibia and humerus of the rat and between their proximal, middle and distal regions. METHODS: The left femur, tibia and humerus from 24 4-6-month-old Sprague-Dawley rats were radiographed using intra-oral film and a charge-coupled device (CCD). Films were digitized at a pixel density comparable to the CCD using a flat-bed scanner. Square regions of interest were selected from proximal, middle, and distal regions of each bone. Fractal dimensions were estimated from the slope of regression lines fitted to plots of log power against log spatial frequency. RESULTS: The fractal dimensions estimates from digitized films were significantly greater than those produced from the CCD (P=0.0008). Estimated fractal dimensions of three types of bone were not significantly different (P=0.0544); however, the three regions of bones were significantly different (P=0.0239). The fractal dimensions estimated from radiographs of the proximal and distal regions of the bones were lower than comparable estimates obtained from the middle region. CONCLUSIONS: Different types of image receptors significantly affect estimates of fractal dimension. There was no difference in the fractal dimensions of the different bones but the three regions differed significantly.

Fractal analysis and its impact factors on pore structure of artificial cores based on the images obtained using magnetic resonance imaging

NASA Astrophysics Data System (ADS)

Wang, Heming; Liu, Yu; Song, Yongchen; Zhao, Yuechao; Zhao, Jiafei; Wang, Dayong

2012-11-01

Pore structure is one of important factors affecting the properties of porous media, but it is difficult to describe the complexity of pore structure exactly. Fractal theory is an effective and available method for quantifying the complex and irregular pore structure. In this paper, the fractal dimension calculated by box-counting method based on fractal theory was applied to characterize the pore structure of artificial cores. The microstructure or pore distribution in the porous material was obtained using the nuclear magnetic resonance imaging (MRI). Three classical fractals and one sand packed bed model were selected as the experimental material to investigate the influence of box sizes, threshold value, and the image resolution when performing fractal analysis. To avoid the influence of box sizes, a sequence of divisors of the image was proposed and compared with other two algorithms (geometric sequence and arithmetic sequence) with its performance of partitioning the image completely and bringing the least fitted error. Threshold value selected manually and automatically showed that it plays an important role during the image binary processing and the minimum-error method can be used to obtain an appropriate or reasonable one. Images obtained under different pixel matrices in MRI were used to analyze the influence of image resolution. Higher image resolution can detect more quantity of pore structure and increase its irregularity. With benefits of those influence factors, fractal analysis on four kinds of artificial cores showed the fractal dimension can be used to distinguish the different kinds of artificial cores and the relationship between fractal dimension and porosity or permeability can be expressed by the model of D = a - bln(x + c).

Fractal scaling of apparent soil moisture estimated from vertical planes of Vertisol pit images

NASA Astrophysics Data System (ADS)

Cumbrera, Ramiro; Tarquis, Ana M.; Gascó, Gabriel; Millán, Humberto

2012-07-01

SummaryImage analysis could be a useful tool for investigating the spatial patterns of apparent soil moisture at multiple resolutions. The objectives of the present work were (i) to define apparent soil moisture patterns from vertical planes of Vertisol pit images and (ii) to describe the scaling of apparent soil moisture distribution using fractal parameters. Twelve soil pits (0.70 m long × 0.60 m width × 0.30 m depth) were excavated on a bare Mazic Pellic Vertisol. Six of them were excavated in April/2011 and six pits were established in May/2011 after 3 days of a moderate rainfall event. Digital photographs were taken from each Vertisol pit using a Kodak™ digital camera. The mean image size was 1600 × 945 pixels with one physical pixel ≈373 μm of the photographed soil pit. Each soil image was analyzed using two fractal scaling exponents, box counting (capacity) dimension (DBC) and interface fractal dimension (Di), and three prefractal scaling coefficients, the total number of boxes intercepting the foreground pattern at a unit scale (A), fractal lacunarity at the unit scale (Λ1) and Shannon entropy at the unit scale (S1). All the scaling parameters identified significant differences between both sets of spatial patterns. Fractal lacunarity was the best discriminator between apparent soil moisture patterns. Soil image interpretation with fractal exponents and prefractal coefficients can be incorporated within a site-specific agriculture toolbox. While fractal exponents convey information on space filling characteristics of the pattern, prefractal coefficients represent the investigated soil property as seen through a higher resolution microscope. In spite of some computational and practical limitations, image analysis of apparent soil moisture patterns could be used in connection with traditional soil moisture sampling, which always renders punctual estimates.

Plant Identification Based on Leaf Midrib Cross-Section Images Using Fractal Descriptors.

PubMed

da Silva, Núbia Rosa; Florindo, João Batista; Gómez, María Cecilia; Rossatto, Davi Rodrigo; Kolb, Rosana Marta; Bruno, Odemir Martinez

2015-01-01

The correct identification of plants is a common necessity not only to researchers but also to the lay public. Recently, computational methods have been employed to facilitate this task, however, there are few studies front of the wide diversity of plants occurring in the world. This study proposes to analyse images obtained from cross-sections of leaf midrib using fractal descriptors. These descriptors are obtained from the fractal dimension of the object computed at a range of scales. In this way, they provide rich information regarding the spatial distribution of the analysed structure and, as a consequence, they measure the multiscale morphology of the object of interest. In Biology, such morphology is of great importance because it is related to evolutionary aspects and is successfully employed to characterize and discriminate among different biological structures. Here, the fractal descriptors are used to identify the species of plants based on the image of their leaves. A large number of samples are examined, being 606 leaf samples of 50 species from Brazilian flora. The results are compared to other imaging methods in the literature and demonstrate that fractal descriptors are precise and reliable in the taxonomic process of plant species identification.

Texture segmentation of non-cooperative spacecrafts images based on wavelet and fractal dimension

NASA Astrophysics Data System (ADS)

Wu, Kanzhi; Yue, Xiaokui

2011-06-01

With the increase of on-orbit manipulations and space conflictions, missions such as tracking and capturing the target spacecrafts are aroused. Unlike cooperative spacecrafts, fixing beacons or any other marks on the targets is impossible. Due to the unknown shape and geometry features of non-cooperative spacecraft, in order to localize the target and obtain the latitude, we need to segment the target image and recognize the target from the background. The data and errors during the following procedures such as feature extraction and matching can also be reduced. Multi-resolution analysis of wavelet theory reflects human beings' recognition towards images from low resolution to high resolution. In addition, spacecraft is the only man-made object in the image compared to the natural background and the differences will be certainly observed between the fractal dimensions of target and background. Combined wavelet transform and fractal dimension, in this paper, we proposed a new segmentation algorithm for the images which contains complicated background such as the universe and planet surfaces. At first, Daubechies wavelet basis is applied to decompose the image in both x axis and y axis, thus obtain four sub-images. Then, calculate the fractal dimensions in four sub-images using different methods; after analyzed the results of fractal dimensions in sub-images, we choose Differential Box Counting in low resolution image as the principle to segment the texture which has the greatest divergences between different sub-images. This paper also presents the results of experiments by using the algorithm above. It is demonstrated that an accurate texture segmentation result can be obtained using the proposed technique.

Time Series Analysis OF SAR Image Fractal Maps: The Somma-Vesuvio Volcanic Complex Case Study

NASA Astrophysics Data System (ADS)

Pepe, Antonio; De Luca, Claudio; Di Martino, Gerardo; Iodice, Antonio; Manzo, Mariarosaria; Pepe, Susi; Riccio, Daniele; Ruello, Giuseppe; Sansosti, Eugenio; Zinno, Ivana

2016-04-01

The fractal dimension is a significant geophysical parameter describing natural surfaces representing the distribution of the roughness over different spatial scale; in case of volcanic structures, it has been related to the specific nature of materials and to the effects of active geodynamic processes. In this work, we present the analysis of the temporal behavior of the fractal dimension estimates generated from multi-pass SAR images relevant to the Somma-Vesuvio volcanic complex (South Italy). To this aim, we consider a Cosmo-SkyMed data-set of 42 stripmap images acquired from ascending orbits between October 2009 and December 2012. Starting from these images, we generate a three-dimensional stack composed by the corresponding fractal maps (ordered according to the acquisition dates), after a proper co-registration. The time-series of the pixel-by-pixel estimated fractal dimension values show that, over invariant natural areas, the fractal dimension values do not reveal significant changes; on the contrary, over urban areas, it correctly assumes values outside the natural surfaces fractality range and show strong fluctuations. As a final result of our analysis, we generate a fractal map that includes only the areas where the fractal dimension is considered reliable and stable (i.e., whose standard deviation computed over the time series is reasonably small). The so-obtained fractal dimension map is then used to identify areas that are homogeneous from a fractal viewpoint. Indeed, the analysis of this map reveals the presence of two distinctive landscape units corresponding to the Mt. Vesuvio and Gran Cono. The comparison with the (simplified) geological map clearly shows the presence in these two areas of volcanic products of different age. The presented fractal dimension map analysis demonstrates the ability to get a figure about the evolution degree of the monitored volcanic edifice and can be profitably extended in the future to other volcanic systems with

Image characterization by fractal descriptors in variational mode decomposition domain: Application to brain magnetic resonance

NASA Astrophysics Data System (ADS)

Lahmiri, Salim

2016-08-01

The main purpose of this work is to explore the usefulness of fractal descriptors estimated in multi-resolution domains to characterize biomedical digital image texture. In this regard, three multi-resolution techniques are considered: the well-known discrete wavelet transform (DWT) and the empirical mode decomposition (EMD), and; the newly introduced; variational mode decomposition mode (VMD). The original image is decomposed by the DWT, EMD, and VMD into different scales. Then, Fourier spectrum based fractal descriptors is estimated at specific scales and directions to characterize the image. The support vector machine (SVM) was used to perform supervised classification. The empirical study was applied to the problem of distinguishing between normal and abnormal brain magnetic resonance images (MRI) affected with Alzheimer disease (AD). Our results demonstrate that fractal descriptors estimated in VMD domain outperform those estimated in DWT and EMD domains; and also those directly estimated from the original image.

Mathematical models used in segmentation and fractal methods of 2-D ultrasound images

NASA Astrophysics Data System (ADS)

Moldovanu, Simona; Moraru, Luminita; Bibicu, Dorin

2012-11-01

Mathematical models are widely used in biomedical computing. The extracted data from images using the mathematical techniques are the "pillar" achieving scientific progress in experimental, clinical, biomedical, and behavioural researches. This article deals with the representation of 2-D images and highlights the mathematical support for the segmentation operation and fractal analysis in ultrasound images. A large number of mathematical techniques are suitable to be applied during the image processing stage. The addressed topics cover the edge-based segmentation, more precisely the gradient-based edge detection and active contour model, and the region-based segmentation namely Otsu method. Another interesting mathematical approach consists of analyzing the images using the Box Counting Method (BCM) to compute the fractal dimension. The results of the paper provide explicit samples performed by various combination of methods.

A Double-Minded Fractal

ERIC Educational Resources Information Center

Simoson, Andrew J.

2009-01-01

This article presents a fun activity of generating a double-minded fractal image for a linear algebra class once the idea of rotation and scaling matrices are introduced. In particular the fractal flip-flops between two words, depending on the level at which the image is viewed. (Contains 5 figures.)

Singular spectrum decomposition of Bouligand-Minkowski fractal descriptors: an application to the classification of texture Images

NASA Astrophysics Data System (ADS)

Florindo, João. Batista

2018-04-01

This work proposes the use of Singular Spectrum Analysis (SSA) for the classification of texture images, more specifically, to enhance the performance of the Bouligand-Minkowski fractal descriptors in this task. Fractal descriptors are known to be a powerful approach to model and particularly identify complex patterns in natural images. Nevertheless, the multiscale analysis involved in those descriptors makes them highly correlated. Although other attempts to address this point was proposed in the literature, none of them investigated the relation between the fractal correlation and the well-established analysis employed in time series. And SSA is one of the most powerful techniques for this purpose. The proposed method was employed for the classification of benchmark texture images and the results were compared with other state-of-the-art classifiers, confirming the potential of this analysis in image classification.

Fractal structures and fractal functions as disease indicators

USGS Publications Warehouse

Escos, J.M; Alados, C.L.; Emlen, J.M.

1995-01-01

Developmental instability is an early indicator of stress, and has been used to monitor the impacts of human disturbance on natural ecosystems. Here we investigate the use of different measures of developmental instability on two species, green peppers (Capsicum annuum), a plant, and Spanish ibex (Capra pyrenaica), an animal. For green peppers we compared the variance in allometric relationship between control plants, and a treatment group infected with the tomato spotted wilt virus. The results show that infected plants have a greater variance about the allometric regression line than the control plants. We also observed a reduction in complexity of branch structure in green pepper with a viral infection. Box-counting fractal dimension of branch architecture declined under stress infection. We also tested the reduction in complexity of behavioral patterns under stress situations in Spanish ibex (Capra pyrenaica). Fractal dimension of head-lift frequency distribution measures predator detection efficiency. This dimension decreased under stressful conditions, such as advanced pregnancy and parasitic infection. Feeding distribution activities reflect food searching efficiency. Power spectral analysis proves to be the most powerful tool for character- izing fractal behavior, revealing a reduction in complexity of time distribution activity under parasitic infection.

A Fractal Analysis of CT Liver Images for the Discrimination of Hepatic Lesions: A Comparative Study

DTIC Science & Technology

2001-10-25

liver images in order to estimate their fractal dimension and to differentiate normal liver parenchyma from hepatocellular carcinoma . Four fractal...methods; thus discriminating up to 93% of the normal parenchyma and up to 82% of the hepatocellular carcinoma , correctly.

Multitemporal and Multiscaled Fractal Analysis of Landsat Satellite Data Using the Image Characterization and Modeling System (ICAMS)

NASA Technical Reports Server (NTRS)

Quattrochi, Dale A.; Emerson, Charles W.; Lam, Nina Siu-Ngan; Laymon, Charles A.

1997-01-01

The Image Characterization And Modeling System (ICAMS) is a public domain software package that is designed to provide scientists with innovative spatial analytical tools to visualize, measure, and characterize landscape patterns so that environmental conditions or processes can be assessed and monitored more effectively. In this study ICAMS has been used to evaluate how changes in fractal dimension, as a landscape characterization index, and resolution, are related to differences in Landsat images collected at different dates for the same area. Landsat Thematic Mapper (TM) data obtained in May and August 1993 over a portion of the Great Basin Desert in eastern Nevada were used for analysis. These data represent contrasting periods of peak "green-up" and "dry-down" for the study area. The TM data sets were converted into Normalized Difference Vegetation Index (NDVI) images to expedite analysis of differences in fractal dimension between the two dates. These NDVI images were also resampled to resolutions of 60, 120, 240, 480, and 960 meters from the original 30 meter pixel size, to permit an assessment of how fractal dimension varies with spatial resolution. Tests of fractal dimension for two dates at various pixel resolutions show that the D values in the August image become increasingly more complex as pixel size increases to 480 meters. The D values in the May image show an even more complex relationship to pixel size than that expressed in the August image. Fractal dimension for a difference image computed for the May and August dates increase with pixel size up to a resolution of 120 meters, and then decline with increasing pixel size. This means that the greatest complexity in the difference images occur around a resolution of 120 meters, which is analogous to the operational domain of changes in vegetation and snow cover that constitute differences between the two dates.

A new version of Visual tool for estimating the fractal dimension of images

NASA Astrophysics Data System (ADS)

Grossu, I. V.; Felea, D.; Besliu, C.; Jipa, Al.; Bordeianu, C. C.; Stan, E.; Esanu, T.

2010-04-01

-dimensional case. Summary of revisions:The application interface was changed from SDI (single document interface) to MDI (multi-document interface). One form was added in order to provide a graphical user interface for the new functionalities (fractal analysis of 2D and 3D images stored in csv files). Additional comments: User friendly graphical interface; Easy deployment mechanism. Running time: In the first approximation, the algorithm is linear. References:[1] I.V. Grossu, C. Besliu, M.V. Rusu, Al. Jipa, C.C. Bordeianu, D. Felea, Comput. Phys. Comm. 180 (2009) 1999-2001.[2] F. Balena, Programming Microsoft Visual Basic 6.0, Microsoft Press, US, 1999.

Classification of diabetic retinopathy using fractal dimension analysis of eye fundus image

NASA Astrophysics Data System (ADS)

Safitri, Diah Wahyu; Juniati, Dwi

2017-08-01

Diabetes Mellitus (DM) is a metabolic disorder when pancreas produce inadequate insulin or a condition when body resist insulin action, so the blood glucose level is high. One of the most common complications of diabetes mellitus is diabetic retinopathy which can lead to a vision problem. Diabetic retinopathy can be recognized by an abnormality in eye fundus. Those abnormalities are characterized by microaneurysms, hemorrhage, hard exudate, cotton wool spots, and venous's changes. The diabetic retinopathy is classified depends on the conditions of abnormality in eye fundus, that is grade 1 if there is a microaneurysm only in the eye fundus; grade 2, if there are a microaneurysm and a hemorrhage in eye fundus; and grade 3: if there are microaneurysm, hemorrhage, and neovascularization in the eye fundus. This study proposed a method and a process of eye fundus image to classify of diabetic retinopathy using fractal analysis and K-Nearest Neighbor (KNN). The first phase was image segmentation process using green channel, CLAHE, morphological opening, matched filter, masking, and morphological opening binary image. After segmentation process, its fractal dimension was calculated using box-counting method and the values of fractal dimension were analyzed to make a classification of diabetic retinopathy. Tests carried out by used k-fold cross validation method with k=5. In each test used 10 different grade K of KNN. The accuracy of the result of this method is 89,17% with K=3 or K=4, it was the best results than others K value. Based on this results, it can be concluded that the classification of diabetic retinopathy using fractal analysis and KNN had a good performance.

Buried mine detection using fractal geometry analysis to the LWIR successive line scan data image

NASA Astrophysics Data System (ADS)

Araki, Kan

2012-06-01

We have engaged in research on buried mine/IED detection by remote sensing method using LWIR camera. A IR image of a ground, containing buried objects can be assumed as a superimposed pattern including thermal scattering which may depend on the ground surface roughness, vegetation canopy, and effect of the sun light, and radiation due to various heat interaction caused by differences in specific heat, size, and buried depth of the objects and local temperature of their surrounding environment. In this cumbersome environment, we introduce fractal geometry for analyzing from an IR image. Clutter patterns due to these complex elements have oftentimes low ordered fractal dimension of Hausdorff Dimension. On the other hand, the target patterns have its tendency of obtaining higher ordered fractal dimension in terms of Information Dimension. Random Shuffle Surrogate method or Fourier Transform Surrogate method is used to evaluate fractional statistics by applying shuffle of time sequence data or phase of spectrum. Fractal interpolation to each line scan was also applied to improve the signal processing performance in order to evade zero division and enhance information of data. Some results of target extraction by using relationship between low and high ordered fractal dimension are to be presented.

The Conundrum of Functional Brain Networks: Small-World Efficiency or Fractal Modularity

PubMed Central

Gallos, Lazaros K.; Sigman, Mariano; Makse, Hernán A.

2012-01-01

The human brain has been studied at multiple scales, from neurons, circuits, areas with well-defined anatomical and functional boundaries, to large-scale functional networks which mediate coherent cognition. In a recent work, we addressed the problem of the hierarchical organization in the brain through network analysis. Our analysis identified functional brain modules of fractal structure that were inter-connected in a small-world topology. Here, we provide more details on the use of network science tools to elaborate on this behavior. We indicate the importance of using percolation theory to highlight the modular character of the functional brain network. These modules present a fractal, self-similar topology, identified through fractal network methods. When we lower the threshold of correlations to include weaker ties, the network as a whole assumes a small-world character. These weak ties are organized precisely as predicted by theory maximizing information transfer with minimal wiring costs. PMID:22586406

A Comparison of Local Variance, Fractal Dimension, and Moran's I as Aids to Multispectral Image Classification

NASA Technical Reports Server (NTRS)

Emerson, Charles W.; Sig-NganLam, Nina; Quattrochi, Dale A.

2004-01-01

The accuracy of traditional multispectral maximum-likelihood image classification is limited by the skewed statistical distributions of reflectances from the complex heterogenous mixture of land cover types in urban areas. This work examines the utility of local variance, fractal dimension and Moran's I index of spatial autocorrelation in segmenting multispectral satellite imagery. Tools available in the Image Characterization and Modeling System (ICAMS) were used to analyze Landsat 7 imagery of Atlanta, Georgia. Although segmentation of panchromatic images is possible using indicators of spatial complexity, different land covers often yield similar values of these indices. Better results are obtained when a surface of local fractal dimension or spatial autocorrelation is combined as an additional layer in a supervised maximum-likelihood multispectral classification. The addition of fractal dimension measures is particularly effective at resolving land cover classes within urbanized areas, as compared to per-pixel spectral classification techniques.

Analysis of the fractal dimension of volcano geomorphology through Synthetic Aperture Radar (SAR) amplitude images acquired in C and X band.

NASA Astrophysics Data System (ADS)

Pepe, S.; Di Martino, G.; Iodice, A.; Manzo, M.; Pepe, A.; Riccio, D.; Ruello, G.; Sansosti, E.; Tizzani, P.; Zinno, I.

2012-04-01

In the last two decades several aspects relevant to volcanic activity have been analyzed in terms of fractal parameters that effectively describe natural objects geometry. More specifically, these researches have been aimed at the identification of (1) the power laws that governed the magma fragmentation processes, (2) the energy of explosive eruptions, and (3) the distribution of the associated earthquakes. In this paper, the study of volcano morphology via satellite images is dealt with; in particular, we use the complete forward model developed by some of the authors (Di Martino et al., 2012) that links the stochastic characterization of amplitude Synthetic Aperture Radar (SAR) images to the fractal dimension of the imaged surfaces, modelled via fractional Brownian motion (fBm) processes. Based on the inversion of such a model, a SAR image post-processing has been implemented (Di Martino et al., 2010), that allows retrieving the fractal dimension of the observed surfaces, dictating the distribution of the roughness over different spatial scales. The fractal dimension of volcanic structures has been related to the specific nature of materials and to the effects of active geodynamic processes. Hence, the possibility to estimate the fractal dimension from a single amplitude-only SAR image is of fundamental importance for the characterization of volcano structures and, moreover, can be very helpful for monitoring and crisis management activities in case of eruptions and other similar natural hazards. The implemented SAR image processing performs the extraction of the point-by-point fractal dimension of the scene observed by the sensor, providing - as an output product - the map of the fractal dimension of the area of interest. In this work, such an analysis is performed on Cosmo-SkyMed, ERS-1/2 and ENVISAT images relevant to active stratovolcanoes in different geodynamic contexts, such as Mt. Somma-Vesuvio, Mt. Etna, Vulcano and Stromboli in Southern Italy, Shinmoe

Fractal characterization and wettability of ion treated silicon surfaces

NASA Astrophysics Data System (ADS)

Yadav, R. P.; Kumar, Tanuj; Baranwal, V.; Vandana, Kumar, Manvendra; Priya, P. K.; Pandey, S. N.; Mittal, A. K.

2017-02-01

Fractal characterization of surface morphology can be useful as a tool for tailoring the wetting properties of solid surfaces. In this work, rippled surfaces of Si (100) are grown using 200 keV Ar+ ion beam irradiation at different ion doses. Relationship between fractal and wetting properties of these surfaces are explored. The height-height correlation function extracted from atomic force microscopic images, demonstrates an increase in roughness exponent with an increase in ion doses. A steep variation in contact angle values is found for low fractal dimensions. Roughness exponent and fractal dimensions are found correlated with the static water contact angle measurement. It is observed that after a crossover of the roughness exponent, the surface morphology has a rippled structure. Larger values of interface width indicate the larger ripples on the surface. The contact angle of water drops on such surfaces is observed to be lowest. Autocorrelation function is used for the measurement of ripple wavelength.

An Evaluation of Fractal Surface Measurement Methods for Characterizing Landscape Complexity from Remote-Sensing Imagery

NASA Technical Reports Server (NTRS)

Lam, Nina Siu-Ngan; Qiu, Hong-Lie; Quattrochi, Dale A.; Emerson, Charles W.; Arnold, James E. (Technical Monitor)

2001-01-01

The rapid increase in digital data volumes from new and existing sensors necessitates the need for efficient analytical tools for extracting information. We developed an integrated software package called ICAMS (Image Characterization and Modeling System) to provide specialized spatial analytical functions for interpreting remote sensing data. This paper evaluates the three fractal dimension measurement methods: isarithm, variogram, and triangular prism, along with the spatial autocorrelation measurement methods Moran's I and Geary's C, that have been implemented in ICAMS. A modified triangular prism method was proposed and implemented. Results from analyzing 25 simulated surfaces having known fractal dimensions show that both the isarithm and triangular prism methods can accurately measure a range of fractal surfaces. The triangular prism method is most accurate at estimating the fractal dimension of higher spatial complexity, but it is sensitive to contrast stretching. The variogram method is a comparatively poor estimator for all of the surfaces, particularly those with higher fractal dimensions. Similar to the fractal techniques, the spatial autocorrelation techniques are found to be useful to measure complex images but not images with low dimensionality. These fractal measurement methods can be applied directly to unclassified images and could serve as a tool for change detection and data mining.

Fractal and Gray Level Cooccurrence Matrix Computational Analysis of Primary Osteosarcoma Magnetic Resonance Images Predicts the Chemotherapy Response.

PubMed

Djuričić, Goran J; Radulovic, Marko; Sopta, Jelena P; Nikitović, Marina; Milošević, Nebojša T

2017-01-01

The prediction of induction chemotherapy response at the time of diagnosis may improve outcomes in osteosarcoma by allowing for personalized tailoring of therapy. The aim of this study was thus to investigate the predictive potential of the so far unexploited computational analysis of osteosarcoma magnetic resonance (MR) images. Fractal and gray level cooccurrence matrix (GLCM) algorithms were employed in retrospective analysis of MR images of primary osteosarcoma localized in distal femur prior to the OsteoSa induction chemotherapy. The predicted and actual chemotherapy response outcomes were then compared by means of receiver operating characteristic (ROC) analysis and accuracy calculation. Dbin, Λ, and SCN were the standard fractal and GLCM features which significantly associated with the chemotherapy outcome, but only in one of the analyzed planes. Our newly developed normalized fractal dimension, called the space-filling ratio (SFR) exerted an independent and much better predictive value with the prediction significance accomplished in two of the three imaging planes, with accuracy of 82% and area under the ROC curve of 0.20 (95% confidence interval 0-0.41). In conclusion, SFR as the newly designed fractal coefficient provided superior predictive performance in comparison to standard image analysis features, presumably by compensating for the tumor size variation in MR images.

Zone specific fractal dimension of retinal images as predictor of stroke incidence.

PubMed

Aliahmad, Behzad; Kumar, Dinesh Kant; Hao, Hao; Unnikrishnan, Premith; Che Azemin, Mohd Zulfaezal; Kawasaki, Ryo; Mitchell, Paul

2014-01-01

Fractal dimensions (FDs) are frequently used for summarizing the complexity of retinal vascular. However, previous techniques on this topic were not zone specific. A new methodology to measure FD of a specific zone in retinal images has been developed and tested as a marker for stroke prediction. Higuchi's fractal dimension was measured in circumferential direction (FDC) with respect to optic disk (OD), in three concentric regions between OD boundary and 1.5 OD diameter from its margin. The significance of its association with future episode of stroke event was tested using the Blue Mountain Eye Study (BMES) database and compared against spectrum fractal dimension (SFD) and box-counting (BC) dimension. Kruskal-Wallis analysis revealed FDC as a better predictor of stroke (H = 5.80, P = 0.016, α = 0.05) compared with SFD (H = 0.51, P = 0.475, α = 0.05) and BC (H = 0.41, P = 0.520, α = 0.05) with overall lower median value for the cases compared to the control group. This work has shown that there is a significant association between zone specific FDC of eye fundus images with future episode of stroke while this difference is not significant when other FD methods are employed.

The fractal heart — embracing mathematics in the cardiology clinic

PubMed Central

Captur, Gabriella; Karperien, Audrey L.; Hughes, Alun D.; Francis, Darrel P.; Moon, James C.

2017-01-01

For clinicians grappling with quantifying the complex spatial and temporal patterns of cardiac structure and function (such as myocardial trabeculae, coronary microvascular anatomy, tissue perfusion, myocyte histology, electrical conduction, heart rate, and blood-pressure variability), fractal analysis is a powerful, but still underused, mathematical tool. In this Perspectives article, we explain some fundamental principles of fractal geometry and place it in a familiar medical setting. We summarize studies in the cardiovascular sciences in which fractal methods have successfully been used to investigate disease mechanisms, and suggest potential future clinical roles in cardiac imaging and time series measurements. We believe that clinical researchers can deploy innovative fractal solutions to common cardiac problems that might ultimately translate into advancements for patient care. PMID:27708281

Comparison of two fractal interpolation methods

NASA Astrophysics Data System (ADS)

Fu, Yang; Zheng, Zeyu; Xiao, Rui; Shi, Haibo

2017-03-01

As a tool for studying complex shapes and structures in nature, fractal theory plays a critical role in revealing the organizational structure of the complex phenomenon. Numerous fractal interpolation methods have been proposed over the past few decades, but they differ substantially in the form features and statistical properties. In this study, we simulated one- and two-dimensional fractal surfaces by using the midpoint displacement method and the Weierstrass-Mandelbrot fractal function method, and observed great differences between the two methods in the statistical characteristics and autocorrelation features. From the aspect of form features, the simulations of the midpoint displacement method showed a relatively flat surface which appears to have peaks with different height as the fractal dimension increases. While the simulations of the Weierstrass-Mandelbrot fractal function method showed a rough surface which appears to have dense and highly similar peaks as the fractal dimension increases. From the aspect of statistical properties, the peak heights from the Weierstrass-Mandelbrot simulations are greater than those of the middle point displacement method with the same fractal dimension, and the variances are approximately two times larger. When the fractal dimension equals to 1.2, 1.4, 1.6, and 1.8, the skewness is positive with the midpoint displacement method and the peaks are all convex, but for the Weierstrass-Mandelbrot fractal function method the skewness is both positive and negative with values fluctuating in the vicinity of zero. The kurtosis is less than one with the midpoint displacement method, and generally less than that of the Weierstrass-Mandelbrot fractal function method. The autocorrelation analysis indicated that the simulation of the midpoint displacement method is not periodic with prominent randomness, which is suitable for simulating aperiodic surface. While the simulation of the Weierstrass-Mandelbrot fractal function method has

Investigation into How 8th Grade Students Define Fractals

ERIC Educational Resources Information Center

Karakus, Fatih

2015-01-01

The analysis of 8th grade students' concept definitions and concept images can provide information about their mental schema of fractals. There is limited research on students' understanding and definitions of fractals. Therefore, this study aimed to investigate the elementary students' definitions of fractals based on concept image and concept…

Characterisation of human non-proliferative diabetic retinopathy using the fractal analysis

PubMed Central

Ţălu, Ştefan; Călugăru, Dan Mihai; Lupaşcu, Carmen Alina

2015-01-01

AIM To investigate and quantify changes in the branching patterns of the retina vascular network in diabetes using the fractal analysis method. METHODS This was a clinic-based prospective study of 172 participants managed at the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and December 2013. A set of 172 segmented and skeletonized human retinal images, corresponding to both normal (24 images) and pathological (148 images) states of the retina were examined. An automatic unsupervised method for retinal vessel segmentation was applied before fractal analysis. The fractal analyses of the retinal digital images were performed using the fractal analysis software ImageJ. Statistical analyses were performed for these groups using Microsoft Office Excel 2003 and GraphPad InStat software. RESULTS It was found that subtle changes in the vascular network geometry of the human retina are influenced by diabetic retinopathy (DR) and can be estimated using the fractal geometry. The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is slightly lower than the corresponding values of mild non-proliferative DR (NPDR) images (segmented and skeletonized versions). The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is higher than the corresponding values of moderate NPDR images (segmented and skeletonized versions). The lowest values were found for the corresponding values of severe NPDR images (segmented and skeletonized versions). CONCLUSION The fractal analysis of fundus photographs may be used for a more complete undeTrstanding of the early and basic pathophysiological mechanisms of diabetes. The architecture of the retinal microvasculature in diabetes can be quantitative quantified by means of the fractal dimension. Microvascular abnormalities on retinal imaging may elucidate early mechanistic pathways for microvascular complications and distinguish patients with DR from

Characterisation of human non-proliferative diabetic retinopathy using the fractal analysis.

PubMed

Ţălu, Ştefan; Călugăru, Dan Mihai; Lupaşcu, Carmen Alina

2015-01-01

To investigate and quantify changes in the branching patterns of the retina vascular network in diabetes using the fractal analysis method. This was a clinic-based prospective study of 172 participants managed at the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and December 2013. A set of 172 segmented and skeletonized human retinal images, corresponding to both normal (24 images) and pathological (148 images) states of the retina were examined. An automatic unsupervised method for retinal vessel segmentation was applied before fractal analysis. The fractal analyses of the retinal digital images were performed using the fractal analysis software ImageJ. Statistical analyses were performed for these groups using Microsoft Office Excel 2003 and GraphPad InStat software. It was found that subtle changes in the vascular network geometry of the human retina are influenced by diabetic retinopathy (DR) and can be estimated using the fractal geometry. The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is slightly lower than the corresponding values of mild non-proliferative DR (NPDR) images (segmented and skeletonized versions). The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is higher than the corresponding values of moderate NPDR images (segmented and skeletonized versions). The lowest values were found for the corresponding values of severe NPDR images (segmented and skeletonized versions). The fractal analysis of fundus photographs may be used for a more complete undeTrstanding of the early and basic pathophysiological mechanisms of diabetes. The architecture of the retinal microvasculature in diabetes can be quantitative quantified by means of the fractal dimension. Microvascular abnormalities on retinal imaging may elucidate early mechanistic pathways for microvascular complications and distinguish patients with DR from healthy individuals.

The analysis of the influence of fractal structure of stimuli on fractal dynamics in fixational eye movements and EEG signal

NASA Astrophysics Data System (ADS)

Namazi, Hamidreza; Kulish, Vladimir V.; Akrami, Amin

2016-05-01

One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders.

Fractal design concepts for stretchable electronics.

PubMed

Fan, Jonathan A; Yeo, Woon-Hong; Su, Yewang; Hattori, Yoshiaki; Lee, Woosik; Jung, Sung-Young; Zhang, Yihui; Liu, Zhuangjian; Cheng, Huanyu; Falgout, Leo; Bajema, Mike; Coleman, Todd; Gregoire, Dan; Larsen, Ryan J; Huang, Yonggang; Rogers, John A

2014-01-01

Stretchable electronics provide a foundation for applications that exceed the scope of conventional wafer and circuit board technologies due to their unique capacity to integrate with soft materials and curvilinear surfaces. The range of possibilities is predicated on the development of device architectures that simultaneously offer advanced electronic function and compliant mechanics. Here we report that thin films of hard electronic materials patterned in deterministic fractal motifs and bonded to elastomers enable unusual mechanics with important implications in stretchable device design. In particular, we demonstrate the utility of Peano, Greek cross, Vicsek and other fractal constructs to yield space-filling structures of electronic materials, including monocrystalline silicon, for electrophysiological sensors, precision monitors and actuators, and radio frequency antennas. These devices support conformal mounting on the skin and have unique properties such as invisibility under magnetic resonance imaging. The results suggest that fractal-based layouts represent important strategies for hard-soft materials integration.

Fractal design concepts for stretchable electronics

NASA Astrophysics Data System (ADS)

Fan, Jonathan A.; Yeo, Woon-Hong; Su, Yewang; Hattori, Yoshiaki; Lee, Woosik; Jung, Sung-Young; Zhang, Yihui; Liu, Zhuangjian; Cheng, Huanyu; Falgout, Leo; Bajema, Mike; Coleman, Todd; Gregoire, Dan; Larsen, Ryan J.; Huang, Yonggang; Rogers, John A.

2014-02-01

Stretchable electronics provide a foundation for applications that exceed the scope of conventional wafer and circuit board technologies due to their unique capacity to integrate with soft materials and curvilinear surfaces. The range of possibilities is predicated on the development of device architectures that simultaneously offer advanced electronic function and compliant mechanics. Here we report that thin films of hard electronic materials patterned in deterministic fractal motifs and bonded to elastomers enable unusual mechanics with important implications in stretchable device design. In particular, we demonstrate the utility of Peano, Greek cross, Vicsek and other fractal constructs to yield space-filling structures of electronic materials, including monocrystalline silicon, for electrophysiological sensors, precision monitors and actuators, and radio frequency antennas. These devices support conformal mounting on the skin and have unique properties such as invisibility under magnetic resonance imaging. The results suggest that fractal-based layouts represent important strategies for hard-soft materials integration.

Fractal analysis of scatter imaging signatures to distinguish breast pathologies

NASA Astrophysics Data System (ADS)

Eguizabal, Alma; Laughney, Ashley M.; Krishnaswamy, Venkataramanan; Wells, Wendy A.; Paulsen, Keith D.; Pogue, Brian W.; López-Higuera, José M.; Conde, Olga M.

2013-02-01

Fractal analysis combined with a label-free scattering technique is proposed for describing the pathological architecture of tumors. Clinicians and pathologists are conventionally trained to classify abnormal features such as structural irregularities or high indices of mitosis. The potential of fractal analysis lies in the fact of being a morphometric measure of the irregular structures providing a measure of the object's complexity and self-similarity. As cancer is characterized by disorder and irregularity in tissues, this measure could be related to tumor growth. Fractal analysis has been probed in the understanding of the tumor vasculature network. This work addresses the feasibility of applying fractal analysis to the scattering power map (as a physical modeling) and principal components (as a statistical modeling) provided by a localized reflectance spectroscopic system. Disorder, irregularity and cell size variation in tissue samples is translated into the scattering power and principal components magnitude and its fractal dimension is correlated with the pathologist assessment of the samples. The fractal dimension is computed applying the box-counting technique. Results show that fractal analysis of ex-vivo fresh tissue samples exhibits separated ranges of fractal dimension that could help classifier combining the fractal results with other morphological features. This contrast trend would help in the discrimination of tissues in the intraoperative context and may serve as a useful adjunct to surgeons.

The fractal geometry of life.

PubMed

Losa, Gabriele A

2009-01-01

The extension of the concepts of Fractal Geometry (Mandelbrot [1983]) toward the life sciences has led to significant progress in understanding complex functional properties and architectural / morphological / structural features characterising cells and tissues during ontogenesis and both normal and pathological development processes. It has even been argued that fractal geometry could provide a coherent description of the design principles underlying living organisms (Weibel [1991]). Fractals fulfil a certain number of theoretical and methodological criteria including a high level of organization, shape irregularity, functional and morphological self-similarity, scale invariance, iterative pathways and a peculiar non-integer fractal dimension [FD]. Whereas mathematical objects are deterministic invariant or self-similar over an unlimited range of scales, biological components are statistically self-similar only within a fractal domain defined by upper and lower limits, called scaling window, in which the relationship between the scale of observation and the measured size or length of the object can be established (Losa and Nonnenmacher [1996]). Selected examples will contribute to depict complex biological shapes and structures as fractal entities, and also to show why the application of the fractal principle is valuable for measuring dimensional, geometrical and functional parameters of cells, tissues and organs occurring within the vegetal and animal realms. If the criteria for a strict description of natural fractals are met, then it follows that a Fractal Geometry of Life may be envisaged and all natural objects and biological systems exhibiting self-similar patterns and scaling properties may be considered as belonging to the new subdiscipline of "fractalomics".

Detection and classification of Breast Cancer in Wavelet Sub-bands of Fractal Segmented Cancerous Zones.

PubMed

Shirazinodeh, Alireza; Noubari, Hossein Ahmadi; Rabbani, Hossein; Dehnavi, Alireza Mehri

2015-01-01

Recent studies on wavelet transform and fractal modeling applied on mammograms for the detection of cancerous tissues indicate that microcalcifications and masses can be utilized for the study of the morphology and diagnosis of cancerous cases. It is shown that the use of fractal modeling, as applied to a given image, can clearly discern cancerous zones from noncancerous areas. In this paper, for fractal modeling, the original image is first segmented into appropriate fractal boxes followed by identifying the fractal dimension of each windowed section using a computationally efficient two-dimensional box-counting algorithm. Furthermore, using appropriate wavelet sub-bands and image Reconstruction based on modified wavelet coefficients, it is shown that it is possible to arrive at enhanced features for detection of cancerous zones. In this paper, we have attempted to benefit from the advantages of both fractals and wavelets by introducing a new algorithm. By using a new algorithm named F1W2, the original image is first segmented into appropriate fractal boxes, and the fractal dimension of each windowed section is extracted. Following from that, by applying a maximum level threshold on fractal dimensions matrix, the best-segmented boxes are selected. In the next step, the segmented Cancerous zones which are candidates are then decomposed by utilizing standard orthogonal wavelet transform and db2 wavelet in three different resolution levels, and after nullifying wavelet coefficients of the image at the first scale and low frequency band of the third scale, the modified reconstructed image is successfully utilized for detection of breast cancer regions by applying an appropriate threshold. For detection of cancerous zones, our simulations indicate the accuracy of 90.9% for masses and 88.99% for microcalcifications detection results using the F1W2 method. For classification of detected mictocalcification into benign and malignant cases, eight features are identified and

Holographic Characterization of Colloidal Fractal Aggregates

NASA Astrophysics Data System (ADS)

Wang, Chen; Cheong, Fook Chiong; Ruffner, David B.; Zhong, Xiao; Ward, Michael D.; Grier, David G.

In-line holographic microscopy images of micrometer-scale fractal aggregates can be interpreted with the Lorenz-Mie theory of light scattering and an effective-sphere model to obtain each aggregate's size and the population-averaged fractal dimension. We demonstrate this technique experimentally using model fractal clusters of polystyrene nanoparticles and fractal protein aggregates composed of bovine serum albumin and bovine pancreas insulin. This technique can characterize several thousand aggregates in ten minutes and naturally distinguishes aggregates from contaminants such as silicone oil droplets. Work supported by the SBIR program of the NSF.

MORPH-II, a software package for the analysis of scanning-electron-micrograph images for the assessment of the fractal dimension of exposed stone surfaces

USGS Publications Warehouse

Mossotti, Victor G.; Eldeeb, A. Raouf

2000-01-01

Turcotte, 1997, and Barton and La Pointe, 1995, have identified many potential uses for the fractal dimension in physicochemical models of surface properties. The image-analysis program described in this report is an extension of the program set MORPH-I (Mossotti and others, 1998), which provided the fractal analysis of electron-microscope images of pore profiles (Mossotti and Eldeeb, 1992). MORPH-II, an integration of the modified kernel of the program MORPH-I with image calibration and editing facilities, was designed to measure the fractal dimension of the exposed surfaces of stone specimens as imaged in cross section in an electron microscope.

Contour fractal analysis of grains

NASA Astrophysics Data System (ADS)

Guida, Giulia; Casini, Francesca; Viggiani, Giulia MB

2017-06-01

Fractal analysis has been shown to be useful in image processing to characterise the shape and the grey-scale complexity in different applications spanning from electronic to medical engineering (e.g. [1]). Fractal analysis consists of several methods to assign a dimension and other fractal characteristics to a dataset describing geometric objects. Limited studies have been conducted on the application of fractal analysis to the classification of the shape characteristics of soil grains. The main objective of the work described in this paper is to obtain, from the results of systematic fractal analysis of artificial simple shapes, the characterization of the particle morphology at different scales. The long term objective of the research is to link the microscopic features of granular media with the mechanical behaviour observed in the laboratory and in situ.

Intelligent fuzzy approach for fast fractal image compression

NASA Astrophysics Data System (ADS)

Nodehi, Ali; Sulong, Ghazali; Al-Rodhaan, Mznah; Al-Dhelaan, Abdullah; Rehman, Amjad; Saba, Tanzila

2014-12-01

Fractal image compression (FIC) is recognized as a NP-hard problem, and it suffers from a high number of mean square error (MSE) computations. In this paper, a two-phase algorithm was proposed to reduce the MSE computation of FIC. In the first phase, based on edge property, range and domains are arranged. In the second one, imperialist competitive algorithm (ICA) is used according to the classified blocks. For maintaining the quality of the retrieved image and accelerating algorithm operation, we divided the solutions into two groups: developed countries and undeveloped countries. Simulations were carried out to evaluate the performance of the developed approach. Promising results thus achieved exhibit performance better than genetic algorithm (GA)-based and Full-search algorithms in terms of decreasing the number of MSE computations. The number of MSE computations was reduced by the proposed algorithm for 463 times faster compared to the Full-search algorithm, although the retrieved image quality did not have a considerable change.

Functional slit lamp biomicroscopy for imaging bulbar conjunctival microvasculature in contact lens wearers

PubMed Central

Jiang, Hong; Zhong, Jianguang; DeBuc, Delia Cabrera; Tao, Aizhu; Xu, Zhe; Lam, Byron L.; Liu, Che; Wang, Jianhua

2014-01-01

Purpose To develop, test and validate functional slit lamp biomicroscopy (FSLB) for generating non-invasive bulbar conjunctival microvascular perfusion maps (nMPMs) and assessing morphometry and hemodyanmics. Methods FSLB was adapted from a traditional slit-lamp microscope by attaching a digital camera to image the bulbar conjunctiva to create nMPMs and measure venular blood flow hemodyanmics. High definition images with a large field of view were obtained on the temporal bulbar conjunctiva for creating nMPMs. A high imaging rate of 60 frame per second and a ~210× high magnification were achieved using the camera inherited high speed setting and movie crop function, for imaging hemodyanmics. Custom software was developed to segment bulbar conjunctival nMPMs for further fractal analysis and quantitatively measure blood vessel diameter, blood flow velocity and flow rate. Six human subjects were imaged before and after 6 hours of wearing contact lenses. Monofractal and multifractal analyses were performed to quantify fractality of the nMPMs. Results The mean bulbar conjunctival vessel diameter was 18.8 ± 2.7 μm at baseline and increased to 19.6 ± 2.4 μm after 6 hours of lens wear (P = 0.020). The blood flow velocity was increased from 0.60 ± 0.12 mm/s to 0.88 ± 0.21 mm/s (P = 0.001). The blood flow rate was also increased from 129.8 ± 59.9 pl/s to 207.2 ± 81.3 pl/s (P = 0.001). Bulbar conjunctival nMPMs showed the intricate details of the bulbar conjunctival microvascular network. At baseline, fractal dimension was 1.63 ± 0.05 and 1.71 ± 0.03 analyzed by monofractal and multifractal analysis, respectively. Significant increases in fractal dimensions were found after 6 hours of lens wear (P < 0.05). Conclusions Microvascular network’s fractality, morphometry and hemodyanmics of the human bulbar conjunctiva can be measured easily and reliably using FSLB. The alternations of the fractal dimensions, morphometry and hemodyanmics during contact lens wear may

Fractal morphology, imaging and mass spectrometry of single aerosol particles in flight (CXIDB ID 16)

DOE Data Explorer

Loh, N. Duane

2012-06-20

This deposition includes the aerosol diffraction images used for phasing, fractal morphology, and time-of-flight mass spectrometry. Files in this deposition are ordered in subdirectories that reflect the specifics.

The Calculation of Fractal Dimension in the Presence of Non-Fractal Clutter

NASA Technical Reports Server (NTRS)

Herren, Kenneth A.; Gregory, Don A.

1999-01-01

The area of information processing has grown dramatically over the last 50 years. In the areas of image processing and information storage the technology requirements have far outpaced the ability of the community to meet demands. The need for faster recognition algorithms and more efficient storage of large quantities of data has forced the user to accept less than lossless retrieval of that data for analysis. In addition to clutter that is not the object of interest in the data set, often the throughput requirements forces the user to accept "noisy" data and to tolerate the clutter inherent in that data. It has been shown that some of this clutter, both the intentional clutter (clouds, trees, etc) as well as the noise introduced on the data by processing requirements can be modeled as fractal or fractal-like. Traditional methods using Fourier deconvolution on these sources of noise in frequency space leads to loss of signal and can, in many cases, completely eliminate the target of interest. The parameters that characterize fractal-like noise (predominately the fractal dimension) have been investigated and a technique to reduce or eliminate noise from real scenes has been developed. Examples of clutter reduced images are presented.

Fractal lacunarity of trabecular bone and magnetic resonance imaging: New perspectives for osteoporotic fracture risk assessment

PubMed Central

Zaia, Annamaria

2015-01-01

Osteoporosis represents one major health condition for our growing elderly population. It accounts for severe morbidity and increased mortality in postmenopausal women and it is becoming an emerging health concern even in aging men. Screening of the population at risk for bone degeneration and treatment assessment of osteoporotic patients to prevent bone fragility fractures represent useful tools to improve quality of life in the elderly and to lighten the related socio-economic impact. Bone mineral density (BMD) estimate by means of dual-energy X-ray absorptiometry is normally used in clinical practice for osteoporosis diagnosis. Nevertheless, BMD alone does not represent a good predictor of fracture risk. From a clinical point of view, bone microarchitecture seems to be an intriguing aspect to characterize bone alteration patterns in aging and pathology. The widening into clinical practice of medical imaging techniques and the impressive advances in information technologies together with enhanced capacity of power calculation have promoted proliferation of new methods to assess changes of trabecular bone architecture (TBA) during aging and osteoporosis. Magnetic resonance imaging (MRI) has recently arisen as a useful tool to measure bone structure in vivo. In particular, high-resolution MRI techniques have introduced new perspectives for TBA characterization by non-invasive non-ionizing methods. However, texture analysis methods have not found favor with clinicians as they produce quite a few parameters whose interpretation is difficult. The introduction in biomedical field of paradigms, such as theory of complexity, chaos, and fractals, suggests new approaches and provides innovative tools to develop computerized methods that, by producing a limited number of parameters sensitive to pathology onset and progression, would speed up their application into clinical practice. Complexity of living beings and fractality of several physio-anatomic structures suggest

Applications of Fractal Analytical Techniques in the Estimation of Operational Scale

NASA Technical Reports Server (NTRS)

Emerson, Charles W.; Quattrochi, Dale A.

2000-01-01

The observational scale and the resolution of remotely sensed imagery are essential considerations in the interpretation process. Many atmospheric, hydrologic, and other natural and human-influenced spatial phenomena are inherently scale dependent and are governed by different physical processes at different spatial domains. This spatial and operational heterogeneity constrains the ability to compare interpretations of phenomena and processes observed in higher spatial resolution imagery to similar interpretations obtained from lower resolution imagery. This is a particularly acute problem, since longterm global change investigations will require high spatial resolution Earth Observing System (EOS), Landsat 7, or commercial satellite data to be combined with lower resolution imagery from older sensors such as Landsat TM and MSS. Fractal analysis is a useful technique for identifying the effects of scale changes on remotely sensed imagery. The fractal dimension of an image is a non-integer value between two and three which indicates the degree of complexity in the texture and shapes depicted in the image. A true fractal surface exhibits self-similarity, a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution, and the slope of the fractal dimension-resolution relationship would be zero. Most geographical phenomena, however, are not self-similar at all scales, but they can be modeled by a stochastic fractal in which the scaling properties of the image exhibit patterns that can be described by statistics such as area-perimeter ratios and autocovariances. Stochastic fractal sets relax the self-similarity assumption and measure many scales and

Fractal dimensions of graph of Weierstrass-type function and local Hölder exponent spectra

NASA Astrophysics Data System (ADS)

Otani, Atsuya

2018-01-01

We study several fractal properties of the Weierstrass-type function where τ :[0, 1)\\to[0, 1) is a cookie cutter map with possibly fractal repeller, and λ and g are functions with proper regularity. In the first part, we determine the box dimension of the graph of W and Hausdorff dimension of its randomised version. In the second part, the Hausdorff spectrum of the local Hölder exponent is characterised in terms of thermodynamic formalism. Furthermore, in the randomised case, a novel formula for the lifted Hausdorff spectrum on the graph is provided.

Pond fractals in a tidal flat.

PubMed

Cael, B B; Lambert, Bennett; Bisson, Kelsey

2015-11-01

Studies over the past decade have reported power-law distributions for the areas of terrestrial lakes and Arctic melt ponds, as well as fractal relationships between their areas and coastlines. Here we report similar fractal structure of ponds in a tidal flat, thereby extending the spatial and temporal scales on which such phenomena have been observed in geophysical systems. Images taken during low tide of a tidal flat in Damariscotta, Maine, reveal a well-resolved power-law distribution of pond sizes over three orders of magnitude with a consistent fractal area-perimeter relationship. The data are consistent with the predictions of percolation theory for unscreened perimeters and scale-free cluster size distributions and are robust to alterations of the image processing procedure. The small spatial and temporal scales of these data suggest this easily observable system may serve as a useful model for investigating the evolution of pond geometries, while emphasizing the generality of fractal behavior in geophysical surfaces.

Pond fractals in a tidal flat

NASA Astrophysics Data System (ADS)

Cael, B. B.; Lambert, Bennett; Bisson, Kelsey

2015-11-01

Studies over the past decade have reported power-law distributions for the areas of terrestrial lakes and Arctic melt ponds, as well as fractal relationships between their areas and coastlines. Here we report similar fractal structure of ponds in a tidal flat, thereby extending the spatial and temporal scales on which such phenomena have been observed in geophysical systems. Images taken during low tide of a tidal flat in Damariscotta, Maine, reveal a well-resolved power-law distribution of pond sizes over three orders of magnitude with a consistent fractal area-perimeter relationship. The data are consistent with the predictions of percolation theory for unscreened perimeters and scale-free cluster size distributions and are robust to alterations of the image processing procedure. The small spatial and temporal scales of these data suggest this easily observable system may serve as a useful model for investigating the evolution of pond geometries, while emphasizing the generality of fractal behavior in geophysical surfaces.

A comparison of the fractal and JPEG algorithms

NASA Technical Reports Server (NTRS)

Cheung, K.-M.; Shahshahani, M.

1991-01-01

A proprietary fractal image compression algorithm and the Joint Photographic Experts Group (JPEG) industry standard algorithm for image compression are compared. In every case, the JPEG algorithm was superior to the fractal method at a given compression ratio according to a root mean square criterion and a peak signal to noise criterion.

Fractal dimension and the navigational information provided by natural scenes.

PubMed

Shamsyeh Zahedi, Moosarreza; Zeil, Jochen

2018-01-01

Recent work on virtual reality navigation in humans has suggested that navigational success is inversely correlated with the fractal dimension (FD) of artificial scenes. Here we investigate the generality of this claim by analysing the relationship between the fractal dimension of natural insect navigation environments and a quantitative measure of the navigational information content of natural scenes. We show that the fractal dimension of natural scenes is in general inversely proportional to the information they provide to navigating agents on heading direction as measured by the rotational image difference function (rotIDF). The rotIDF determines the precision and accuracy with which the orientation of a reference image can be recovered or maintained and the range over which a gradient descent in image differences will find the minimum of the rotIDF, that is the reference orientation. However, scenes with similar fractal dimension can differ significantly in the depth of the rotIDF, because FD does not discriminate between the orientations of edges, while the rotIDF is mainly affected by edge orientation parallel to the axis of rotation. We present a new equation for the rotIDF relating navigational information to quantifiable image properties such as contrast to show (1) that for any given scene the maximum value of the rotIDF (its depth) is proportional to pixel variance and (2) that FD is inversely proportional to pixel variance. This contrast dependence, together with scene differences in orientation statistics, explains why there is no strict relationship between FD and navigational information. Our experimental data and their numerical analysis corroborate these results.

Fractal-based wideband invisibility cloak

NASA Astrophysics Data System (ADS)

Cohen, Nathan; Okoro, Obinna; Earle, Dan; Salkind, Phil; Unger, Barry; Yen, Sean; McHugh, Daniel; Polterzycki, Stefan; Shelman-Cohen, A. J.

2015-03-01

A wideband invisibility cloak (IC) at microwave frequencies is described. Using fractal resonators in closely spaced (sub wavelength) arrays as a minimal number of cylindrical layers (rings), the IC demonstrates that it is physically possible to attain a `see through' cloaking device with: (a) wideband coverage; (b) simple and attainable fabrication; (c) high fidelity emulation of the free path; (d) minimal side scattering; (d) a near absence of shadowing in the scattering. Although not a practical device, this fractal-enabled technology demonstrator opens up new opportunities for diverted-image (DI) technology and use of fractals in wideband optical, infrared, and microwave applications.

Turbulence Enhancement by Fractal Square Grids: Effects of the Number of Fractal Scales

NASA Astrophysics Data System (ADS)

Omilion, Alexis; Ibrahim, Mounir; Zhang, Wei

2017-11-01

Fractal square grids offer a unique solution for passive flow control as they can produce wakes with a distinct turbulence intensity peak and a prolonged turbulence decay region at the expense of only minimal pressure drop. While previous studies have solidified this characteristic of fractal square grids, how the number of scales (or fractal iterations N) affect turbulence production and decay of the induced wake is still not well understood. The focus of this research is to determine the relationship between the fractal iteration N and the turbulence produced in the wake flow using well-controlled water-tunnel experiments. Particle Image Velocimetry (PIV) is used to measure the instantaneous velocity fields downstream of four different fractal grids with increasing number of scales (N = 1, 2, 3, and 4) and a conventional single-scale grid. By comparing the turbulent scales and statistics of the wake, we are able to determine how each iteration affects the peak turbulence intensity and the production/decay of turbulence from the grid. In light of the ability of these fractal grids to increase turbulence intensity with low pressure drop, this work can potentially benefit a wide variety of applications where energy efficient mixing or convective heat transfer is a key process.

A Brief Historical Introduction to Fractals and Fractal Geometry

ERIC Educational Resources Information Center

Debnath, Lokenath

2006-01-01

This paper deals with a brief historical introduction to fractals, fractal dimension and fractal geometry. Many fractals including the Cantor fractal, the Koch fractal, the Minkowski fractal, the Mandelbrot and Given fractal are described to illustrate self-similar geometrical figures. This is followed by the discovery of dynamical systems and…

Depth of focus enhancement of a modified imaging quasi-fractal zone plate.

PubMed

Zhang, Qinqin; Wang, Jingang; Wang, Mingwei; Bu, Jing; Zhu, Siwei; Gao, Bruce Z; Yuan, Xiaocong

2012-10-01

We propose a new parameter w for optimization of foci distribution of conventional fractal zone plates (FZPs) with a greater depth of focus (DOF) in imaging. Numerical simulations of DOF distribution on axis directions indicate that the values of DOF can be extended by a factor of 1.5 or more by a modified quasi-FZP. In experiments, we employ a simple object-lens-image-plane arrangement to pick up images at various positions within the DOF of a conventional FZP and a quasi-FZP, respectively. Experimental results show that the parameter w improves foci distribution of FZPs in good agreement with theoretical predictions.

Depth of focus enhancement of a modified imaging quasi-fractal zone plate

PubMed Central

Zhang, Qinqin; Wang, Jingang; Wang, Mingwei; Bu, Jing; Zhu, Siwei; Gao, Bruce Z.; Yuan, Xiaocong

2013-01-01

We propose a new parameter w for optimization of foci distribution of conventional fractal zone plates (FZPs) with a greater depth of focus (DOF) in imaging. Numerical simulations of DOF distribution on axis directions indicate that the values of DOF can be extended by a factor of 1.5 or more by a modified quasi-FZP. In experiments, we employ a simple object–lens–image-plane arrangement to pick up images at various positions within the DOF of a conventional FZP and a quasi-FZP, respectively. Experimental results show that the parameter w improves foci distribution of FZPs in good agreement with theoretical predictions. PMID:24285908

GENERATING FRACTAL PATTERNS BY USING p-CIRCLE INVERSION

NASA Astrophysics Data System (ADS)

Ramírez, José L.; Rubiano, Gustavo N.; Zlobec, Borut Jurčič

2015-10-01

In this paper, we introduce the p-circle inversion which generalizes the classical inversion with respect to a circle (p = 2) and the taxicab inversion (p = 1). We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the p-circle inversion.

Analysis and classification of commercial ham slice images using directional fractal dimension features.

PubMed

Mendoza, Fernando; Valous, Nektarios A; Allen, Paul; Kenny, Tony A; Ward, Paddy; Sun, Da-Wen

2009-02-01

This paper presents a novel and non-destructive approach to the appearance characterization and classification of commercial pork, turkey and chicken ham slices. Ham slice images were modelled using directional fractal (DF(0°;45°;90°;135°)) dimensions and a minimum distance classifier was adopted to perform the classification task. Also, the role of different colour spaces and the resolution level of the images on DF analysis were investigated. This approach was applied to 480 wafer thin ham slices from four types of hams (120 slices per type): i.e., pork (cooked and smoked), turkey (smoked) and chicken (roasted). DF features were extracted from digitalized intensity images in greyscale, and R, G, B, L(∗), a(∗), b(∗), H, S, and V colour components for three image resolution levels (100%, 50%, and 25%). Simulation results show that in spite of the complexity and high variability in colour and texture appearance, the modelling of ham slice images with DF dimensions allows the capture of differentiating textural features between the four commercial ham types. Independent DF features entail better discrimination than that using the average of four directions. However, DF dimensions reveal a high sensitivity to colour channel, orientation and image resolution for the fractal analysis. The classification accuracy using six DF dimension features (a(90°)(∗),a(135°)(∗),H(0°),H(45°),S(0°),H(90°)) was 93.9% for training data and 82.2% for testing data.

Focusing behavior of the fractal vector optical fields designed by fractal lattice growth model.

PubMed

Gao, Xu-Zhen; Pan, Yue; Zhao, Meng-Dan; Zhang, Guan-Lin; Zhang, Yu; Tu, Chenghou; Li, Yongnan; Wang, Hui-Tian

2018-01-22

We introduce a general fractal lattice growth model, significantly expanding the application scope of the fractal in the realm of optics. This model can be applied to construct various kinds of fractal "lattices" and then to achieve the design of a great diversity of fractal vector optical fields (F-VOFs) combinating with various "bases". We also experimentally generate the F-VOFs and explore their universal focusing behaviors. Multiple focal spots can be flexibly enginnered, and the optical tweezers experiment validates the simulated tight focusing fields, which means that this model allows the diversity of the focal patterns to flexibly trap and manipulate micrometer-sized particles. Furthermore, the recovery performance of the F-VOFs is also studied when the input fields and spatial frequency spectrum are obstructed, and the results confirm the robustness of the F-VOFs in both focusing and imaging processes, which is very useful in information transmission.

Radial distribution function for hard spheres in fractal dimensions: A heuristic approximation.

PubMed

Santos, Andrés; de Haro, Mariano López

2016-06-01

Analytic approximations for the radial distribution function, the structure factor, and the equation of state of hard-core fluids in fractal dimension d (1≤d≤3) are developed as heuristic interpolations from the knowledge of the exact and Percus-Yevick results for the hard-rod and hard-sphere fluids, respectively. In order to assess their value, such approximate results are compared with those of recent Monte Carlo simulations and numerical solutions of the Percus-Yevick equation for a fractal dimension [M. Heinen et al., Phys. Rev. Lett. 115, 097801 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.097801], a good agreement being observed.

FAST TRACK COMMUNICATION: Weyl law for fat fractals

NASA Astrophysics Data System (ADS)

Spina, María E.; García-Mata, Ignacio; Saraceno, Marcos

2010-10-01

It has been conjectured that for a class of piecewise linear maps the closure of the set of images of the discontinuity has the structure of a fat fractal, that is, a fractal with positive measure. An example of such maps is the sawtooth map in the elliptic regime. In this work we analyze this problem quantum mechanically in the semiclassical regime. We find that the fraction of states localized on the unstable set satisfies a modified fractal Weyl law, where the exponent is given by the exterior dimension of the fat fractal.

[Modeling continuous scaling of NDVI based on fractal theory].

PubMed

Luan, Hai-Jun; Tian, Qing-Jiu; Yu, Tao; Hu, Xin-Li; Huang, Yan; Du, Ling-Tong; Zhao, Li-Min; Wei, Xi; Han, Jie; Zhang, Zhou-Wei; Li, Shao-Peng

2013-07-01

Scale effect was one of the very important scientific problems of remote sensing. The scale effect of quantitative remote sensing can be used to study retrievals' relationship between different-resolution images, and its research became an effective way to confront the challenges, such as validation of quantitative remote sensing products et al. Traditional up-scaling methods cannot describe scale changing features of retrievals on entire series of scales; meanwhile, they are faced with serious parameters correction issues because of imaging parameters' variation of different sensors, such as geometrical correction, spectral correction, etc. Utilizing single sensor image, fractal methodology was utilized to solve these problems. Taking NDVI (computed by land surface radiance) as example and based on Enhanced Thematic Mapper Plus (ETM+) image, a scheme was proposed to model continuous scaling of retrievals. Then the experimental results indicated that: (a) For NDVI, scale effect existed, and it could be described by fractal model of continuous scaling; (2) The fractal method was suitable for validation of NDVI. All of these proved that fractal was an effective methodology of studying scaling of quantitative remote sensing.

The fractal dimension of cell membrane correlates with its capacitance: A new fractal single-shell model

PubMed Central

Wang, Xujing; Becker, Frederick F.; Gascoyne, Peter R. C.

2010-01-01

The scale-invariant property of the cytoplasmic membrane of biological cells is examined by applying the Minkowski–Bouligand method to digitized scanning electron microscopy images of the cell surface. The membrane is found to exhibit fractal behavior, and the derived fractal dimension gives a good description of its morphological complexity. Furthermore, we found that this fractal dimension correlates well with the specific membrane dielectric capacitance derived from the electrorotation measurements. Based on these findings, we propose a new fractal single-shell model to describe the dielectrics of mammalian cells, and compare it with the conventional single-shell model (SSM). We found that while both models fit with experimental data well, the new model is able to eliminate the discrepancy between the measured dielectric property of cells and that predicted by the SSM. PMID:21198103

Fractal analysis of phasic laser images of the myocardium for the purpose of diagnostics of acute coronary insufficiency

NASA Astrophysics Data System (ADS)

Wanchuliak, O. Y.; Bachinskyi, V. T.

2011-09-01

In this work on the base of Mueller-matrix description of optical anisotropy, the possibility of monitoring of time changes of myocardium tissue birefringence, has been considered. The optical model of polycrystalline networks of myocardium is suggested. The results of investigating the interrelation between the values correlation (correlation area, asymmetry coefficient and autocorrelation function excess) and fractal (dispersion of logarithmic dependencies of power spectra) parameters are presented. They characterize the distributions of Mueller matrix elements in the points of laser images of myocardium histological sections. The criteria of differentiation of death coming reasons are determined.

Fractal evaluation of drug amorphicity from optical and scanning electron microscope images

NASA Astrophysics Data System (ADS)

Gavriloaia, Bogdan-Mihai G.; Vizireanu, Radu C.; Neamtu, Catalin I.; Gavriloaia, Gheorghe V.

2013-09-01

Amorphous materials are metastable, more reactive than the crystalline ones, and have to be evaluated before pharmaceutical compound formulation. Amorphicity is interpreted as a spatial chaos, and patterns of molecular aggregates of dexamethasone, D, were investigated in this paper by using fractal dimension, FD. Images having three magnifications of D were taken from an optical microscope, OM, and with eight magnifications, from a scanning electron microscope, SEM, were analyzed. The average FD for pattern irregularities of OM images was 1.538, and about 1.692 for SEM images. The FDs of the two kinds of images are less sensitive of threshold level. 3D images were shown to illustrate dependence of FD of threshold and magnification level. As a result, optical image of single scale is enough to characterize the drug amorphicity. As a result, the OM image at a single scale is enough to characterize the amorphicity of D.

Modeling liver physiology: combining fractals, imaging and animation.

PubMed

Lin, Debbie W; Johnson, Scott; Hunt, C Anthony

2004-01-01

Physiological modeling of vascular and microvascular networks in several key human organ systems is critical for a deeper understanding of pharmacology and the effect of pharmacotherapies on disease. Like the lung and the kidney, the morphology of its vascular and microvascular system plays a major role in its functional capability. To understand liver function in absorption and metabolism of food and drugs, one must examine the morphology and physiology at both higher and lower level liver function. We have developed validated virtualized dynamic three dimensional (3D) models of liver secondary units and primary units by combining a number of different methods: three-dimensional rendering, fractals, and animation. We have simulated particle dynamics in the liver secondary unit. The resulting models are suitable for use in helping researchers easily visualize and gain intuition on results of in silico liver experiments.

Jupiter Fractal Art

NASA Image and Video Library

2017-08-10

See Jupiter's Great Red Spot as you've never seen it before in this new Jovian work of art. Artist Mik Petter created this unique, digital artwork using data from the JunoCam imager on NASA's Juno spacecraft. The art form, known as fractals, uses mathematical formulas to create art with an infinite variety of form, detail, color and light. The tumultuous atmospheric zones in and around the Great Red Spot are highlighted by the author's use of colorful fractals. Vibrant colors of various tints and hues, combined with the almost organic-seeming shapes, make this image seem to be a colorized and crowded petri dish of microorganisms, or a close-up view of microscopic and wildly-painted seashells. The original JunoCam image was taken on July 10, 2017 at 7:10 p.m. PDT (10:10 p.m. EDT), as the Juno spacecraft performed its seventh close flyby of Jupiter. The spacecraft captured the image from about 8,648 miles (13,917 kilometers) above the tops of the clouds of the planet at a latitude of -32.6 degrees. https://photojournal.jpl.nasa.gov/catalog/PIA21777

Fractal analysis of bone structure with applications to osteoporosis and microgravity effects

NASA Astrophysics Data System (ADS)

Acharya, Raj S.; LeBlanc, Adrian; Shackelford, Linda; Swarnakar, Vivek; Krishnamurthy, Ram; Hausman, E.; Lin, Chin-Shoou

1995-05-01

We characterize the trabecular structure with the aid of fractal dimension. We use alternating sequential filters (ASF) to generate a nonlinear pyramid for fractal dimension computations. We do not make any assumptions of the statistical distributions of the underlying fractal bone structure. The only assumption of our scheme is the rudimentary definition of self-similarity. This allows us the freedom of not being constrained by statistical estimation schemes. With mathematical simulations, we have shown that the ASF methods outperform other existing methods for fractal dimension estimation. We have shown that the fractal dimension remains the same when computed with both the x-ray images and the MRI images of the patella. We have shown that the fractal dimension of osteoporotic subjects is lower than that of the normal subjects. In animal models, we have shown that the fractal dimension of osteoporotic rats was lower than that of the normal rats. In a 17 week bedrest study, we have shown that the subject's prebedrest fractal dimension is higher than that of the postbedrest fractal dimension.

Fractal analysis as a potential tool for surface morphology of thin films

NASA Astrophysics Data System (ADS)

Soumya, S.; Swapna, M. S.; Raj, Vimal; Mahadevan Pillai, V. P.; Sankararaman, S.

2017-12-01

Fractal geometry developed by Mandelbrot has emerged as a potential tool for analyzing complex systems in the diversified fields of science, social science, and technology. Self-similar objects having the same details in different scales are referred to as fractals and are analyzed using the mathematics of non-Euclidean geometry. The present work is an attempt to correlate fractal dimension for surface characterization by Atomic Force Microscopy (AFM). Taking the AFM images of zinc sulphide (ZnS) thin films prepared by pulsed laser deposition (PLD) technique, under different annealing temperatures, the effect of annealing temperature and surface roughness on fractal dimension is studied. The annealing temperature and surface roughness show a strong correlation with fractal dimension. From the regression equation set, the surface roughness at a given annealing temperature can be calculated from the fractal dimension. The AFM images are processed using Photoshop and fractal dimension is calculated by box-counting method. The fractal dimension decreases from 1.986 to 1.633 while the surface roughness increases from 1.110 to 3.427, for a change of annealing temperature 30 ° C to 600 ° C. The images are also analyzed by power spectrum method to find the fractal dimension. The study reveals that the box-counting method gives better results compared to the power spectrum method.

Image encryption based on fractal-structured phase mask in fractional Fourier transform domain

NASA Astrophysics Data System (ADS)

Zhao, Meng-Dan; Gao, Xu-Zhen; Pan, Yue; Zhang, Guan-Lin; Tu, Chenghou; Li, Yongnan; Wang, Hui-Tian

2018-04-01

We present an optical encryption approach based on the combination of fractal Fresnel lens (FFL) and fractional Fourier transform (FrFT). Our encryption approach is in fact a four-fold encryption scheme, including the random phase encoding produced by the Gerchberg–Saxton algorithm, a FFL, and two FrFTs. A FFL is composed of a Sierpinski carpet fractal plate and a Fresnel zone plate. In our encryption approach, the security is enhanced due to the more expandable key spaces and the use of FFL overcomes the alignment problem of the optical axis in optical system. Only using the perfectly matched parameters of the FFL and the FrFT, the plaintext can be recovered well. We present an image encryption algorithm that from the ciphertext we can get two original images by the FrFT with two different phase distribution keys, obtained by performing 100 iterations between the two plaintext and ciphertext, respectively. We test the sensitivity of our approach to various parameters such as the wavelength of light, the focal length of FFL, and the fractional orders of FrFT. Our approach can resist various attacks.

Fractal characteristic in the wearing of cutting tool

NASA Astrophysics Data System (ADS)

Mei, Anhua; Wang, Jinghui

1995-11-01

This paper studies the cutting tool wear with fractal geometry. The wearing image of the flank has been collected by machine vision which consists of CCD camera and personal computer. After being processed by means of preserving smoothing, binary making and edge extracting, the clear boundary enclosing the worn area has been obtained. The fractal dimension of the worn surface is calculated by the methods called `Slit Island' and `Profile'. The experiments and calciating give the conclusion that the worn surface is enclosed by a irregular boundary curve with some fractal dimension and characteristics of self-similarity. Furthermore, the relation between the cutting velocity and the fractal dimension of the worn region has been submitted. This paper presents a series of methods for processing and analyzing the fractal information in the blank wear, which can be applied to research the projective relation between the fractal structure and the wear state, and establish the fractal model of the cutting tool wear.

Fractal cometary dust - a window into the early Solar system

NASA Astrophysics Data System (ADS)

Mannel, T.; Bentley, M. S.; Schmied, R.; Jeszenszky, H.; Levasseur-Regourd, A. C.; Romstedt, J.; Torkar, K.

2016-11-01

The properties of dust in the protoplanetary disc are key to understanding the formation of planets in our Solar system. Many models of dust growth predict the development of fractal structures which evolve into non-fractal, porous dust pebbles representing the main component for planetesimal accretion. In order to understand comets and their origins, the Rosetta orbiter followed comet 67P/Churyumov-Gerasimenko for over two years and carried a dedicated instrument suite for dust analysis. One of these instruments, the MIDAS (Micro-Imaging Dust Analysis System) atomic force microscope, recorded the 3D topography of micro- to nanometre-sized dust. All particles analysed to date have been found to be hierarchical agglomerates. Most show compact packing; however, one is extremely porous. This paper contains a structural description of a compact aggregate and the outstanding porous one. Both particles are tens of micrometres in size and show rather narrow subunit size distributions with noticeably similar mean values of 1.48^{+0.13}_{-0.59} μm for the porous particle and 1.36^{+0.15}_{-0.59} μm for the compact. The porous particle allows a fractal analysis, where a density-density correlation function yields a fractal dimension of Df = 1.70 ± 0.1. GIADA, another dust analysis instrument on board Rosetta, confirms the existence of a dust population with a similar fractal dimension. The fractal particles are interpreted as pristine agglomerates built in the protoplanetary disc and preserved in the comet. The similar subunits of both fractal and compact dust indicate a common origin which is, given the properties of the fractal, dominated by slow agglomeration of equally sized aggregates known as cluster-cluster agglomeration.

Hyper-Fractal Analysis: A visual tool for estimating the fractal dimension of 4D objects

NASA Astrophysics Data System (ADS)

Grossu, I. V.; Grossu, I.; Felea, D.; Besliu, C.; Jipa, Al.; Esanu, T.; Bordeianu, C. C.; Stan, E.

2013-04-01

This work presents a new version of a Visual Basic 6.0 application for estimating the fractal dimension of images and 3D objects (Grossu et al. (2010) [1]). The program was extended for working with four-dimensional objects stored in comma separated values files. This might be of interest in biomedicine, for analyzing the evolution in time of three-dimensional images. New version program summaryProgram title: Hyper-Fractal Analysis (Fractal Analysis v03) Catalogue identifier: AEEG_v3_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEG_v3_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 745761 No. of bytes in distributed program, including test data, etc.: 12544491 Distribution format: tar.gz Programming language: MS Visual Basic 6.0 Computer: PC Operating system: MS Windows 98 or later RAM: 100M Classification: 14 Catalogue identifier of previous version: AEEG_v2_0 Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 831-832 Does the new version supersede the previous version? Yes Nature of problem: Estimating the fractal dimension of 4D images. Solution method: Optimized implementation of the 4D box-counting algorithm. Reasons for new version: Inspired by existing applications of 3D fractals in biomedicine [3], we extended the optimized version of the box-counting algorithm [1, 2] to the four-dimensional case. This might be of interest in analyzing the evolution in time of 3D images. The box-counting algorithm was extended in order to support 4D objects, stored in comma separated values files. A new form was added for generating 2D, 3D, and 4D test data. The application was tested on 4D objects with known dimension, e.g. the Sierpinski hypertetrahedron gasket, Df=ln(5)/ln(2) (Fig. 1). The algorithm could be extended, with minimum effort, to

A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging

NASA Astrophysics Data System (ADS)

Liang, Yingjie; Ye, Allen Q.; Chen, Wen; Gatto, Rodolfo G.; Colon-Perez, Luis; Mareci, Thomas H.; Magin, Richard L.

2016-10-01

Non-Gaussian (anomalous) diffusion is wide spread in biological tissues where its effects modulate chemical reactions and membrane transport. When viewed using magnetic resonance imaging (MRI), anomalous diffusion is characterized by a persistent or 'long tail' behavior in the decay of the diffusion signal. Recent MRI studies have used the fractional derivative to describe diffusion dynamics in normal and post-mortem tissue by connecting the order of the derivative with changes in tissue composition, structure and complexity. In this study we consider an alternative approach by introducing fractal time and space derivatives into Fick's second law of diffusion. This provides a more natural way to link sub-voxel tissue composition with the observed MRI diffusion signal decay following the application of a diffusion-sensitive pulse sequence. Unlike previous studies using fractional order derivatives, here the fractal derivative order is directly connected to the Hausdorff fractal dimension of the diffusion trajectory. The result is a simpler, computationally faster, and more direct way to incorporate tissue complexity and microstructure into the diffusional dynamics. Furthermore, the results are readily expressed in terms of spectral entropy, which provides a quantitative measure of the overall complexity of the heterogeneous and multi-scale structure of biological tissues. As an example, we apply this new model for the characterization of diffusion in fixed samples of the mouse brain. These results are compared with those obtained using the mono-exponential, the stretched exponential, the fractional derivative, and the diffusion kurtosis models. Overall, we find that the order of the fractal time derivative, the diffusion coefficient, and the spectral entropy are potential biomarkers to differentiate between the microstructure of white and gray matter. In addition, we note that the fractal derivative model has practical advantages over the existing models from the

Fractal analysis of bone structure with applications to osteoporosis and microgravity effects

DOE Office of Scientific and Technical Information (OSTI.GOV)

Acharya, R.S.; Swarnarkar, V.; Krishnamurthy, R.

1995-12-31

The authors characterize the trabecular structure with the aid of fractal dimension. The authors use Alternating Sequential filters to generate a nonlinear pyramid for fractal dimension computations. The authors do not make any assumptions of the statistical distributions of the underlying fractal bone structure. The only assumption of the scheme is the rudimentary definition of self similarity. This allows them the freedom of not being constrained by statistical estimation schemes. With mathematical simulations, the authors have shown that the ASF methods outperform other existing methods for fractal dimension estimation. They have shown that the fractal dimension remains the same whenmore » computed with both the X-Ray images and the MRI images of the patella. They have shown that the fractal dimension of osteoporotic subjects is lower than that of the normal subjects. In animal models, the authors have shown that the fractal dimension of osteoporotic rats was lower than that of the normal rats. In a 17 week bedrest study, they have shown that the subject`s prebedrest fractal dimension is higher than that of the postbedrest fractal dimension.« less

Fractal Dimensions of Umbral and Penumbral Regions of Sunspots

NASA Astrophysics Data System (ADS)

Rajkumar, B.; Haque, S.; Hrudey, W.

2017-11-01

The images of sunspots in 16 active regions taken at the University College of the Cayman Islands (UCCI) Observatory on Grand Cayman during June-November 2015 were used to determine their fractal dimensions using the perimeter-area method for the umbral and the penumbral region. Scale-free fractal dimensions of 2.09 ±0.42 and 1.72 ±0.4 were found, respectively. This value was higher than the value determined by Chumak and Chumak ( Astron. Astrophys. Trans. 10, 329, 1996), who used a similar method, but only for the penumbral region of their sample set. The umbral and penumbral fractal dimensions for the specific sunspots are positively correlated with r = 0.58. Furthermore, a similar time-series analysis was performed on eight images of AR 12403, from 21 August 2015 to 28 August 2015 taken from the Debrecen Photoheliographic Data (DPD). The correlation is r = 0.623 between the umbral and penumbral fractal dimensions in the time series, indicating that the complexity in morphology indicated by the fractal dimension between the umbra and penumbra followed each other in time as well.

Do-It-Yourself Fractal Functions

ERIC Educational Resources Information Center

Shriver, Janet; Willard, Teri; McDaniel, Mandy

2017-01-01

In the set of fractal activities described in this article, students will accomplish much more than just creating a fun set of cards that simply resemble an art project. Goals of this activity, designed for an algebra 1 class, are to encourage students to generate data, look for and analyze patterns, and create their own models--all from a set of…

Paradigms of Complexity: Fractals and Structures in the Sciences

NASA Astrophysics Data System (ADS)

Novak, Miroslav M.

The Table of Contents for the book is as follows: * Preface * The Origin of Complexity (invited talk) * On the Existence of Spatially Uniform Scaling Laws in the Climate System * Multispectral Backscattering: A Fractal-Structure Probe * Small-Angle Multiple Scattering on a Fractal System of Point Scatterers * Symmetric Fractals Generated by Cellular Automata * Bispectra and Phase Correlations for Chaotic Dynamical Systems * Self-Organized Criticality Models of Neural Development * Altered Fractal and Irregular Heart Rate Behavior in Sick Fetuses * Extract Multiple Scaling in Long-Term Heart Rate Variability * A Semi-Continous Box Counting Method for Fractal Dimension Measurement of Short Single Dimension Temporal Signals - Preliminary Study * A Fractional Brownian Motion Model of Cracking * Self-Affine Scaling Studies on Fractography * Coarsening of Fractal Interfaces * A Fractal Model of Ocean Surface Superdiffusion * Stochastic Subsurface Flow and Transport in Fractal Fractal Conductivity Fields * Rendering Through Iterated Function Systems * The σ-Hull - The Hull Where Fractals Live - Calculating a Hull Bounded by Log Spirals to Solve the Inverse IFS-Problem by the Detected Orbits * On the Multifractal Properties of Passively Convected Scalar Fields * New Statistical Textural Transforms for Non-Stationary Signals: Application to Generalized Mutlifractal Analysis * Laplacian Growth of Parallel Needles: Their Mullins-Sekerka Instability * Entropy Dynamics Associated with Self-Organization * Fractal Properties in Economics (invited talk) * Fractal Approach to the Regional Seismic Event Discrimination Problem * Fractal and Topological Complexity of Radioactive Contamination * Pattern Selection: Nonsingular Saffman-Taylor Finger and Its Dynamic Evolution with Zero Surface Tension * A Family of Complex Wavelets for the Characterization of Singularities * Stabilization of Chaotic Amplitude Fluctuations in Multimode, Intracavity-Doubled Solid-State Lasers * Chaotic

Applications of fractals in ecology.

PubMed

Sugihara, G; M May, R

1990-03-01

Fractal models describe the geometry of a wide variety of natural objects such as coastlines, island chains, coral reefs, satellite ocean-color images and patches of vegetation. Cast in the form of modified diffusion models, they can mimic natural and artificial landscapes having different types of complexity of shape. This article provides a brief introduction to fractals and reports on how they can be used by ecologists to answer a variety of basic questions, about scale, measurement and hierarchy in, ecological systems. Copyright © 1990. Published by Elsevier Ltd.

Fractal Physiology and the Fractional Calculus: A Perspective

PubMed Central

West, Bruce J.

2010-01-01

This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a

Fractal analysis of the ischemic transition region in chronic ischemic heart disease using magnetic resonance imaging.

PubMed

Michallek, Florian; Dewey, Marc

2017-04-01

To introduce a novel hypothesis and method to characterise pathomechanisms underlying myocardial ischemia in chronic ischemic heart disease by local fractal analysis (FA) of the ischemic myocardial transition region in perfusion imaging. Vascular mechanisms to compensate ischemia are regulated at various vascular scales with their superimposed perfusion pattern being hypothetically self-similar. Dedicated FA software ("FraktalWandler") has been developed. Fractal dimensions during first-pass (FD first-pass ) and recirculation (FD recirculation ) are hypothesised to indicate the predominating pathomechanism and ischemic severity, respectively. Twenty-six patients with evidence of myocardial ischemia in 108 ischemic myocardial segments on magnetic resonance imaging (MRI) were analysed. The 40th and 60th percentiles of FD first-pass were used for pathomechanical classification, assigning lesions with FD first-pass  ≤ 2.335 to predominating coronary microvascular dysfunction (CMD) and ≥2.387 to predominating coronary artery disease (CAD). Optimal classification point in ROC analysis was FD first-pass  = 2.358. FD recirculation correlated moderately with per cent diameter stenosis in invasive coronary angiography in lesions classified CAD (r = 0.472, p = 0.001) but not CMD (r = 0.082, p = 0.600). The ischemic transition region may provide information on pathomechanical composition and severity of myocardial ischemia. FA of this region is feasible and may improve diagnosis compared to traditional noninvasive myocardial perfusion analysis. • A novel hypothesis and method is introduced to pathophysiologically characterise myocardial ischemia. • The ischemic transition region appears a meaningful diagnostic target in perfusion imaging. • Fractal analysis may characterise pathomechanical composition and severity of myocardial ischemia.

Fractals, Coherence and Brain Dynamics

NASA Astrophysics Data System (ADS)

Vitiello, Giuseppe

2010-11-01

I show that the self-similarity property of deterministic fractals provides a direct connection with the space of the entire analytical functions. Fractals are thus described in terms of coherent states in the Fock-Bargmann representation. Conversely, my discussion also provides insights on the geometrical properties of coherent states: it allows to recognize, in some specific sense, fractal properties of coherent states. In particular, the relation is exhibited between fractals and q-deformed coherent states. The connection with the squeezed coherent states is also displayed. In this connection, the non-commutative geometry arising from the fractal relation with squeezed coherent states is discussed and the fractal spectral properties are identified. I also briefly discuss the description of neuro-phenomenological data in terms of squeezed coherent states provided by the dissipative model of brain and consider the fact that laboratory observations have shown evidence that self-similarity characterizes the brain background activity. This suggests that a connection can be established between brain dynamics and the fractal self-similarity properties on the basis of the relation discussed in this report between fractals and squeezed coherent states. Finally, I do not consider in this paper the so-called random fractals, namely those fractals obtained by randomization processes introduced in their iterative generation. Since self-similarity is still a characterizing property in many of such random fractals, my conjecture is that also in such cases there must exist a connection with the coherent state algebraic structure. In condensed matter physics, in many cases the generation by the microscopic dynamics of some kind of coherent states is involved in the process of the emergence of mesoscopic/macroscopic patterns. The discussion presented in this paper suggests that also fractal generation may provide an example of emergence of global features, namely long range

Cell type classifiers for breast cancer microscopic images based on fractal dimension texture analysis of image color layers.

PubMed

Jitaree, Sirinapa; Phinyomark, Angkoon; Boonyaphiphat, Pleumjit; Phukpattaranont, Pornchai

2015-01-01

Having a classifier of cell types in a breast cancer microscopic image (BCMI), obtained with immunohistochemical staining, is required as part of a computer-aided system that counts the cancer cells in such BCMI. Such quantitation by cell counting is very useful in supporting decisions and planning of the medical treatment of breast cancer. This study proposes and evaluates features based on texture analysis by fractal dimension (FD), for the classification of histological structures in a BCMI into either cancer cells or non-cancer cells. The cancer cells include positive cells (PC) and negative cells (NC), while the normal cells comprise stromal cells (SC) and lymphocyte cells (LC). The FD feature values were calculated with the box-counting method from binarized images, obtained by automatic thresholding with Otsu's method of the grayscale images for various color channels. A total of 12 color channels from four color spaces (RGB, CIE-L*a*b*, HSV, and YCbCr) were investigated, and the FD feature values from them were used with decision tree classifiers. The BCMI data consisted of 1,400, 1,200, and 800 images with pixel resolutions 128 × 128, 192 × 192, and 256 × 256, respectively. The best cross-validated classification accuracy was 93.87%, for distinguishing between cancer and non-cancer cells, obtained using the Cr color channel with window size 256. The results indicate that the proposed algorithm, based on fractal dimension features extracted from a color channel, performs well in the automatic classification of the histology in a BCMI. This might support accurate automatic cell counting in a computer-assisted system for breast cancer diagnosis. © Wiley Periodicals, Inc.

Music and fractals

NASA Astrophysics Data System (ADS)

Wuorinen, Charles

2015-03-01

Any of the arts may produce exemplars that have fractal characteristics. There may be fractal painting, fractal poetry, and the like. But these will always be specific instances, not necessarily displaying intrinsic properties of the art-medium itself. Only music, I believe, of all the arts possesses an intrinsically fractal character, so that its very nature is fractally determined. Thus, it is reasonable to assert that any instance of music is fractal...

Fractal Bread.

ERIC Educational Resources Information Center

Esbenshade, Donald H., Jr.

1991-01-01

Develops the idea of fractals through a laboratory activity that calculates the fractal dimension of ordinary white bread. Extends use of the fractal dimension to compare other complex structures as other breads and sponges. (MDH)

Exploring fractal behaviour of blood oxygen saturation in preterm babies

NASA Astrophysics Data System (ADS)

Zahari, Marina; Hui, Tan Xin; Zainuri, Nuryazmin Ahmat; Darlow, Brian A.

2017-04-01

Recent evidence has been emerging that oxygenation instability in preterm babies could lead to an increased risk of retinal injury such as retinopathy of prematurity. There is a potential that disease severity could be better understood using nonlinear methods for time series data such as fractal theories [1]. Theories on fractal behaviours have been employed by researchers in various disciplines who were motivated to look into the behaviour or structure of irregular fluctuations in temporal data. In this study, an investigation was carried out to examine whether fractal behaviour could be detected in blood oxygen time series. Detection for the presence of fractals in oxygen data of preterm infants was performed using the methods of power spectrum, empirical probability distribution function and autocorrelation function. The results from these fractal identification methods indicate the possibility that these data exhibit fractal nature. Subsequently, a fractal framework for future research was suggested for oxygen time series.

Fractal Characteristics of the Pore Network in Diatomites Using Mercury Porosimetry and Image Analysis

NASA Astrophysics Data System (ADS)

Stańczak, Grażyna; Rembiś, Marek; Figarska-Warchoł, Beata; Toboła, Tomasz

The complex pore space considerably affects the unique properties of diatomite and its significant potential for many industrial applications. The pore network in the diatomite from the Lower Miocene strata of the Skole nappe (the Jawornik deposit, SE Poland) has been investigated using a fractal approach. The fractal dimension of the pore-space volume was calculated using the Menger sponge as a model of a porous body and the mercury porosimetry data in a pore-throat diameter range between 10,000 and 10 nm. Based on the digital analyses of the two-dimensional images from thin sections taken under a scanning electron microscope at the backscattered electron mode at different magnifications, the authors tried to quantify the pore spaces of the diatomites using the box counting method. The results derived from the analyses of the pore-throat diameter distribution using mercury porosimetry have revealed that the pore space of the diatomite has the bifractal structure in two separated ranges of the pore-throat diameters considerably smaller than the pore-throat sizes corresponding to threshold pressures. Assuming that the fractal dimensions identified for the ranges of the smaller pore-throat diameters characterize the overall pore-throat network in the Jawornik diatomite, we can set apart the distribution of the pore-throat volume (necks) and the pore volume from the distribution of the pore-space volume (pores and necks together).

Spatial and radiometric characterization of multi-spectrum satellite images through multi-fractal analysis

NASA Astrophysics Data System (ADS)

Alonso, Carmelo; Tarquis, Ana M.; Zúñiga, Ignacio; Benito, Rosa M.

2017-03-01

Several studies have shown that vegetation indexes can be used to estimate root zone soil moisture. Earth surface images, obtained by high-resolution satellites, presently give a lot of information on these indexes, based on the data of several wavelengths. Because of the potential capacity for systematic observations at various scales, remote sensing technology extends the possible data archives from the present time to several decades back. Because of this advantage, enormous efforts have been made by researchers and application specialists to delineate vegetation indexes from local scale to global scale by applying remote sensing imagery. In this work, four band images have been considered, which are involved in these vegetation indexes, and were taken by satellites Ikonos-2 and Landsat-7 of the same geographic location, to study the effect of both spatial (pixel size) and radiometric (number of bits coding the image) resolution on these wavelength bands as well as two vegetation indexes: the Normalized Difference Vegetation Index (NDVI) and the Enhanced Vegetation Index (EVI). In order to do so, a multi-fractal analysis of these multi-spectral images was applied in each of these bands and the two indexes derived. The results showed that spatial resolution has a similar scaling effect in the four bands, but radiometric resolution has a larger influence in blue and green bands than in red and near-infrared bands. The NDVI showed a higher sensitivity to the radiometric resolution than EVI. Both were equally affected by the spatial resolution. From both factors, the spatial resolution has a major impact in the multi-fractal spectrum for all the bands and the vegetation indexes. This information should be taken in to account when vegetation indexes based on different satellite sensors are obtained.

Self-Similarity of Plasmon Edge Modes on Koch Fractal Antennas.

PubMed

Bellido, Edson P; Bernasconi, Gabriel D; Rossouw, David; Butet, Jérémy; Martin, Olivier J F; Botton, Gianluigi A

2017-11-28

We investigate the plasmonic behavior of Koch snowflake fractal geometries and their possible application as broadband optical antennas. Lithographically defined planar silver Koch fractal antennas were fabricated and characterized with high spatial and spectral resolution using electron energy loss spectroscopy. The experimental data are supported by numerical calculations carried out with a surface integral equation method. Multiple surface plasmon edge modes supported by the fractal structures have been imaged and analyzed. Furthermore, by isolating and reproducing self-similar features in long silver strip antennas, the edge modes present in the Koch snowflake fractals are identified. We demonstrate that the fractal response can be obtained by the sum of basic self-similar segments called characteristic edge units. Interestingly, the plasmon edge modes follow a fractal-scaling rule that depends on these self-similar segments formed in the structure after a fractal iteration. As the size of a fractal structure is reduced, coupling of the modes in the characteristic edge units becomes relevant, and the symmetry of the fractal affects the formation of hybrid modes. This analysis can be utilized not only to understand the edge modes in other planar structures but also in the design and fabrication of fractal structures for nanophotonic applications.

Hands-On Fractals and the Unexpected in Mathematics

ERIC Educational Resources Information Center

Gluchoff, Alan

2006-01-01

This article describes a hands-on project in which unusual fractal images are produced using only a photocopy machine and office supplies. The resulting images are an example of the contraction mapping principle.

Application of fractal dimension method of functional MRI time-series to limbic dysregulation in anxiety study.

PubMed

Olejarczyk, Elzbieta

2007-01-01

Functional magnetic resonance imaging (fMRI) allows to investigate the amplitude of activation in neural networks of brain. In this work we present the results of fMRI time-series analysis performed to identify the process of dysregulation of dynamic interaction between different limbic system regions in healthy adults in state of increased anxiety. The results obtain for 65 healthy adults using nonlinear dynamics methods like fractal dimension confirm the key roles of the bilateral amygdala, bilateral hippocampus, BA9 (dorsolateral prefrontal cortex), and BA45 (ventromedial prefrontal cortex) in modulating emotional response in healthy adults. For different regions of interest (ROIs) significant correlations were found not only for the neutral respective rest but also for fear and angry contrasts.

Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis

PubMed Central

George, Anuh T.; Jeon, Tina; Hynan, Linda S.; Youn, Teddy S.; Kennedy, David N.; Dickerson, Bradford

2010-01-01

The purpose of this project is to apply a modified fractal analysis technique to high-resolution T1 weighted magnetic resonance images in order to quantify the alterations in the shape of the cerebral cortex that occur in patients with Alzheimer’s disease. Images were selected from the Alzheimer’s Disease Neuroimaging Initiative database (Control N=15, Mild-Moderate AD N=15). The images were segmented using a semi-automated analysis program. Four coronal and three axial profiles of the cerebral cortical ribbon were created. The fractal dimensions (Df) of the cortical ribbons were then computed using a box-counting algorithm. The mean Df of the cortical ribbons from AD patients were lower than age-matched controls on six of seven profiles. The fractal measure has regional variability which reflects local differences in brain structure. Fractal dimension is complementary to volumetric measures and may assist in identifying disease state or disease progression. PMID:20740072

Fractal nematic colloids

NASA Astrophysics Data System (ADS)

Hashemi, S. M.; Jagodič, U.; Mozaffari, M. R.; Ejtehadi, M. R.; Muševič, I.; Ravnik, M.

2017-01-01

Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter.

Fractal nematic colloids

PubMed Central

Hashemi, S. M.; Jagodič, U.; Mozaffari, M. R.; Ejtehadi, M. R.; Muševič, I.; Ravnik, M.

2017-01-01

Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter. PMID:28117325

Characterization of branch complexity by fractal analyses and detect plant functional adaptations

USGS Publications Warehouse

Alados, C.L.; Escos, J.; Emlen, J.M.; Freeman, D.C.

1999-01-01

The comparison between complexity in the sense of space occupancy (box-counting fractal dimension Dc and information dimension DI ) and heterogeneity in the sense of space distribution (average evenness index and evenness variation coefficient JCV) were investigated in mathematical fractal objects and natural branch ¯ J structures. In general, increased fractal dimension was paired with low heterogeneity. Comparisons between branch architecture in Anthyllis cytisoides under different slope exposure and grazing impact revealed that branches were more complex and more homogeneously distributed for plants on northern exposures than southern, while grazing had no impact during a wet year. Developmental instability was also investigated by the statistical noise of the allometric relation between internode length and node order. In conclusion, our study demonstrated that fractal dimension of branch structure can be used to analyze the structural organization of plants, especially if we consider not only fractal dimension but also shoot distribution within the canopy (lacunarity). These indexes together with developmental instability analyses are good indicators of growth responses to the environment.

Fractal analysis of Xylella fastidiosa biofilm formation

NASA Astrophysics Data System (ADS)

Moreau, A. L. D.; Lorite, G. S.; Rodrigues, C. M.; Souza, A. A.; Cotta, M. A.

2009-07-01

We have investigated the growth process of Xylella fastidiosa biofilms inoculated on a glass. The size and the distance between biofilms were analyzed by optical images; a fractal analysis was carried out using scaling concepts and atomic force microscopy images. We observed that different biofilms show similar fractal characteristics, although morphological variations can be identified for different biofilm stages. Two types of structural patterns are suggested from the observed fractal dimensions Df. In the initial and final stages of biofilm formation, Df is 2.73±0.06 and 2.68±0.06, respectively, while in the maturation stage, Df=2.57±0.08. These values suggest that the biofilm growth can be understood as an Eden model in the former case, while diffusion-limited aggregation (DLA) seems to dominate the maturation stage. Changes in the correlation length parallel to the surface were also observed; these results were correlated with the biofilm matrix formation, which can hinder nutrient diffusion and thus create conditions to drive DLA growth.

Fractal Theory and Field Cover Experiments: Implications for the Fractal Characteristics and Radon Diffusion Behavior of Soils and Rocks.

PubMed

Tan, Wanyu; Li, Yongmei; Tan, Kaixuan; Duan, Xianzhe; Liu, Dong; Liu, Zehua

2016-12-01

Radon diffusion and transport through different media is a complex process affected by many factors. In this study, the fractal theories and field covering experiments were used to study the fractal characteristics of particle size distribution (PSD) of six kinds of geotechnical materials (e.g., waste rock, sand, laterite, kaolin, mixture of sand and laterite, and mixture of waste rock and laterite) and their effects on radon diffusion. In addition, the radon diffusion coefficient and diffusion length were calculated. Moreover, new formulas for estimating diffusion coefficient and diffusion length functional of fractal dimension d of PSD were proposed. These results demonstrate the following points: (1) the fractal dimension d of the PSD can be used to characterize the property of soils and rocks in the studies of radon diffusion behavior; (2) the diffusion coefficient and diffusion length decrease with increasing fractal dimension of PSD; and (3) the effectiveness of final covers in reducing radon exhalation of uranium tailings impoundments can be evaluated on the basis of the fractal dimension of PSD of materials.

Fractal dynamics of earthquakes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bak, P.; Chen, K.

1995-05-01

Many objects in nature, from mountain landscapes to electrical breakdown and turbulence, have a self-similar fractal spatial structure. It seems obvious that to understand the origin of self-similar structures, one must understand the nature of the dynamical processes that created them: temporal and spatial properties must necessarily be completely interwoven. This is particularly true for earthquakes, which have a variety of fractal aspects. The distribution of energy released during earthquakes is given by the Gutenberg-Richter power law. The distribution of epicenters appears to be fractal with dimension D {approx} 1--1.3. The number of after shocks decay as a function ofmore » time according to the Omori power law. There have been several attempts to explain the Gutenberg-Richter law by starting from a fractal distribution of faults or stresses. But this is a hen-and-egg approach: to explain the Gutenberg-Richter law, one assumes the existence of another power-law--the fractal distribution. The authors present results of a simple stick slip model of earthquakes, which evolves to a self-organized critical state. Emphasis is on demonstrating that empirical power laws for earthquakes indicate that the Earth`s crust is at the critical state, with no typical time, space, or energy scale. Of course the model is tremendously oversimplified; however in analogy with equilibrium phenomena they do not expect criticality to depend on details of the model (universality).« less

Fractal analysis of the susceptibility weighted imaging patterns in malignant brain tumors during antiangiogenic treatment: technical report on four cases serially imaged by 7 T magnetic resonance during a period of four weeks.

PubMed

Di Ieva, Antonio; Matula, Christian; Grizzi, Fabio; Grabner, Günther; Trattnig, Siegfried; Tschabitscher, Manfred

2012-01-01

The need for new and objective indexes for the neuroradiologic follow-up of brain tumors and for monitoring the effects of antiangiogenic strategies in vivo led us to perform a technical study on four patients who received computerized analysis of tumor-associated vasculature with ultra-high-field (7 T) magnetic resonance imaging (MRI). The image analysis involved the application of susceptibility weighted imaging (SWI) to evaluate vascular structures. Four patients affected by recurrent malignant brain tumors were enrolled in the present study. After the first 7-T SWI MRI procedure, the patients underwent antiangiogenic treatment with bevacizumab. The imaging was repeated every 2 weeks for a period of 4 weeks. The SWI patterns visualized in the three MRI temporal sequences were analyzed by means of a computer-aided fractal-based method to objectively quantify their geometric complexity. In two clinically deteriorating patients we found an increase of the geometric complexity of the space-filling properties of the SWI patterns over time despite the antiangiogenic treatment. In one patient, who showed improvement with the therapy, the fractal dimension of the intratumoral structure decreased, whereas in the fourth patient, no differences were found. The qualitative changes of the intratumoral SWI patterns during a period of 4 weeks were quantified with the fractal dimension. Because SWI patterns are also related to the presence of vascular structures, the quantification of their space-filling properties with fractal dimension seemed to be a valid tool for the in vivo neuroradiologic follow-up of brain tumors. Copyright © 2012 Elsevier Inc. All rights reserved.

LETTER TO THE EDITOR: Fractal diffusion coefficient from dynamical zeta functions

NASA Astrophysics Data System (ADS)

Cristadoro, Giampaolo

2006-03-01

Dynamical zeta functions provide a powerful method to analyse low-dimensional dynamical systems when the underlying symbolic dynamics is under control. On the other hand, even simple one-dimensional maps can show an intricate structure of the grammar rules that may lead to a non-smooth dependence of global observables on parameters changes. A paradigmatic example is the fractal diffusion coefficient arising in a simple piecewise linear one-dimensional map of the real line. Using the Baladi-Ruelle generalization of the Milnor-Thurnston kneading determinant, we provide the exact dynamical zeta function for such a map and compute the diffusion coefficient from its smallest zero.

Fractal vector optical fields.

PubMed

Pan, Yue; Gao, Xu-Zhen; Cai, Meng-Qiang; Zhang, Guan-Lin; Li, Yongnan; Tu, Chenghou; Wang, Hui-Tian

2016-07-15

We introduce the concept of a fractal, which provides an alternative approach for flexibly engineering the optical fields and their focal fields. We propose, design, and create a new family of optical fields-fractal vector optical fields, which build a bridge between the fractal and vector optical fields. The fractal vector optical fields have polarization states exhibiting fractal geometry, and may also involve the phase and/or amplitude simultaneously. The results reveal that the focal fields exhibit self-similarity, and the hierarchy of the fractal has the "weeding" role. The fractal can be used to engineer the focal field.

Taxonomy of Individual Variations in Aesthetic Responses to Fractal Patterns

PubMed Central

Spehar, Branka; Walker, Nicholas; Taylor, Richard P.

2016-01-01

In two experiments, we investigate group and individual preferences in a range of different types of patterns with varying fractal-like scaling characteristics. In Experiment 1, we used 1/f filtered grayscale images as well as their thresholded (black and white) and edges only counterparts. Separate groups of observers viewed different types of images varying in slope of their amplitude spectra. Although with each image type, the groups exhibited the “universal” pattern of preference for intermediate amplitude spectrum slopes, we identified 4 distinct sub-groups in each case. Sub-group 1 exhibited a typical peak preference for intermediate amplitude spectrum slopes (“intermediate”; approx. 50%); sub-group 2 exhibited a linear increase in preference with increasing amplitude spectrum slope (“smooth”; approx. 20%), while sub-group 3 exhibited a linear decrease in preference as a function of the amplitude spectrum slope (“sharp”; approx. 20%). Sub-group 4 revealed no significant preference (“other”; approx. 10%). In Experiment 2, we extended the range of different image types and investigated preferences within the same observers. We replicate the results of our first experiment and show that individual participants exhibit stable patterns of preference across a wide range of image types. In both experiments, Q-mode factor analysis identified two principal factors that were able to explain more than 80% of interindividual variations in preference across all types of images, suggesting a highly similar dimensional structure of interindividual variations in preference for fractal-like scaling characteristics. PMID:27458365

The fractal geometry of Hartree-Fock

NASA Astrophysics Data System (ADS)

Theel, Friethjof; Karamatskou, Antonia; Santra, Robin

2017-12-01

The Hartree-Fock method is an important approximation for the ground-state electronic wave function of atoms and molecules so that its usage is widespread in computational chemistry and physics. The Hartree-Fock method is an iterative procedure in which the electronic wave functions of the occupied orbitals are determined. The set of functions found in one step builds the basis for the next iteration step. In this work, we interpret the Hartree-Fock method as a dynamical system since dynamical systems are iterations where iteration steps represent the time development of the system, as encountered in the theory of fractals. The focus is put on the convergence behavior of the dynamical system as a function of a suitable control parameter. In our case, a complex parameter λ controls the strength of the electron-electron interaction. An investigation of the convergence behavior depending on the parameter λ is performed for helium, neon, and argon. We observe fractal structures in the complex λ-plane, which resemble the well-known Mandelbrot set, determine their fractal dimension, and find that with increasing nuclear charge, the fragmentation increases as well.

Not just fractal surfaces, but surface fractal aggregates: Derivation of the expression for the structure factor and its applications

NASA Astrophysics Data System (ADS)

Besselink, R.; Stawski, T. M.; Van Driessche, A. E. S.; Benning, L. G.

2016-12-01

Densely packed surface fractal aggregates form in systems with high local volume fractions of particles with very short diffusion lengths, which effectively means that particles have little space to move. However, there are no prior mathematical models, which would describe scattering from such surface fractal aggregates and which would allow the subdivision between inter- and intraparticle interferences of such aggregates. Here, we show that by including a form factor function of the primary particles building the aggregate, a finite size of the surface fractal interfacial sub-surfaces can be derived from a structure factor term. This formalism allows us to define both a finite specific surface area for fractal aggregates and the fraction of particle interfacial sub-surfaces at the perimeter of an aggregate. The derived surface fractal model is validated by comparing it with an ab initio approach that involves the generation of a "brick-in-a-wall" von Koch type contour fractals. Moreover, we show that this approach explains observed scattering intensities from in situ experiments that followed gypsum (CaSO4 ṡ 2H2O) precipitation from highly supersaturated solutions. Our model of densely packed "brick-in-a-wall" surface fractal aggregates may well be the key precursor step in the formation of several types of mosaic- and meso-crystals.

Towards thermomechanics of fractal media

NASA Astrophysics Data System (ADS)

Ostoja-Starzewski, Martin

2007-11-01

Hans Ziegler’s thermomechanics [1,2,3], established half a century ago, is extended to fractal media on the basis of a recently introduced continuum mechanics due to Tarasov [14,15]. Employing the concept of internal (kinematic) variables and internal stresses, as well as the quasiconservative and dissipative stresses, a field form of the second law of thermodynamics is derived. In contradistinction to the conventional Clausius Duhem inequality, it involves generalized rates of strain and internal variables. Upon introducing a dissipation function and postulating the thermodynamic orthogonality on any lengthscale, constitutive laws of elastic-dissipative fractal media naturally involving generalized derivatives of strain and stress can then be derived. This is illustrated on a model viscoelastic material. Also generalized to fractal bodies is the Hill condition necessary for homogenization of their constitutive responses.

a New Method for Calculating Fractal Dimensions of Porous Media Based on Pore Size Distribution

NASA Astrophysics Data System (ADS)

Xia, Yuxuan; Cai, Jianchao; Wei, Wei; Hu, Xiangyun; Wang, Xin; Ge, Xinmin

Fractal theory has been widely used in petrophysical properties of porous rocks over several decades and determination of fractal dimensions is always the focus of researches and applications by means of fractal-based methods. In this work, a new method for calculating pore space fractal dimension and tortuosity fractal dimension of porous media is derived based on fractal capillary model assumption. The presented work establishes relationship between fractal dimensions and pore size distribution, which can be directly used to calculate the fractal dimensions. The published pore size distribution data for eight sandstone samples are used to calculate the fractal dimensions and simultaneously compared with prediction results from analytical expression. In addition, the proposed fractal dimension method is also tested through Micro-CT images of three sandstone cores, and are compared with fractal dimensions by box-counting algorithm. The test results also prove a self-similar fractal range in sandstone when excluding smaller pores.

Polymer Self-Assembly into Unique Fractal Nanostructures in Solution by a One-Shot Synthetic Procedure.

PubMed

Shin, Suyong; Gu, Ming-Long; Yu, Chin-Yang; Jeon, Jongseol; Lee, Eunji; Choi, Tae-Lim

2018-01-10

A fractal nanostructure having a high surface area is potentially useful in sensors, catalysts, functional coatings, and biomedical and electronic applications. Preparation of fractal nanostructures on solid substrates has been reported using various inorganic or organic compounds. However, achieving such a process using polymers in solution has been extremely challenging. Here, we report a simple one-shot preparation of polymer fractal nanostructures in solution via an unprecedented assembly mechanism controlled by polymerization and self-assembly kinetics. This was possible only because one monomer was significantly more reactive than the other, thereby easily forming a diblock copolymer microstructure. Then, the second insoluble block containing poly(p-phenylenevinylene) (PPV) without any side chains spontaneously underwent self-assembly during polymerization by an in situ nanoparticlization of conjugated polymers (INCP) method. The formation of fractal structures in solution was confirmed by various imaging techniques such as atomic force microscopy, transmission electron microscopy (TEM), and cryogenic TEM. The diffusion-limited aggregation theory was adopted to explain the branching patterns of the fractal nanostructures according to the changes in polymerization conditions such as the monomer concentration and the presence of additives. Finally, after detailed kinetic analyses, we proposed a plausible mechanism for the formation of unique fractal nanostructures, where the gradual formation and continuous growth of micelles in a chain-growth-like manner were accounted for.

[Recent progress of research and applications of fractal and its theories in medicine].

PubMed

Cai, Congbo; Wang, Ping

2014-10-01

Fractal, a mathematics concept, is used to describe an image of self-similarity and scale invariance. Some organisms have been discovered with the fractal characteristics, such as cerebral cortex surface, retinal vessel structure, cardiovascular network, and trabecular bone, etc. It has been preliminarily confirmed that the three-dimensional structure of cells cultured in vitro could be significantly enhanced by bionic fractal surface. Moreover, fractal theory in clinical research will help early diagnosis and treatment of diseases, reducing the patient's pain and suffering. The development process of diseases in the human body can be expressed by the fractal theories parameter. It is of considerable significance to retrospectively review the preparation and application of fractal surface and its diagnostic value in medicine. This paper gives an application of fractal and its theories in the medical science, based on the research achievements in our laboratory.

Plasmon confinement in fractal quantum systems

NASA Astrophysics Data System (ADS)

Westerhout, Tom; van Veen, Edo; Katsnelson, Mikhail I.; Yuan, Shengjun

2018-05-01

Recent progress in the fabrication of materials has made it possible to create arbitrary nonperiodic two-dimensional structures in the quantum plasmon regime. This paves the way for exploring the quantum plasmonic properties of electron gases in complex geometries. In this work we study systems with a fractal dimension. We calculate the full dielectric functions of two prototypical fractals with different ramification numbers, namely the Sierpinski carpet and gasket. We show that the Sierpinski carpet has a dispersion comparable to a square lattice, but the Sierpinski gasket features highly localized plasmon modes with a flat dispersion. This strong plasmon confinement in finitely ramified fractals can provide a novel setting for manipulating light at the quantum level.

The fractal forest: fractal geometry and applications in forest science.

Treesearch

Nancy D. Lorimer; Robert G. Haight; Rolfe A. Leary

1994-01-01

Fractal geometry is a tool for describing and analyzing irregularity. Because most of what we measure in the forest is discontinuous, jagged, and fragmented, fractal geometry has potential for improving the precision of measurement and description. This study reviews the literature on fractal geometry and its applications to forest measurements.

Heterogeneity of Glucose Metabolism in Esophageal Cancer Measured by Fractal Analysis of Fluorodeoxyglucose Positron Emission Tomography Image: Correlation between Metabolic Heterogeneity and Survival.

PubMed

Tochigi, Toru; Shuto, Kiyohiko; Kono, Tsuguaki; Ohira, Gaku; Tohma, Takayuki; Gunji, Hisashi; Hayano, Koichi; Narushima, Kazuo; Fujishiro, Takeshi; Hanaoka, Toshiharu; Akutsu, Yasunori; Okazumi, Shinichi; Matsubara, Hisahiro

2017-01-01

Intratumoral heterogeneity is a well-recognized characteristic feature of cancer. The purpose of this study is to assess the heterogeneity of the intratumoral glucose metabolism using fractal analysis, and evaluate its prognostic value in patients with esophageal squamous cell carcinoma (ESCC). 18F-fluorodeoxyglucose positron emission tomography (FDG-PET) studies of 79 patients who received curative surgery were evaluated. FDG-PET images were analyzed using fractal analysis software, where differential box-counting method was employed to calculate the fractal dimension (FD) of the tumor lesion. Maximum standardized uptake value (SUVmax) and FD were compared with overall survival (OS). The median SUVmax and FD of ESCCs in this cohort were 13.8 and 1.95, respectively. In univariate analysis performed using Cox's proportional hazard model, T stage and FD showed significant associations with OS (p = 0.04, p < 0.0001, respectively), while SUVmax did not (p = 0.1). In Kaplan-Meier analysis, the low FD tumor (<1.95) showed a significant association with favorable OS (p < 0.0001). In wthe multivariate analysis among TNM staging, serum tumor markers, FD, and SUVmax, the FD was identified as the only independent prognostic factor for OS (p = 0.0006; hazards ratio 0.251, 95% CI 0.104-0.562). Metabolic heterogeneity measured by fractal analysis can be a novel imaging biomarker for survival in patients with ESCC. © 2016 S. Karger AG, Basel.

Experimental criteria for the determination of fractal parameters of premixed turbulent flames

NASA Astrophysics Data System (ADS)

Shepherd, I. G.; Cheng, Robert K.; Talbot, L.

1992-10-01

The influence of spatial resolution, digitization noise, the number of records used for averaging, and the method of analysis on the determination of the fractal parameters of a high Damköhler number, methane/air, premixed, turbulent stagnation-point flame are investigated in this paper. The flow exit velocity was 5 m/s and the turbulent Reynolds number was 70 based on a integral scale of 3 mm and a turbulent intensity of 7%. The light source was a copper vapor laser which delivered 20 nsecs, 5 mJ pulses at 4 kHz and the tomographic cross-sections of the flame were recorded by a high speed movie camera. The spatial resolution of the images is 155 × 121 μm/pixel with a field of view of 50 × 65 mm. The stepping caliper technique for obtaining the fractal parameters is found to give the clearest indication of the cutoffs and the effects of noise. It is necessary to ensemble average the results from more than 25 statistically independent images to reduce sufficiently the scatter in the fractal parameters. The effects of reduced spatial resolution on fractal plots are estimated by artificial degradation of the resolution of the digitized flame boundaries. The effect of pixel resolution, an apparent increase in flame length below the inner scale rolloff, appears in the fractal plots when the measurent scale is less than approximately twice the pixel resolution. Although a clearer determination of fractal parameters is obtained by local averaging of the flame boundaries which removes digitization noise, at low spatial resolution this technique can reduce the fractal dimension. The degree of fractal isotropy of the flame surface can have a significant effect on the estimation of the flame surface area and hence burning rate from two-dimensional images. To estimate this isotropy a determination of the outer cutoff is required and three-dimensional measurements are probably also necessary.

A Tutorial Review on Fractal Spacetime and Fractional Calculus

NASA Astrophysics Data System (ADS)

He, Ji-Huan

2014-11-01

This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz's notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie's mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.

Method for non-referential defect characterization using fractal encoding and active contours

DOEpatents

Gleason, Shaun S [Knoxville, TN; Sari-Sarraf, Hamed [Lubbock, TX

2007-05-15

A method for identification of anomalous structures, such as defects, includes the steps of providing a digital image and applying fractal encoding to identify a location of at least one anomalous portion of the image. The method does not require a reference image to identify the location of the anomalous portion. The method can further include the step of initializing an active contour based on the location information obtained from the fractal encoding step and deforming an active contour to enhance the boundary delineation of the anomalous portion.

Trabecular Bone Mechanical Properties and Fractal Dimension

NASA Technical Reports Server (NTRS)

Hogan, Harry A.

1996-01-01

Countermeasures for reducing bone loss and muscle atrophy due to extended exposure to the microgravity environment of space are continuing to be developed and improved. An important component of this effort is finite element modeling of the lower extremity and spinal column. These models will permit analysis and evaluation specific to each individual and thereby provide more efficient and effective exercise protocols. Inflight countermeasures and post-flight rehabilitation can then be customized and targeted on a case-by-case basis. Recent Summer Faculty Fellowship participants have focused upon finite element mesh generation, muscle force estimation, and fractal calculations of trabecular bone microstructure. Methods have been developed for generating the three-dimensional geometry of the femur from serial section magnetic resonance images (MRI). The use of MRI as an imaging modality avoids excessive exposure to radiation associated with X-ray based methods. These images can also detect trabecular bone microstructure and architecture. The goal of the current research is to determine the degree to which the fractal dimension of trabecular architecture can be used to predict the mechanical properties of trabecular bone tissue. The elastic modulus and the ultimate strength (or strain) can then be estimated from non-invasive, non-radiating imaging and incorporated into the finite element models to more accurately represent the bone tissue of each individual of interest. Trabecular bone specimens from the proximal tibia are being studied in this first phase of the work. Detailed protocols and procedures have been developed for carrying test specimens through all of the steps of a multi-faceted test program. The test program begins with MRI and X-ray imaging of the whole bones before excising a smaller workpiece from the proximal tibia region. High resolution MRI scans are then made and the piece further cut into slabs (roughly 1 cm thick). The slabs are X-rayed again

Microvascular fractal dimension predicts prognosis and response to chemotherapy in glioblastoma: an automatic image analysis study.

PubMed

Chen, Cong; He, Zhi-Cheng; Shi, Yu; Zhou, Wenchao; Zhang, Xia; Xiao, Hua-Liang; Wu, Hai-Bo; Yao, Xiao-Hong; Luo, Wan-Chun; Cui, You-Hong; Bao, Shideng; Kung, Hsiang-Fu; Bian, Xiu-Wu; Ping, Yi-Fang

2018-05-15

The microvascular profile has been included in the WHO glioma grading criteria. Nevertheless, microvessels in gliomas of the same WHO grade, e.g., WHO IV glioblastoma (GBM), exhibit heterogeneous and polymorphic morphology, whose possible clinical significance remains to be determined. In this study, we employed a fractal geometry-derived parameter, microvascular fractal dimension (mvFD), to quantify microvessel complexity and developed a home-made macro in Image J software to automatically determine mvFD from the microvessel-stained immunohistochemical images of GBM. We found that mvFD effectively quantified the morphological complexity of GBM microvasculature. Furthermore, high mvFD favored the survival of GBM patients as an independent prognostic indicator and predicted a better response to chemotherapy of GBM patients. When investigating the underlying relations between mvFD and tumor growth by deploying Ki67/mvFD as an index for microvasculature-normalized tumor proliferation, we discovered an inverse correlation between mvFD and Ki67/mvFD. Furthermore, mvFD inversely correlated with the expressions of a glycolytic marker, LDHA, which indicated poor prognosis of GBM patients. Conclusively, we developed an automatic approach for mvFD measurement, and demonstrated that mvFD could predict the prognosis and response to chemotherapy of GBM patients.

A new set of wavelet- and fractals-based features for Gleason grading of prostate cancer histopathology images

NASA Astrophysics Data System (ADS)

Mosquera Lopez, Clara; Agaian, Sos

2013-02-01

Prostate cancer detection and staging is an important step towards patient treatment selection. Advancements in digital pathology allow the application of new quantitative image analysis algorithms for computer-assisted diagnosis (CAD) on digitized histopathology images. In this paper, we introduce a new set of features to automatically grade pathological images using the well-known Gleason grading system. The goal of this study is to classify biopsy images belonging to Gleason patterns 3, 4, and 5 by using a combination of wavelet and fractal features. For image classification we use pairwise coupling Support Vector Machine (SVM) classifiers. The accuracy of the system, which is close to 97%, is estimated through three different cross-validation schemes. The proposed system offers the potential for automating classification of histological images and supporting prostate cancer diagnosis.

Random-fractal Ansatz for the configurations of two-dimensional critical systems

NASA Astrophysics Data System (ADS)

Lee, Ching Hua; Ozaki, Dai; Matsueda, Hiroaki

2016-12-01

Critical systems have always intrigued physicists and precipitated the development of new techniques. Recently, there has been renewed interest in the information contained in the configurations of classical critical systems, whose computation do not require full knowledge of the wave function. Inspired by holographic duality, we investigated the entanglement properties of the classical configurations (snapshots) of the Potts model by introducing an Ansatz ensemble of random fractal images. By virtue of the central limit theorem, our Ansatz accurately reproduces the entanglement spectra of actual Potts snapshots without any fine tuning of parameters or artificial restrictions on ensemble choice. It provides a microscopic interpretation of the results of previous studies, which established a relation between the scaling behavior of snapshot entropy and the critical exponent. More importantly, it elucidates the role of ensemble disorder in restoring conformal invariance, an aspect previously ignored. Away from criticality, the breakdown of scale invariance leads to a renormalization of the parameter Σ in the random fractal Ansatz, whose variation can be used as an alternative determination of the critical exponent. We conclude by providing a recipe for the explicit construction of fractal unit cells consistent with a given scaling exponent.

Fractal astronomy.

NASA Astrophysics Data System (ADS)

Beech, M.

1989-02-01

The author discusses some of the more recent research on fractal astronomy and results presented in several astronomical studies. First, the large-scale structure of the universe is considered, while in another section one drops in scale to examine some of the smallest bodies in our solar system; the comets and meteoroids. The final section presents some thoughts on what influence the fractal ideology might have on astronomy, focusing particularly on the question recently raised by Kadanoff, "Fractals: where's the physics?"

A tale of two fractals: The Hofstadter butterfly and the integral Apollonian gaskets

NASA Astrophysics Data System (ADS)

Satija, Indubala I.

2016-11-01

This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden D3 symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions.

Anomalous relaxation in fractal structures

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fujiwara, S.; Yonezawa, F.

1995-03-01

For the purpose of studying some interesting properties of anomalous relaxation in fractal structures, we carry out Monte Carlo simulations of random walks on two-dimensional fractal structures (Sierpinski carpets with different cutouts and site-percolation clusters in a square lattice at the critical concentration). We find that the relaxation is of the Cole-Cole type [J. Chem. Phys. 9, 341 (1941)], which is one of the empirical laws of anomalous relaxation. Scaling properties are found in the relaxation function as well as in the particle density. We also find that, in strucures with almost the same fractal dimension, relaxation in structures withmore » dead ends is slower than that in structures without them. This paper ascertains that the essential aspects of the anomalous relaxation due to many-body effects can be explained in the framework of the one-body model.« less

Microtopographic Inspection and Fractal Analysis of Skin Neoplasia

NASA Astrophysics Data System (ADS)

Costa, Manuel F. M.; Hipolito, Alberto Valencia; Gutierrez, Gustavo Fidel; Chanona, Jorge; Gallegos, Eva Ramón

2008-04-01

Early detection of skin cancer is fundamental to a successful treatment. Changes in the shape, including the relief, of skin lesions are an indicator of a possible malignity. Optical microtopographic inspection of skin lesions can be used to identify diagnostic patterns of benign and malign skin' lesions. Statistical parameters like the mean roughness (Ra) may allow the discrimination between different types of lesions and degree of malignity. Fractal analysis of bi-dimensional and 3D images of skin lesions can validate or complement that assessment by calculation of its fractal dimensions (FD). On the study herein reported the microtopographic inspection of the skin lesions were performed using the optical triangulation based microtopographer developed at the Physics Department of the University of Minho, MICROTOP.03.MFC. The patients that participated in this research work were men and women older than 15 years with the clinical and histopathology diagnoses of: melanoma, basocellular carcinoma, epidermoide carcinoma, actinic keratosis, keratoacantosis and benign nevus. Latex impressions of the lesions were taken and microtopographically analyzed. Characteristic information for each type of studied lesion was obtained. For melanoma it was observed that on the average these tumors present an increased roughness of around 67 percent compared to the roughness of the healthy skin. This feature allows the distinction from other tumors as basocellular carcinoma (were the roughness increase was in the average of 49 percent) and benign lesions as the epidermoide cyst (37 percent) or the seborrhea keratosis (4 percent). Tumor size and roughness are directly proportional to the grade of malignality. The characterization of the fractal geometry of 2D (histological slides) and 3D images of skin lesions was performed by obtaining its FD evaluated by means of the Box counting method. Results obtained showed that the average fractal dimension of histological slide images (FDh

Fractal analysis of mandibular trabecular bone: optimal tile sizes for the tile counting method.

PubMed

Huh, Kyung-Hoe; Baik, Jee-Seon; Yi, Won-Jin; Heo, Min-Suk; Lee, Sam-Sun; Choi, Soon-Chul; Lee, Sun-Bok; Lee, Seung-Pyo

2011-06-01

This study was performed to determine the optimal tile size for the fractal dimension of the mandibular trabecular bone using a tile counting method. Digital intraoral radiographic images were obtained at the mandibular angle, molar, premolar, and incisor regions of 29 human dry mandibles. After preprocessing, the parameters representing morphometric characteristics of the trabecular bone were calculated. The fractal dimensions of the processed images were analyzed in various tile sizes by the tile counting method. The optimal range of tile size was 0.132 mm to 0.396 mm for the fractal dimension using the tile counting method. The sizes were closely related to the morphometric parameters. The fractal dimension of mandibular trabecular bone, as calculated with the tile counting method, can be best characterized with a range of tile sizes from 0.132 to 0.396 mm.

Three-Dimensional Surface Parameters and Multi-Fractal Spectrum of Corroded Steel

PubMed Central

Shanhua, Xu; Songbo, Ren; Youde, Wang

2015-01-01

To study multi-fractal behavior of corroded steel surface, a range of fractal surfaces of corroded surfaces of Q235 steel were constructed by using the Weierstrass-Mandelbrot method under a high total accuracy. The multi-fractal spectrum of fractal surface of corroded steel was calculated to study the multi-fractal characteristics of the W-M corroded surface. Based on the shape feature of the multi-fractal spectrum of corroded steel surface, the least squares method was applied to the quadratic fitting of the multi-fractal spectrum of corroded surface. The fitting function was quantitatively analyzed to simplify the calculation of multi-fractal characteristics of corroded surface. The results showed that the multi-fractal spectrum of corroded surface was fitted well with the method using quadratic curve fitting, and the evolution rules and trends were forecasted accurately. The findings can be applied to research on the mechanisms of corroded surface formation of steel and provide a new approach for the establishment of corrosion damage constitutive models of steel. PMID:26121468

Three-Dimensional Surface Parameters and Multi-Fractal Spectrum of Corroded Steel.

PubMed

Shanhua, Xu; Songbo, Ren; Youde, Wang

2015-01-01

To study multi-fractal behavior of corroded steel surface, a range of fractal surfaces of corroded surfaces of Q235 steel were constructed by using the Weierstrass-Mandelbrot method under a high total accuracy. The multi-fractal spectrum of fractal surface of corroded steel was calculated to study the multi-fractal characteristics of the W-M corroded surface. Based on the shape feature of the multi-fractal spectrum of corroded steel surface, the least squares method was applied to the quadratic fitting of the multi-fractal spectrum of corroded surface. The fitting function was quantitatively analyzed to simplify the calculation of multi-fractal characteristics of corroded surface. The results showed that the multi-fractal spectrum of corroded surface was fitted well with the method using quadratic curve fitting, and the evolution rules and trends were forecasted accurately. The findings can be applied to research on the mechanisms of corroded surface formation of steel and provide a new approach for the establishment of corrosion damage constitutive models of steel.

Fractal dimension assessment of brain white matter structural complexity post stroke in relation to upper-extremity motor function

PubMed Central

Zhang, Luduan; Butler, Andrew J.; Sun, Chang-Kai; Sahgal, Vinod; Wittenberg, George F.; Yue, Guang H.

2008-01-01

Little is known about the association between brain white matter (WM) structure and motor function in humans. This study investigated complexity of brain WM interior shape as determined by magnetic resonance imaging (MRI) and its relationship with upper-extremity (UE) motor function in patients post stroke. We hypothesized that (1) the WM complexity would decrease following stroke, and (2) higher WM complexity in non-affected cortical areas would be related to greater UE motor function. Thirty-eight stroke patients (16 with left-hemisphere lesions) underwent MRI anatomical brain scans. Fractal dimension (FD), a quantitative shape metric, was applied onto skeletonized brain WM images to evaluate WM internal structural complexity. Wolf Motor Function Test (WMFT) and Fugl-Meyer Motor Assessment (FM) scores were measured to assess motor function of the affected limb. The WM complexity was lower in the stroke-affected hemisphere. The FD was associated with better motor function in two subgroups: with left-subcortical lesions, FD values of the lesion-free areas of the left hemisphere were associated with better FM scores; with right-cortical lesions, FD values of lesion-free regions were robustly associated with better WMFT scores. These findings suggest that greater residual WM complexity is associated with less impaired UE motor function, which is more robust in patients with right-hemisphere lesions. No correlations were found between lesion volume and WMFT or FM scores. This study addressed WM complexity in stroke patients and its relationship with UE motor function. Measurement of brain WM reorganization may be a sensitive correlate of UE function in people recovering from stroke. PMID:18590710

Fractals for Geoengineering

NASA Astrophysics Data System (ADS)

Oleshko, Klaudia; de Jesús Correa López, María; Romero, Alejandro; Ramírez, Victor; Pérez, Olga

2016-04-01

The effectiveness of fractal toolbox to capture the scaling or fractal probability distribution, and simply fractal statistics of main hydrocarbon reservoir attributes, was highlighted by Mandelbrot (1995) and confirmed by several researchers (Zhao et al., 2015). Notwithstanding, after more than twenty years, it's still common the opinion that fractals are not useful for the petroleum engineers and especially for Geoengineering (Corbett, 2012). In spite of this negative background, we have successfully applied the fractal and multifractal techniques to our project entitled "Petroleum Reservoir as a Fractal Reactor" (2013 up to now). The distinguishable feature of Fractal Reservoir is the irregular shapes and rough pore/solid distributions (Siler, 2007), observed across a broad range of scales (from SEM to seismic). At the beginning, we have accomplished the detailed analysis of Nelson and Kibler (2003) Catalog of Porosity and Permeability, created for the core plugs of siliciclastic rocks (around ten thousand data were compared). We enriched this Catalog by more than two thousand data extracted from the last ten years publications on PoroPerm (Corbett, 2012) in carbonates deposits, as well as by our own data from one of the PEMEX, Mexico, oil fields. The strong power law scaling behavior was documented for the major part of these data from the geological deposits of contrasting genesis. Based on these results and taking into account the basic principles and models of the Physics of Fractals, introduced by Per Back and Kan Chen (1989), we have developed new software (Muukíl Kaab), useful to process the multiscale geological and geophysical information and to integrate the static geological and petrophysical reservoir models to dynamic ones. The new type of fractal numerical model with dynamical power law relations among the shapes and sizes of mesh' cells was designed and calibrated in the studied area. The statistically sound power law relations were established

The museum of unnatural form: a visual and tactile experience of fractals.

PubMed

Della-Bosca, D; Taylor, R P

2009-01-01

A remarkable computer technology is revolutionizing the world of design, allowing intricate patterns to be created with mathematical precision and then 'printed' as physical objects. Contour crafting is a fabrication process capable of assembling physical structures the sizes of houses, firing the imagination of a new generation of architects and artists (Khoshnevisat, 2008). Daniel Della-Bosca has jumped at this opportunity to create the 'Museum of Unnatural Form' at Griffith University. Della-Bosca's museum is populated with fractals sculptures - his own versions of nature's complex objects - that have been printed with the new technology. His sculptures bridge the historical divide in fractal studies between the abstract images of mathematics and the physical objects of Nature (Mandelbrot, 1982). Four of his fractal images will be featured on the cover of NDPLS in 2009.

The role of the circadian system in fractal neurophysiological control

PubMed Central

Pittman-Polletta, Benjamin R.; Scheer, Frank A.J.L.; Butler, Matthew P.; Shea, Steven A.; Hu, Kun

2013-01-01

Many neurophysiological variables such as heart rate, motor activity, and neural activity are known to exhibit intrinsic fractal fluctuations - similar temporal fluctuation patterns at different time scales. These fractal patterns contain information about health, as many pathological conditions are accompanied by their alteration or absence. In physical systems, such fluctuations are characteristic of critical states on the border between randomness and order, frequently arising from nonlinear feedback interactions between mechanisms operating on multiple scales. Thus, the existence of fractal fluctuations in physiology challenges traditional conceptions of health and disease, suggesting that high levels of integrity and adaptability are marked by complex variability, not constancy, and are properties of a neurophysiological network, not individual components. Despite the subject's theoretical and clinical interest, the neurophysiological mechanisms underlying fractal regulation remain largely unknown. The recent discovery that the circadian pacemaker (suprachiasmatic nucleus) plays a crucial role in generating fractal patterns in motor activity and heart rate sheds an entirely new light on both fractal control networks and the function of this master circadian clock, and builds a bridge between the fields of circadian biology and fractal physiology. In this review, we sketch the emerging picture of the developing interdisciplinary field of fractal neurophysiology by examining the circadian system’s role in fractal regulation. PMID:23573942

Using fractal analysis of thermal signatures for thyroid disease evaluation

NASA Astrophysics Data System (ADS)

Gavriloaia, Gheorghe; Sofron, Emil; Gavriloaia, Mariuca-Roxana; Ghemigean, Adina-Mariana

2010-11-01

The skin is the largest organ of the body and it protects against heat, light, injury and infection. Skin temperature is an important parameter for diagnosing diseases. Thermal analysis is non-invasive, painless, and relatively inexpensive, showing a great potential research. Since the thyroid regulates metabolic rate it is intimately connected to body temperature, more than, any modification of its function generates a specific thermal image on the neck skin. The shapes of thermal signatures are often irregular in size and shape. Euclidean geometry is not able to evaluate their shape for different thyroid diseases, and fractal geometry is used in this paper. Different thyroid diseases generate different shapes, and their complexity are evaluated by specific mathematical approaches, fractal analysis, in order to the evaluate selfsimilarity and lacunarity. Two kinds of thyroid diseases, hyperthyroidism and papillary cancer are analyzed in this paper. The results are encouraging and show the ability to continue research for thermal signature to be used in early diagnosis of thyroid diseases.

Fractal morphometry of cell complexity.

PubMed

Losa, Gabriele A

2002-01-01

Irregularity and self-similarity under scale changes are the main attributes of the morphological complexity of both normal and abnormal cells and tissues. In other words, the shape of a self-similar object does not change when the scale of measurement changes, because each part of it looks similar to the original object. However, the size and geometrical parameters of an irregular object do differ when it is examined at increasing resolution, which reveals more details. Significant progress has been made over the past three decades in understanding how irregular shapes and structures in the physical and biological sciences can be analysed. Dominant influences have been the discovery of a new practical geometry of Nature, now known as fractal geometry, and the continuous improvements in computation capabilities. Unlike conventional Euclidean geometry, which was developed to describe regular and ideal geometrical shapes which are practically unknown in nature, fractal geometry can be used to measure the fractal dimension, contour length, surface area and other dimension parameters of almost all irregular and complex biological tissues. We have used selected examples to illustrate the application of the fractal principle to measuring irregular and complex membrane ultrastructures of cells at specific functional and pathological stage.

Fractal analysis of mandibular trabecular bone: optimal tile sizes for the tile counting method

PubMed Central

Huh, Kyung-Hoe; Baik, Jee-Seon; Heo, Min-Suk; Lee, Sam-Sun; Choi, Soon-Chul; Lee, Sun-Bok; Lee, Seung-Pyo

2011-01-01

Purpose This study was performed to determine the optimal tile size for the fractal dimension of the mandibular trabecular bone using a tile counting method. Materials and Methods Digital intraoral radiographic images were obtained at the mandibular angle, molar, premolar, and incisor regions of 29 human dry mandibles. After preprocessing, the parameters representing morphometric characteristics of the trabecular bone were calculated. The fractal dimensions of the processed images were analyzed in various tile sizes by the tile counting method. Results The optimal range of tile size was 0.132 mm to 0.396 mm for the fractal dimension using the tile counting method. The sizes were closely related to the morphometric parameters. Conclusion The fractal dimension of mandibular trabecular bone, as calculated with the tile counting method, can be best characterized with a range of tile sizes from 0.132 to 0.396 mm. PMID:21977478

Fractal analysis of seafloor textures for target detection in synthetic aperture sonar imagery

NASA Astrophysics Data System (ADS)

Nabelek, T.; Keller, J.; Galusha, A.; Zare, A.

2018-04-01

Fractal analysis of an image is a mathematical approach to generate surface related features from an image or image tile that can be applied to image segmentation and to object recognition. In undersea target countermeasures, the targets of interest can appear as anomalies in a variety of contexts, visually different textures on the seafloor. In this paper, we evaluate the use of fractal dimension as a primary feature and related characteristics as secondary features to be extracted from synthetic aperture sonar (SAS) imagery for the purpose of target detection. We develop three separate methods for computing fractal dimension. Tiles with targets are compared to others from the same background textures without targets. The different fractal dimension feature methods are tested with respect to how well they can be used to detect targets vs. false alarms within the same contexts. These features are evaluated for utility using a set of image tiles extracted from a SAS data set generated by the U.S. Navy in conjunction with the Office of Naval Research. We find that all three methods perform well in the classification task, with a fractional Brownian motion model performing the best among the individual methods. We also find that the secondary features are just as useful, if not more so, in classifying false alarms vs. targets. The best classification accuracy overall, in our experimentation, is found when the features from all three methods are combined into a single feature vector.

Fractality of sensations and the brain health: the theory linking neurodegenerative disorder with distortion of spatial and temporal scale-invariance and fractal complexity of the visible world

PubMed Central

Zueva, Marina V.

2015-01-01

The theory that ties normal functioning and pathology of the brain and visual system with the spatial–temporal structure of the visual and other sensory stimuli is described for the first time in the present study. The deficit of fractal complexity of environmental influences can lead to the distortion of fractal complexity in the visual pathways of the brain and abnormalities of development or aging. The use of fractal light stimuli and fractal stimuli of other modalities can help to restore the functions of the brain, particularly in the elderly and in patients with neurodegenerative disorders or amblyopia. Non-linear dynamics of these physiological processes have a strong base of evidence, which is seen in the impaired fractal regulation of rhythmic activity in aged and diseased brains. From birth to old age, we live in a non-linear world, in which objects and processes with the properties of fractality and non-linearity surround us. Against this background, the evolution of man took place and all periods of life unfolded. Works of art created by man may also have fractal properties. The positive influence of music on cognitive functions is well-known. Insufficiency of sensory experience is believed to play a crucial role in the pathogenesis of amblyopia and age-dependent diseases. The brain is very plastic in its early development, and the plasticity decreases throughout life. However, several studies showed the possibility to reactivate the adult’s neuroplasticity in a variety of ways. We propose that a non-linear structure of sensory information on many spatial and temporal scales is crucial to the brain health and fractal regulation of physiological rhythms. Theoretical substantiation of the author’s theory is presented. Possible applications and the future research that can experimentally confirm or refute the theoretical concept are considered. PMID:26236232

Fractal pharmacokinetics.

PubMed

Pereira, Luis M

2010-06-01

Pharmacokinetics (PK) has been traditionally dealt with under the homogeneity assumption. However, biological systems are nowadays comprehensively understood as being inherently fractal. Specifically, the microenvironments where drug molecules interact with membrane interfaces, metabolic enzymes or pharmacological receptors, are unanimously recognized as unstirred, space-restricted, heterogeneous and geometrically fractal. Therefore, classical Fickean diffusion and the notion of the compartment as a homogeneous kinetic space must be revisited. Diffusion in fractal spaces has been studied for a long time making use of fractional calculus and expanding on the notion of dimension. Combining this new paradigm with the need to describe and explain experimental data results in defining time-dependent rate constants with a characteristic fractal exponent. Under the one-compartment simplification this strategy is straightforward. However, precisely due to the heterogeneity of the underlying biology, often at least a two-compartment model is required to address macroscopic data such as drug concentrations. This simple modelling step-up implies significant analytical and numerical complications. However, a few methods are available that make possible the original desideratum. In fact, exploring the full range of parametric possibilities and looking at different drugs and respective biological concentrations, it may be concluded that all PK modelling approaches are indeed particular cases of the fractal PK theory.

Skin cancer texture analysis of OCT images based on Haralick, fractal dimension and the complex directional field features

NASA Astrophysics Data System (ADS)

Raupov, Dmitry S.; Myakinin, Oleg O.; Bratchenko, Ivan A.; Kornilin, Dmitry V.; Zakharov, Valery P.; Khramov, Alexander G.

2016-04-01

Optical coherence tomography (OCT) is usually employed for the measurement of tumor topology, which reflects structural changes of a tissue. We investigated the possibility of OCT in detecting changes using a computer texture analysis method based on Haralick texture features, fractal dimension and the complex directional field method from different tissues. These features were used to identify special spatial characteristics, which differ healthy tissue from various skin cancers in cross-section OCT images (B-scans). Speckle reduction is an important pre-processing stage for OCT image processing. In this paper, an interval type-II fuzzy anisotropic diffusion algorithm for speckle noise reduction in OCT images was used. The Haralick texture feature set includes contrast, correlation, energy, and homogeneity evaluated in different directions. A box-counting method is applied to compute fractal dimension of investigated tissues. Additionally, we used the complex directional field calculated by the local gradient methodology to increase of the assessment quality of the diagnosis method. The complex directional field (as well as the "classical" directional field) can help describe an image as set of directions. Considering to a fact that malignant tissue grows anisotropically, some principal grooves may be observed on dermoscopic images, which mean possible existence of principal directions on OCT images. Our results suggest that described texture features may provide useful information to differentiate pathological from healthy patients. The problem of recognition melanoma from nevi is decided in this work due to the big quantity of experimental data (143 OCT-images include tumors as Basal Cell Carcinoma (BCC), Malignant Melanoma (MM) and Nevi). We have sensitivity about 90% and specificity about 85%. Further research is warranted to determine how this approach may be used to select the regions of interest automatically.

a New Improved Threshold Segmentation Method for Scanning Images of Reservoir Rocks Considering Pore Fractal Characteristics

NASA Astrophysics Data System (ADS)

Lin, Wei; Li, Xizhe; Yang, Zhengming; Lin, Lijun; Xiong, Shengchun; Wang, Zhiyuan; Wang, Xiangyang; Xiao, Qianhua

Based on the basic principle of the porosity method in image segmentation, considering the relationship between the porosity of the rocks and the fractal characteristics of the pore structures, a new improved image segmentation method was proposed, which uses the calculated porosity of the core images as a constraint to obtain the best threshold. The results of comparative analysis show that the porosity method can best segment images theoretically, but the actual segmentation effect is deviated from the real situation. Due to the existence of heterogeneity and isolated pores of cores, the porosity method that takes the experimental porosity of the whole core as the criterion cannot achieve the desired segmentation effect. On the contrary, the new improved method overcomes the shortcomings of the porosity method, and makes a more reasonable binary segmentation for the core grayscale images, which segments images based on the actual porosity of each image by calculated. Moreover, the image segmentation method based on the calculated porosity rather than the measured porosity also greatly saves manpower and material resources, especially for tight rocks.

Reply to "Comment on 'Hydrodynamics of fractal continuum flow' and 'Map of fluid flow in fractal porous medium into fractal continuum flow'".

PubMed

Balankin, Alexander S; Elizarraraz, Benjamin Espinoza

2013-11-01

The aim of this Reply is to elucidate the difference between the fractal continuum models used in the preceding Comment and the models of fractal continuum flow which were put forward in our previous articles [Phys. Rev. E 85, 025302(R) (2012); 85, 056314 (2012)]. In this way, some drawbacks of the former models are highlighted. Specifically, inconsistencies in the definitions of the fractal derivative, the Jacobian of transformation, the displacement vector, and angular momentum are revealed. The proper forms of the Reynolds' transport theorem and angular momentum principle for the fractal continuum are reaffirmed in a more illustrative manner. Consequently, we emphasize that in the absence of any internal angular momentum, body couples, and couple stresses, the Cauchy stress tensor in the fractal continuum should be symmetric. Furthermore, we stress that the approach based on the Cartesian product measured and used in the preceding Comment cannot be employed to study the path-connected fractals, such as a flow in a fractally permeable medium. Thus, all statements of our previous works remain unchallenged.

Scattering from fractals

NASA Astrophysics Data System (ADS)

Hurd, Alan J.

The realization that structures in Nature often can be described by Mandelbrot's fractals has led to a revolution in many areas of physics. The interaction of waves with fractal systems has, understandably, become intensely studied since scattering is the method of choice to probe delicate fractal structures such as chainlike particle aggregates. Not all of these waves are electromagnetic. Neutron scattering, for example, is an important complementary tool to structural studies by X-ray and light scattering. Since the phenomenology of small-angle neutron scattering (SANS), as it is applied to fractal systems, is identical to that of small-angle X-ray scattering (SAXS), it falls within the scope of this paper.

Correlating brain blood oxygenation level dependent (BOLD) fractal dimension mapping with magnetic resonance spectroscopy (MRS) in Alzheimer's disease.

PubMed

Warsi, Mohammed A; Molloy, William; Noseworthy, Michael D

2012-10-01

To correlate temporal fractal structure of resting state blood oxygen level dependent (rsBOLD) functional magnetic resonance imaging (fMRI) with in vivo proton magnetic resonance spectroscopy ((1)H-MRS), in Alzheimer's disease (AD) and healthy age-matched normal controls (NC). High temporal resolution (4 Hz) rsBOLD signal and single voxel (left putamen) magnetic resonance spectroscopy data was acquired in 33 AD patients and 13 NC. The rsBOLD data was analyzed using two types of fractal dimension (FD) analysis based on relative dispersion and frequency power spectrum. Comparisons in FD were performed between AD and NC, and FD measures were correlated with (1)H-MRS findings. Temporal fractal analysis of rsBOLD, was able to differentiate AD from NC subjects (P = 0.03). Low FD correlated with markers of AD severity including decreased concentrations of N-acetyl aspartate (R = 0.44, P = 0.015) and increased myoinositol (mI) (R = -0.45, P = 0.012). Based on these results we suggest fractal analysis of rsBOLD could provide an early marker of AD.

Temporal fractal analysis of the rs-BOLD signal identifies brain abnormalities in autism spectrum disorder.

PubMed

Dona, Olga; Hall, Geoffrey B; Noseworthy, Michael D

2017-01-01

Brain connectivity in autism spectrum disorders (ASD) has proven difficult to characterize due to the heterogeneous nature of the spectrum. Connectivity in the brain occurs in a complex, multilevel and multi-temporal manner, driving the fluctuations observed in local oxygen demand. These fluctuations can be characterized as fractals, as they auto-correlate at different time scales. In this study, we propose a model-free complexity analysis based on the fractal dimension of the rs-BOLD signal, acquired with magnetic resonance imaging. The fractal dimension can be interpreted as measure of signal complexity and connectivity. Previous studies have suggested that reduction in signal complexity can be associated with disease. Therefore, we hypothesized that a detectable difference in rs-BOLD signal complexity could be observed between ASD patients and Controls. Anatomical and functional data from fifty-five subjects with ASD (12.7 ± 2.4 y/o) and 55 age-matched (14.1 ± 3.1 y/o) healthy controls were accessed through the NITRC database and the ABIDE project. Subjects were scanned using a 3T GE Signa MRI and a 32-channel RF-coil. Axial FSPGR-3D images were used to prescribe rs-BOLD (TE/TR = 30/2000ms) where 300 time points were acquired. Motion correction was performed on the functional data and anatomical and functional images were aligned and spatially warped to the N27 standard brain atlas. Fractal analysis, performed on a grey matter mask, was done by estimating the Hurst exponent in the frequency domain using a power spectral density approach and refining the estimation in the time domain with de-trended fluctuation analysis and signal summation conversion methods. Voxel-wise fractal dimension (FD) was calculated for every subject in the control group and in the ASD group to create ROI-based Z-scores for the ASD patients. Voxel-wise validation of FD normality across controls was confirmed, and non-Gaussian voxels were eliminated from subsequent analysis. To maintain

Fractal Electronic Circuits Assembled From Nanoclusters

NASA Astrophysics Data System (ADS)

Fairbanks, M. S.; McCarthy, D.; Taylor, R. P.; Brown, S. A.

2009-07-01

Many patterns in nature can be described using fractal geometry. The effect of this fractal character is an array of properties that can include high internal connectivity, high dispersivity, and enhanced surface area to volume ratios. These properties are often desirable in applications and, consequently, fractal geometry is increasingly employed in technologies ranging from antenna to storm barriers. In this paper, we explore the application of fractal geometry to electrical circuits, inspired by the pervasive fractal structure of neurons in the brain. We show that, under appropriate growth conditions, nanoclusters of Sb form into islands on atomically flat substrates via a process close to diffusion-limited aggregation (DLA), establishing fractal islands that will form the basis of our fractal circuits. We perform fractal analysis of the islands to determine the spatial scaling properties (characterized by the fractal dimension, D) of the proposed circuits and demonstrate how varying growth conditions can affect D. We discuss fabrication approaches for establishing electrical contact to the fractal islands. Finally, we present fractal circuit simulations, which show that the fractal character of the circuit translates into novel, non-linear conduction properties determined by the circuit's D value.

Flat bands in fractal-like geometry

NASA Astrophysics Data System (ADS)

Pal, Biplab; Saha, Kush

2018-05-01

We report the presence of multiple flat bands in a class of two-dimensional lattices formed by Sierpinski gasket (SPG) fractal geometries as the basic unit cells. Solving the tight-binding Hamiltonian for such lattices with different generations of a SPG network, we find multiple degenerate and nondegenerate completely flat bands, depending on the configuration of parameters of the Hamiltonian. Moreover, we establish a generic formula to determine the number of such bands as a function of the generation index ℓ of the fractal geometry. We show that the flat bands and their neighboring dispersive bands have remarkable features, the most interesting one being the spin-1 conical-type spectrum at the band center without any staggered magnetic flux, in contrast to the kagome lattice. We furthermore investigate the effect of magnetic flux in these lattice settings and show that different combinations of fluxes through such fractal unit cells lead to a richer spectrum with a single isolated flat band or gapless electron- or holelike flat bands. Finally, we discuss a possible experimental setup to engineer such a fractal flat-band network using single-mode laser-induced photonic waveguides.

Surface Fractal Analysis for Estimating the Fracture Energy Absorption of Nanoparticle Reinforced Composites

PubMed Central

Pramanik, Brahmananda; Tadepalli, Tezeswi; Mantena, P. Raju

2012-01-01

In this study, the fractal dimensions of failure surfaces of vinyl ester based nanocomposites are estimated using two classical methods, Vertical Section Method (VSM) and Slit Island Method (SIM), based on the processing of 3D digital microscopic images. Self-affine fractal geometry has been observed in the experimentally obtained failure surfaces of graphite platelet reinforced nanocomposites subjected to quasi-static uniaxial tensile and low velocity punch-shear loading. Fracture energy and fracture toughness are estimated analytically from the surface fractal dimensionality. Sensitivity studies show an exponential dependency of fracture energy and fracture toughness on the fractal dimensionality. Contribution of fracture energy to the total energy absorption of these nanoparticle reinforced composites is demonstrated. For the graphite platelet reinforced nanocomposites investigated, surface fractal analysis has depicted the probable ductile or brittle fracture propagation mechanism, depending upon the rate of loading. PMID:28817017

Fractals and foods.

PubMed

Peleg, M

1993-01-01

Fractal geometry and related concepts have had only a very minor impact on food research. The very few reported food applications deal mainly with the characterization of the contours of agglomerated instant coffee particles, the surface morphology of treated starch particles, the microstructure of casein gels viewed as a product limited diffusion aggregation, and the jagged mechanical signatures of crunchy dry foods. Fractal geometry describes objects having morphological features that are scale invariant. A demonstration of the self-similarity of fractal objects can be found in the familiar morphology of cauliflower and broccoli, both foods. Processes regulated by nonlinear dynamics can exhibit a chaotic behavior that has fractal characteristics. Examples are mixing of viscous fluids, turbulence, crystallization, agglomeration, diffusion, and possibly food spoilage.

Effect of angle of deposition on the Fractal properties of ZnO thin film surface

NASA Astrophysics Data System (ADS)

Yadav, R. P.; Agarwal, D. C.; Kumar, Manvendra; Rajput, Parasmani; Tomar, D. S.; Pandey, S. N.; Priya, P. K.; Mittal, A. K.

2017-09-01

Zinc oxide (ZnO) thin films were prepared by atom beam sputtering at various deposition angles in the range of 20-75°. The deposited thin films were examined by glancing angle X-ray diffraction and atomic force microscopy (AFM). Scaling law analysis was performed on AFM images to show that the thin film surfaces are self-affine. Fractal dimension of each of the 256 vertical sections along the fast scan direction of a discretized surface, obtained from the AFM height data, was estimated using the Higuchi's algorithm. Hurst exponent was computed from the fractal dimension. The grain sizes, as determined by applying self-correlation function on AFM micrographs, varied with the deposition angle in the same manner as the Hurst exponent.

[Lithology feature extraction of CASI hyperspectral data based on fractal signal algorithm].

PubMed

Tang, Chao; Chen, Jian-Ping; Cui, Jing; Wen, Bo-Tao

2014-05-01

Hyperspectral data is characterized by combination of image and spectrum and large data volume dimension reduction is the main research direction. Band selection and feature extraction is the primary method used for this objective. In the present article, the authors tested methods applied for the lithology feature extraction from hyperspectral data. Based on the self-similarity of hyperspectral data, the authors explored the application of fractal algorithm to lithology feature extraction from CASI hyperspectral data. The "carpet method" was corrected and then applied to calculate the fractal value of every pixel in the hyperspectral data. The results show that fractal information highlights the exposed bedrock lithology better than the original hyperspectral data The fractal signal and characterized scale are influenced by the spectral curve shape, the initial scale selection and iteration step. At present, research on the fractal signal of spectral curve is rare, implying the necessity of further quantitative analysis and investigation of its physical implications.

Fractals in geology and geophysics

NASA Technical Reports Server (NTRS)

Turcotte, Donald L.

1989-01-01

The definition of a fractal distribution is that the number of objects N with a characteristic size greater than r scales with the relation N of about r exp -D. The frequency-size distributions for islands, earthquakes, fragments, ore deposits, and oil fields often satisfy this relation. This application illustrates a fundamental aspect of fractal distributions, scale invariance. The requirement of an object to define a scale in photograhs of many geological features is one indication of the wide applicability of scale invariance to geological problems; scale invariance can lead to fractal clustering. Geophysical spectra can also be related to fractals; these are self-affine fractals rather than self-similar fractals. Examples include the earth's topography and geoid.

Fractals in the Classroom

ERIC Educational Resources Information Center

Fraboni, Michael; Moller, Trisha

2008-01-01

Fractal geometry offers teachers great flexibility: It can be adapted to the level of the audience or to time constraints. Although easily explained, fractal geometry leads to rich and interesting mathematical complexities. In this article, the authors describe fractal geometry, explain the process of iteration, and provide a sample exercise.…

Efficient fractal-based mutation in evolutionary algorithms from iterated function systems

NASA Astrophysics Data System (ADS)

Salcedo-Sanz, S.; Aybar-Ruíz, A.; Camacho-Gómez, C.; Pereira, E.

2018-03-01

In this paper we present a new mutation procedure for Evolutionary Programming (EP) approaches, based on Iterated Function Systems (IFSs). The new mutation procedure proposed consists of considering a set of IFS which are able to generate fractal structures in a two-dimensional phase space, and use them to modify a current individual of the EP algorithm, instead of using random numbers from different probability density functions. We test this new proposal in a set of benchmark functions for continuous optimization problems. In this case, we compare the proposed mutation against classical Evolutionary Programming approaches, with mutations based on Gaussian, Cauchy and chaotic maps. We also include a discussion on the IFS-based mutation in a real application of Tuned Mass Dumper (TMD) location and optimization for vibration cancellation in buildings. In both practical cases, the proposed EP with the IFS-based mutation obtained extremely competitive results compared to alternative classical mutation operators.

MORPH-I (Ver 1.0) a software package for the analysis of scanning electron micrograph (binary formatted) images for the assessment of the fractal dimension of enclosed pore surfaces

USGS Publications Warehouse

Mossotti, Victor G.; Eldeeb, A. Raouf; Oscarson, Robert

1998-01-01

MORPH-I is a set of C-language computer programs for the IBM PC and compatible minicomputers. The programs in MORPH-I are used for the fractal analysis of scanning electron microscope and electron microprobe images of pore profiles exposed in cross-section. The program isolates and traces the cross-sectional profiles of exposed pores and computes the Richardson fractal dimension for each pore. Other programs in the set provide for image calibration, display, and statistical analysis of the computed dimensions for highly complex porous materials. Requirements: IBM PC or compatible; minimum 640 K RAM; mathcoprocessor; SVGA graphics board providing mode 103 display.

Predicting beauty: fractal dimension and visual complexity in art.

PubMed

Forsythe, A; Nadal, M; Sheehy, N; Cela-Conde, C J; Sawey, M

2011-02-01

Visual complexity has been known to be a significant predictor of preference for artistic works for some time. The first study reported here examines the extent to which perceived visual complexity in art can be successfully predicted using automated measures of complexity. Contrary to previous findings the most successful predictor of visual complexity was Gif compression. The second study examined the extent to which fractal dimension could account for judgments of perceived beauty. The fractal dimension measure accounts for more of the variance in judgments of perceived beauty in visual art than measures of visual complexity alone, particularly for abstract and natural images. Results also suggest that when colour is removed from an artistic image observers are unable to make meaningful judgments as to its beauty. ©2010 The British Psychological Society.

Assessment of disintegrant efficacy with fractal dimensions from real-time MRI.

PubMed

Quodbach, Julian; Moussavi, Amir; Tammer, Roland; Frahm, Jens; Kleinebudde, Peter

2014-11-20

An efficient disintegrant is capable of breaking up a tablet in the smallest possible particles in the shortest time. Until now, comparative data on the efficacy of different disintegrants is based on dissolution studies or the disintegration time. Extending these approaches, this study introduces a method, which defines the evolution of fractal dimensions of tablets as surrogate parameter for the available surface area. Fractal dimensions are a measure for the tortuosity of a line, in this case the upper surface of a disintegrating tablet. High-resolution real-time MRI was used to record videos of disintegrating tablets. The acquired video images were processed to depict the upper surface of the tablets and a box-counting algorithm was used to estimate the fractal dimensions. The influence of six different disintegrants, of different relative tablet density, and increasing disintegrant concentration was investigated to evaluate the performance of the novel method. Changing relative densities hardly affect the progression of fractal dimensions, whereas an increase in disintegrant concentration causes increasing fractal dimensions during disintegration, which are also reached quicker. Different disintegrants display only minor differences in the maximal fractal dimension, yet the kinetic in which the maximum is reached allows a differentiation and classification of disintegrants. Copyright © 2014 Elsevier B.V. All rights reserved.

Fundamental Fractal Antenna Design Process

NASA Astrophysics Data System (ADS)

Zhu, L. P.; Kim, T. C.; Kakas, G. D.

2017-12-01

Antenna designers are always looking to come up with new ideas to push the envelope for new antennas, using a smaller volume while striving for higher bandwidth, wider bandwidth, and antenna gain. One proposed method of increasing bandwidth or shrinking antenna size is via the use of fractal geometry, which gives rise to fractal antennas. Fractals are those fun shapes that if one zooms in or zoom out, the structure is always the same. Design a new type of antenna based on fractal antenna design by utilize the Design of Experiment (DOE) will be shown in fractal antenna design process. Investigate conformal fractal antenna design for patterns, dimensions, and size, of the antenna but maintaining or improving the antenna performance. Research shows an antenna designer how to create basic requirements of the fractal antenna through a step by step process, and provides how to optimize the antenna design with the model prediction, lab measurement, and actual results from the compact range measurement on the antenna patterns.

Fractality and growth of He bubbles in metals

NASA Astrophysics Data System (ADS)

Kajita, Shin; Ito, Atsushi M.; Ohno, Noriyasu

2017-08-01

Pinholes are formed on surfaces of metals by the exposure to helium plasmas, and they are regarded as the initial process of the growth of fuzzy nanostructures. In this study, number density of the pinholes is investigated in detail from the scanning electron microscope (SEM) micrographs of tungsten and tantalum exposed to the helium plasmas. A power law relation was identified between the number density and the size of pinholes. From the slope and the region where the power law was satisfied, the fractal dimension D and smin, which characterize the SEM images, are deduced. Parametric dependences and material dependence of D and smin are revealed. To explain the fractality, simple Monte-Carlo simulations including random walks of He atoms and absorption on bubble was introduced. It is shown that the initial position of the random walk is one of the key factors to deduce the fractality. The results indicated that new nucleations of bubbles are necessary to reproduce the number-density distribution of bubbles.

a Fractal Permeability Model Coupling Boundary-Layer Effect for Tight Oil Reservoirs

NASA Astrophysics Data System (ADS)

Wang, Fuyong; Liu, Zhichao; Jiao, Liang; Wang, Congle; Guo, Hu

A fractal permeability model coupling non-flowing boundary-layer effect for tight oil reservoirs was proposed. Firstly, pore structures of tight formations were characterized with fractal theory. Then, with the empirical equation of boundary-layer thickness, Hagen-Poiseuille equation and fractal theory, a fractal torturous capillary tube model coupled with boundary-layer effect was developed, and verified with experimental data. Finally, the parameters influencing effective liquid permeability were quantitatively investigated. The research results show that effective liquid permeability of tight formations is not only decided by pore structures, but also affected by boundary-layer distributions, and effective liquid permeability is the function of fluid type, fluid viscosity, pressure gradient, fractal dimension, tortuosity fractal dimension, minimum pore radius and maximum pore radius. For the tight formations dominated with nanoscale pores, boundary-layer effect can significantly reduce effective liquid permeability, especially under low pressure gradient.

Fractals: To Know, to Do, to Simulate.

ERIC Educational Resources Information Center

Talanquer, Vicente; Irazoque, Glinda

1993-01-01

Discusses the development of fractal theory and suggests fractal aggregates as an attractive alternative for introducing fractal concepts. Describes methods for producing metallic fractals and a computer simulation for drawing fractals. (MVL)

Using Fractal And Morphological Criteria For Automatic Classification Of Lung Diseases

NASA Astrophysics Data System (ADS)

Vehel, Jacques Levy

1989-11-01

Medical Images are difficult to analyze by means of classical image processing tools because they are very complex and irregular. Such shapes are obtained for instance in Nuclear Medecine with the spatial distribution of activity for organs such as lungs, liver, and heart. We have tried to apply two different theories to these signals: - Fractal Geometry deals with the analysis of complex irregular shapes which cannot well be described by the classical Euclidean geometry. - Integral Geometry treats sets globally and allows to introduce robust measures. We have computed three parameters on three kinds of Lung's SPECT images: normal, pulmonary embolism and chronic desease: - The commonly used fractal dimension (FD), that gives a measurement of the irregularity of the 3D shape. - The generalized lacunarity dimension (GLD), defined as the variance of the ratio of the local activity by the mean activity, which is only sensitive to the distribution and the size of gaps in the surface. - The Favard length that gives an approximation of the surface of a 3-D shape. The results show that each slice of the lung, considered as a 3D surface, is fractal and that the fractal dimension is the same for each slice and for the three kind of lungs; as for the lacunarity and Favard length, they are clearly different for normal lungs, pulmonary embolisms and chronic diseases. These results indicate that automatic classification of Lung's SPECT can be achieved, and that a quantitative measurement of the evolution of the disease could be made.

On the fractal morphology of combustion-generated soot aggregates

DOE Office of Scientific and Technical Information (OSTI.GOV)

Koylu, U.O.

1995-12-31

The fractal properties of soot aggregates were investigated using ex-situ and in-situ experimental methods as well as computer simulations. Ex-situ experiments involved thermophoretic sampling and analysis by transmission electron microscopy (TEM), while in-situ measurements employed angular static light scattering and data inversion based on Rayleigh-Debye-Gans (RDG) approximation. Computer simulations used a sequential algorithm which mimics mass fractal-like structures. So from a variety of hydrocarbon-fueled laminar and turbulent nonpremixed flame environments were considered in the present study. The TEM analysis of projected soot images sampled from fuel-rich conditions of buoyant and weakly-buoyant laminar flames indicated that the fractal dimension of sootmore » was relatively independent of position in flames, fuel type and flame condition. These measurements yielded an average fractal dimension of 1.8, although other structure parameters such as the primary particle diameters and number of primary particles in aggregates had wide range of values. Fractal prefactor (lacunarity) was also measured for soot sampled from the fuel-lean conditions of turbulent flames, considering the actual morphology by tilting the samples during TEM analysis. These measurements yielded a fractal dimension of 1.65 and a lacunarity of 8.5, with experimental uncertainties (95% confidence) of 0.08 and 0.5, respectively. Relationships between the actual and projected structure properties of soot were also developed by combining TEM observations with numerical simulations. Practical approximate formulae were suggested to find radius of gyration of an aggregate from its maximum dimension, and number of primary particles in an aggregate from projected area. Finally, the fractal dimension and lacunarity of soot were obtained using light scattering for the same conditions of the above TEM measurements.« less

Electromagnetism on anisotropic fractal media

NASA Astrophysics Data System (ADS)

Ostoja-Starzewski, Martin

2013-04-01

Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell's electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.

Fractals, malware, and data models

NASA Astrophysics Data System (ADS)

Jaenisch, Holger M.; Potter, Andrew N.; Williams, Deborah; Handley, James W.

2012-06-01

We examine the hypothesis that the decision boundary between malware and non-malware is fractal. We introduce a novel encoding method derived from text mining for converting disassembled programs first into opstrings and then filter these into a reduced opcode alphabet. These opcodes are enumerated and encoded into real floating point number format and used for characterizing frequency of occurrence and distribution properties of malware functions to compare with non-malware functions. We use the concept of invariant moments to characterize the highly non-Gaussian structure of the opcode distributions. We then derive Data Model based classifiers from identified features and interpolate and extrapolate the parameter sample space for the derived Data Models. This is done to examine the nature of the parameter space classification boundary between families of malware and the general non-malware category. Preliminary results strongly support the fractal boundary hypothesis, and a summary of our methods and results are presented here.

Alpha-spectrometry and fractal analysis of surface micro-images for characterisation of porous materials used in manufacture of targets for laser plasma experiments

DOE Office of Scientific and Technical Information (OSTI.GOV)

Aushev, A A; Barinov, S P; Vasin, M G

2015-06-30

We present the results of employing the alpha-spectrometry method to determine the characteristics of porous materials used in targets for laser plasma experiments. It is shown that the energy spectrum of alpha-particles, after their passage through porous samples, allows one to determine the distribution of their path length in the foam skeleton. We describe the procedure of deriving such a distribution, excluding both the distribution broadening due to statistical nature of the alpha-particle interaction with an atomic structure (straggling) and hardware effects. The fractal analysis of micro-images is applied to the same porous surface samples that have been studied bymore » alpha-spectrometry. The fractal dimension and size distribution of the number of the foam skeleton grains are obtained. Using the data obtained, a distribution of the total foam skeleton thickness along a chosen direction is constructed. It roughly coincides with the path length distribution of alpha-particles within a range of larger path lengths. It is concluded that the combined use of the alpha-spectrometry method and fractal analysis of images will make it possible to determine the size distribution of foam skeleton grains (or pores). The results can be used as initial data in theoretical studies on propagation of the laser and X-ray radiation in specific porous samples. (laser plasma)« less

Is the co-seismic slip distribution fractal?

NASA Astrophysics Data System (ADS)

Milliner, Christopher; Sammis, Charles; Allam, Amir; Dolan, James

2015-04-01

Co-seismic along-strike slip heterogeneity is widely observed for many surface-rupturing earthquakes as revealed by field and high-resolution geodetic methods. However, this co-seismic slip variability is currently a poorly understood phenomenon. Key unanswered questions include: What are the characteristics and underlying causes of along-strike slip variability? Do the properties of slip variability change from fault-to-fault, along-strike or at different scales? We cross-correlate optical, pre- and post-event air photos using the program COSI-Corr to measure the near-field, surface deformation pattern of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes in high-resolution. We produce the co-seismic slip profiles of both events from over 1,000 displacement measurements and observe consistent along-strike slip variability. Although the observed slip heterogeneity seems apparently complex and disordered, a spectral analysis reveals that the slip distributions are indeed self-affine fractal i.e., slip exhibits a consistent degree of irregularity at all observable length scales, with a 'short-memory' and is not random. We find a fractal dimension of 1.58 and 1.75 for the Landers and Hector Mine earthquakes, respectively, indicating that slip is more heterogeneous for the Hector Mine event. Fractal slip is consistent with both dynamic and quasi-static numerical simulations that use non-planar faults, which in turn causes heterogeneous along-strike stress, and we attribute the observed fractal slip to fault surfaces of fractal roughness. As fault surfaces are known to smooth over geologic time due to abrasional wear and fracturing, we also test whether the fractal properties of slip distributions alters between earthquakes from immature to mature fault systems. We will present results that test this hypothesis by using the optical image correlation technique to measure historic, co-seismic slip distributions of earthquakes from structurally mature, large

Analysis of Fractional Flow for Transient Two-Phase Flow in Fractal Porous Medium

NASA Astrophysics Data System (ADS)

Lu, Ting; Duan, Yonggang; Fang, Quantang; Dai, Xiaolu; Wu, Jinsui

2016-03-01

Prediction of fractional flow in fractal porous medium is important for reservoir engineering and chemical engineering as well as hydrology. A physical conceptual fractional flow model of transient two-phase flow is developed in fractal porous medium based on the fractal characteristics of pore-size distribution and on the approximation that porous medium consist of a bundle of tortuous capillaries. The analytical expression for fractional flow for wetting phase is presented, and the proposed expression is the function of structural parameters (such as tortuosity fractal dimension, pore fractal dimension, maximum and minimum diameters of capillaries) and fluid properties (such as contact angle, viscosity and interfacial tension) in fractal porous medium. The sensitive parameters that influence fractional flow and its derivative are formulated, and their impacts on fractional flow are discussed.

Static friction between rigid fractal surfaces

NASA Astrophysics Data System (ADS)

Alonso-Marroquin, Fernando; Huang, Pengyu; Hanaor, Dorian A. H.; Flores-Johnson, E. A.; Proust, Gwénaëlle; Gan, Yixiang; Shen, Luming

2015-09-01

Using spheropolygon-based simulations and contact slope analysis, we investigate the effects of surface topography and atomic scale friction on the macroscopically observed friction between rigid blocks with fractal surface structures. From our mathematical derivation, the angle of macroscopic friction is the result of the sum of the angle of atomic friction and the slope angle between the contact surfaces. The latter is obtained from the determination of all possible contact slopes between the two surface profiles through an alternative signature function. Our theory is validated through numerical simulations of spheropolygons with fractal Koch surfaces and is applied to the description of frictional properties of Weierstrass-Mandelbrot surfaces. The agreement between simulations and theory suggests that for interpreting macroscopic frictional behavior, the descriptors of surface morphology should be defined from the signature function rather than from the slopes of the contacting surfaces.

Fractal dimension of trabecular bone projection texture is related to three-dimensional microarchitecture.

PubMed

Pothuaud, L; Benhamou, C L; Porion, P; Lespessailles, E; Harba, R; Levitz, P

2000-04-01

The purpose of this work was to understand how fractal dimension of two-dimensional (2D) trabecular bone projection images could be related to three-dimensional (3D) trabecular bone properties such as porosity or connectivity. Two alteration processes were applied to trabecular bone images obtained by magnetic resonance imaging: a trabeculae dilation process and a trabeculae removal process. The trabeculae dilation process was applied from the 3D skeleton graph to the 3D initial structure with constant connectivity. The trabeculae removal process was applied from the initial structure to an altered structure having 99% of porosity, in which both porosity and connectivity were modified during this second process. Gray-level projection images of each of the altered structures were simply obtained by summation of voxels, and fractal dimension (Df) was calculated. Porosity (phi) and connectivity per unit volume (Cv) were calculated from the 3D structure. Significant relationships were found between Df, phi, and Cv. Df values increased when porosity increased (dilation and removal processes) and when connectivity decreased (only removal process). These variations were in accordance with all previous clinical studies, suggesting that fractal evaluation of trabecular bone projection has real meaning in terms of porosity and connectivity of the 3D architecture. Furthermore, there was a statistically significant linear dependence between Df and Cv when phi remained constant. Porosity is directly related to bone mineral density and fractal dimension can be easily evaluated in clinical routine. These two parameters could be associated to evaluate the connectivity of the structure.

Fractals in physiology and medicine

NASA Technical Reports Server (NTRS)

Goldberger, Ary L.; West, Bruce J.

1987-01-01

The paper demonstrates how the nonlinear concepts of fractals, as applied in physiology and medicine, can provide an insight into the organization of such complex structures as the tracheobronchial tree and heart, as well as into the dynamics of healthy physiological variability. Particular attention is given to the characteristics of computer-generated fractal lungs and heart and to fractal pathologies in these organs. It is shown that alterations in fractal scaling may underlie a number of pathophysiological disturbances, including sudden cardiac death syndromes.

A New Fractal Model of Chromosome and DNA Processes

NASA Astrophysics Data System (ADS)

Bouallegue, K.

Dynamic chromosome structure remains unknown. Can fractals and chaos be used as new tools to model, identify and generate a structure of chromosomes?Fractals and chaos offer a rich environment for exploring and modeling the complexity of nature. In a sense, fractal geometry is used to describe, model, and analyze the complex forms found in nature. Fractals have also been widely not only in biology but also in medicine. To this effect, a fractal is considered an object that displays self-similarity under magnification and can be constructed using a simple motif (an image repeated on ever-reduced scales).It is worth noting that the problem of identifying a chromosome has become a challenge to find out which one of the models it belongs to. Nevertheless, the several different models (a hierarchical coiling, a folded fiber, and radial loop) have been proposed for mitotic chromosome but have not reached a dynamic model yet.This paper is an attempt to solve topological problems involved in the model of chromosome and DNA processes. By combining the fractal Julia process and the numerical dynamical system, we have finally found out four main points. First, we have developed not only a model of chromosome but also a model of mitosis and one of meiosis. Equally important, we have identified the centromere position through the numerical model captured below. More importantly, in this paper, we have discovered the processes of the cell divisions of both mitosis and meiosis. All in all, the results show that this work could have a strong impact on the welfare of humanity and can lead to a cure of genetic diseases.

Fractal and Multifractal Analysis of Human Gait

NASA Astrophysics Data System (ADS)

Muñoz-Diosdado, A.; del Río Correa, J. L.; Angulo-Brown, F.

2003-09-01

We carried out a fractal and multifractal analysis of human gait time series of young and old individuals, and adults with three illnesses that affect the march: The Parkinson's and Huntington's diseases and the amyotrophic lateral sclerosis (ALS). We obtained cumulative plots of events, the correlation function, the Hurst exponent and the Higuchi's fractal dimension of these time series and found that these fractal markers could be a factor to characterize the march, since we obtained different values of these quantities for youths and adults and they are different also for healthy and ill persons and the most anomalous values belong to ill persons. In other physiological signals there is complexity lost related with the age and the illness, in the case of the march the opposite occurs. The multifractal analysis could be also a useful tool to understand the dynamics of these and other complex systems.

Vector calculus in non-integer dimensional space and its applications to fractal media

NASA Astrophysics Data System (ADS)

Tarasov, Vasily E.

2015-02-01

We suggest a generalization of vector calculus for the case of non-integer dimensional space. The first and second orders operations such as gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional space are defined. For simplification we consider scalar and vector fields that are independent of angles. We formulate a generalization of vector calculus for rotationally covariant scalar and vector functions. This generalization allows us to describe fractal media and materials in the framework of continuum models with non-integer dimensional space. As examples of application of the suggested calculus, we consider elasticity of fractal materials (fractal hollow ball and fractal cylindrical pipe with pressure inside and outside), steady distribution of heat in fractal media, electric field of fractal charged cylinder. We solve the correspondent equations for non-integer dimensional space models.

Fractal correlation properties of R-R interval dynamics and mortality in patients with depressed left ventricular function after an acute myocardial infarction

NASA Technical Reports Server (NTRS)

Huikuri, H. V.; Makikallio, T. H.; Peng, C. K.; Goldberger, A. L.; Hintze, U.; Moller, M.

2000-01-01

BACKGROUND: Preliminary data suggest that the analysis of R-R interval variability by fractal analysis methods may provide clinically useful information on patients with heart failure. The purpose of this study was to compare the prognostic power of new fractal and traditional measures of R-R interval variability as predictors of death after acute myocardial infarction. METHODS AND RESULTS: Time and frequency domain heart rate (HR) variability measures, along with short- and long-term correlation (fractal) properties of R-R intervals (exponents alpha(1) and alpha(2)) and power-law scaling of the power spectra (exponent beta), were assessed from 24-hour Holter recordings in 446 survivors of acute myocardial infarction with a depressed left ventricular function (ejection fraction fractal measures of R-R interval variability were significant univariate predictors of all-cause mortality. Reduced short-term scaling exponent alpha(1) was the most powerful R-R interval variability measure as a predictor of all-cause mortality (alpha(1) <0.75, relative risk 3.0, 95% confidence interval 2.5 to 4.2, P<0.001). It remained an independent predictor of death (P<0.001) after adjustment for other postinfarction risk markers, such as age, ejection fraction, NYHA class, and medication. Reduced alpha(1) predicted both arrhythmic death (P<0.001) and nonarrhythmic cardiac death (P<0.001). CONCLUSIONS: Analysis of the fractal characteristics of short-term R-R interval dynamics yields more powerful prognostic information than the traditional measures of HR variability among patients with depressed left ventricular function after an acute myocardial infarction.

Fractal analysis of MRI data for the characterization of patients with schizophrenia and bipolar disorder.

PubMed

Squarcina, Letizia; De Luca, Alberto; Bellani, Marcella; Brambilla, Paolo; Turkheimer, Federico E; Bertoldo, Alessandra

2015-02-21

Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders.

Fractal analysis of MRI data for the characterization of patients with schizophrenia and bipolar disorder

NASA Astrophysics Data System (ADS)

Squarcina, Letizia; De Luca, Alberto; Bellani, Marcella; Brambilla, Paolo; Turkheimer, Federico E.; Bertoldo, Alessandra

2015-02-01

Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders.

Collisions of ideal gas molecules with a rough/fractal surface. A computational study.

PubMed

Panczyk, Tomasz

2007-02-01

The frequency of collisions of ideal gas molecules (argon) with a rough surface has been studied. The rough/fractal surface was created using random deposition technique. By applying various depositions, the roughness of the surface was controlled and, as a measure of the irregularity, the fractal dimensions of the surfaces were determined. The surfaces were next immersed in argon (under pressures 2 x 10(3) to 2 x 10(5) Pa) and the numbers of collisions with these surfaces were counted. The calculations were carried out using a simplified molecular dynamics simulation technique (only hard core repulsions were assumed). As a result, it was stated that the frequency of collisions is a linear function of pressure for all fractal dimensions studied (D = 2, ..., 2.5). The frequency per unit pressure is quite complex function of the fractal dimension; however, the changes of that frequency with the fractal dimension are not strong. It was found that the frequency of collisions is controlled by the number of weakly folded sites on the surfaces and there is some mapping between the shape of adsorption energy distribution functions and this number of weakly folded sites. The results for the rough/fractal surfaces were compared with the prediction given by the Langmuir-Hertz equation (valid for smooth surface), generally the departure from the Langmuir-Hertz equation is not higher than 48% for the studied systems (i.e. for the surfaces created using the random deposition technique).

Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric

NASA Astrophysics Data System (ADS)

Balankin, Alexander S.; Bory-Reyes, Juan; Shapiro, Michael

2016-02-01

One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil-Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals.

SU-D-BRA-04: Fractal Dimension Analysis of Edge-Detected Rectal Cancer CTs for Outcome Prediction

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhong, H; Wang, J; Hu, W

2015-06-15

Purpose: To extract the fractal dimension features from edge-detected rectal cancer CTs, and to examine the predictability of fractal dimensions to outcomes of primary rectal cancer patients. Methods: Ninety-seven rectal cancer patients treated with neo-adjuvant chemoradiation were enrolled in this study. CT images were obtained before chemoradiotherapy. The primary lesions of the rectal cancer were delineated by experienced radiation oncologists. These images were extracted and filtered by six different Laplacian of Gaussian (LoG) filters with different filter values (0.5–3.0: from fine to coarse) to achieve primary lesions in different anatomical scales. Edges of the original images were found at zero-crossingsmore » of the filtered images. Three different fractal dimensions (box-counting dimension, Minkowski dimension, mass dimension) were calculated upon the image slice with the largest cross-section of the primary lesion. The significance of these fractal dimensions in survival, recurrence and metastasis were examined by Student’s t-test. Results: For a follow-up time of two years, 18 of 97 patients had experienced recurrence, 24 had metastasis, and 18 were dead. Minkowski dimensions under large filter values (2.0, 2.5, 3.0) were significantly larger (p=0.014, 0.006, 0.015) in patients with recurrence than those without. For metastasis, only box-counting dimensions under a single filter value (2.5) showed differences (p=0.016) between patients with and without. For overall survival, box-counting dimensions (filter values = 0.5, 1.0, 1.5), Minkowski dimensions (filter values = 0.5, 1.5, 2.0, 2,5) and mass dimensions (filter values = 1.5, 2.0) were all significant (p<0.05). Conclusion: It is feasible to extract shape information by edge detection and fractal dimensions analysis in neo-adjuvant rectal cancer patients. This information can be used to prognosis prediction.« less

Order-fractal transitions in abstract paintings

DOE Office of Scientific and Technical Information (OSTI.GOV)

Calleja, E.M. de la, E-mail: elsama79@gmail.com; Cervantes, F.; Calleja, J. de la

2016-08-15

In this study, we determined the degree of order for 22 Jackson Pollock paintings using the Hausdorff–Besicovitch fractal dimension. Based on the maximum value of each multi-fractal spectrum, the artworks were classified according to the year in which they were painted. It has been reported that Pollock’s paintings are fractal and that this feature was more evident in his later works. However, our results show that the fractal dimension of these paintings ranges among values close to two. We characterize this behavior as a fractal-order transition. Based on the study of disorder-order transition in physical systems, we interpreted the fractal-ordermore » transition via the dark paint strokes in Pollock’s paintings as structured lines that follow a power law measured by the fractal dimension. We determined self-similarity in specific paintings, thereby demonstrating an important dependence on the scale of observations. We also characterized the fractal spectrum for the painting entitled Teri’s Find. We obtained similar spectra for Teri’s Find and Number 5, thereby suggesting that the fractal dimension cannot be rejected completely as a quantitative parameter for authenticating these artworks. -- Highlights: •We determined the degree of order in Jackson Pollock paintings using the Hausdorff–Besicovitch dimension. •We detected a fractal-order transition from Pollock’s paintings between 1947 and 1951. •We suggest that Jackson Pollock could have painted Teri’s Find.« less

A novel fractal image compression scheme with block classification and sorting based on Pearson's correlation coefficient.

PubMed

Wang, Jianji; Zheng, Nanning

2013-09-01

Fractal image compression (FIC) is an image coding technology based on the local similarity of image structure. It is widely used in many fields such as image retrieval, image denoising, image authentication, and encryption. FIC, however, suffers from the high computational complexity in encoding. Although many schemes are published to speed up encoding, they do not easily satisfy the encoding time or the reconstructed image quality requirements. In this paper, a new FIC scheme is proposed based on the fact that the affine similarity between two blocks in FIC is equivalent to the absolute value of Pearson's correlation coefficient (APCC) between them. First, all blocks in the range and domain pools are chosen and classified using an APCC-based block classification method to increase the matching probability. Second, by sorting the domain blocks with respect to APCCs between these domain blocks and a preset block in each class, the matching domain block for a range block can be searched in the selected domain set in which these APCCs are closer to APCC between the range block and the preset block. Experimental results show that the proposed scheme can significantly speed up the encoding process in FIC while preserving the reconstructed image quality well.

Entrainment to a real time fractal visual stimulus modulates fractal gait dynamics.

PubMed

Rhea, Christopher K; Kiefer, Adam W; D'Andrea, Susan E; Warren, William H; Aaron, Roy K

2014-08-01

Fractal patterns characterize healthy biological systems and are considered to reflect the ability of the system to adapt to varying environmental conditions. Previous research has shown that fractal patterns in gait are altered following natural aging or disease, and this has potential negative consequences for gait adaptability that can lead to increased risk of injury. However, the flexibility of a healthy neurological system to exhibit different fractal patterns in gait has yet to be explored, and this is a necessary step toward understanding human locomotor control. Fifteen participants walked for 15min on a treadmill, either in the absence of a visual stimulus or while they attempted to couple the timing of their gait with a visual metronome that exhibited a persistent fractal pattern (contained long-range correlations) or a random pattern (contained no long-range correlations). The stride-to-stride intervals of the participants were recorded via analog foot pressure switches and submitted to detrended fluctuation analysis (DFA) to determine if the fractal patterns during the visual metronome conditions differed from the baseline (no metronome) condition. DFA α in the baseline condition was 0.77±0.09. The fractal patterns in the stride-to-stride intervals were significantly altered when walking to the fractal metronome (DFA α=0.87±0.06) and to the random metronome (DFA α=0.61±0.10) (both p<.05 when compared to the baseline condition), indicating that a global change in gait dynamics was observed. A variety of strategies were identified at the local level with a cross-correlation analysis, indicating that local behavior did not account for the consistent global changes. Collectively, the results show that a gait dynamics can be shifted in a prescribed manner using a visual stimulus and the shift appears to be a global phenomenon. Copyright © 2014 Elsevier B.V. All rights reserved.

Fitting Curves by Fractal Interpolation: AN Application to the Quantification of Cognitive Brain Processes

NASA Astrophysics Data System (ADS)

Navascues, M. A.; Sebastian, M. V.

Fractal interpolants of Barnsley are defined for any continuous function defined on a real compact interval. The uniform distance between the function and its approximant is bounded in terms of the vertical scale factors. As a general result, the density of the affine fractal interpolation functions of Barnsley in the space of continuous functions in a compact interval is proved. A method of data fitting by means of fractal interpolation functions is proposed. The procedure is applied to the quantification of cognitive brain processes. In particular, the increase in the complexity of the electroencephalographic signal produced by the execution of a test of visual attention is studied. The experiment was performed on two types of children: a healthy control group and a set of children diagnosed with an attention deficit disorder.

A fractal concentration area method for assigning a color palette for image representation

NASA Astrophysics Data System (ADS)

Cheng, Qiuming; Li, Qingmou

2002-05-01

Displaying the remotely sensed image with a proper color palette is the first task in any kind of image processing and pattern recognition in GIS and image processing environments. The purpose of displaying the image should be not only to provide a visual representation of the variance of the image, although this has been the primary objective of most conventional methods, but also the color palette should reflect real-world features on the ground which must be the primary objective of employing remotely sensed data. Although most conventional methods focus only on the first purpose of image representation, the concentration-area ( C- A plot) fractal method proposed in this paper aims to meet both purposes on the basis of pixel values and pixel value frequency distribution as well as spatial and geometrical properties of image patterns. The C- A method can be used to establish power-law relationships between the area A(⩾ s) with the pixel values greater than s and the pixel value s itself after plotting these values on log-log paper. A number of straight-line segments can be manually or automatically fitted to the points on the log-log paper, each representing a power-law relationship between the area A and the cutoff pixel value for s in a particular range. These straight-line segments can yield a group of cutoff values on the basis of which the image can be classified into discrete classes or zones. These zones usually correspond to the real-world features on the ground. A Windows program has been prepared in ActiveX format for implementing the C- A method and integrating it into other GIS and image processing systems. A case study of Landsat TM band 5 has been used to demonstrate the application of the method and the flexibility of the computer program.

Are fractal dimensions of the spatial distribution of mineral deposits meaningful?

USGS Publications Warehouse

Raines, G.L.

2008-01-01

It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand, a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on datasets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5-1000 km. In particular, one typically observes a local dimension at spatial scales less than 30-60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly underexplored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the

Effect of Fractal Dimension on the Strain Behavior of Particulate Media

NASA Astrophysics Data System (ADS)

Altun, Selim; Sezer, Alper; Goktepe, A. Burak

2016-12-01

In this study, the influence of several fractal identifiers of granular materials on dynamic behavior of a flexible pavement structure as a particulate stratum is considered. Using experimental results and numerical methods as well, 15 different grain-shaped sands obtained from 5 different sources were analyzed as pavement base course materials. Image analyses were carried out by use of a stereomicroscope on 15 different samples to obtain quantitative particle shape information. Furthermore, triaxial compression tests were conducted to determine stress-strain and shear strength parameters of sands. Additionally, the dynamic response of the particulate media to standard traffic loads was computed using finite element modeling (FEM) technique. Using area-perimeter, line divider and box counting methods, over a hundred grains for each sand type were subjected to fractal analysis. Relationships among fractal dimension descriptors and dynamic strain levels were established for assessment of importance of shape descriptors of sands at various scales on the dynamic behavior. In this context, the advantage of fractal geometry concept to describe irregular and fractured shapes was used to characterize the sands used as base course materials. Results indicated that fractal identifiers can be preferred to analyze the effect of shape properties of sands on dynamic behavior of pavement base layers.

On the Gompertzian growth in the fractal space-time.

PubMed

Molski, Marcin; Konarski, Jerzy

2008-06-01

An analytical approach to determination of time-dependent temporal fractal dimension b(t)(t) and scaling factor a(t)(t) for the Gompertzian growth in the fractal space-time is presented. The derived formulae take into account the proper boundary conditions and permit a calculation of the mean values b(t)(t) and a(t)(t) at any period of time. The formulae derived have been tested on experimental data obtained by Schrek for the Brown-Pearce rabbit's tumor growth. The results obtained confirm a possibility of successful mapping of the experimental Gompertz curve onto the fractal power-law scaling function y(t)=a(t)tb(t) and support a thesis that Gompertzian growth is a self-similar and allometric process of a holistic nature.

Launching the chaotic realm of iso-fractals: A short remark

DOE Office of Scientific and Technical Information (OSTI.GOV)

O'Schmidt, Nathan; Katebi, Reza; Corda, Christian

In this brief note, we introduce the new, emerging sub-discipline of iso-fractals by highlighting and discussing the preliminary results of recent works. First, we note the abundance of fractal, chaotic, non-linear, and self-similar structures in nature while emphasizing the importance of studying such systems because fractal geometry is the language of chaos. Second, we outline the iso-fractal generalization of the Mandelbrot set to exemplify the newly generated Mandelbrot iso-sets. Third, we present the cutting-edge notion of dynamic iso-spaces and explain how a mathematical space can be iso-topically lifted with iso-unit functions that (continuously or discretely) change; in the discrete casemore » examples, we mention that iteratively generated sequences like Fibonacci’s numbers and (the complex moduli of) Mandelbrot’s numbers can supply a deterministic chain of iso-units to construct an ordered series of (magnified and/or de-magnified) iso-spaces that are locally iso-morphic. Fourth, we consider the initiation of iso-fractals with Inopin’s holographic ring (IHR) topology and fractional statistics for 2D and 3D iso-spaces. In total, the reviewed iso-fractal results are a significant improvement over traditional fractals because the application of Santilli’s iso-mathematics arms us an extra degree of freedom for attacking problems in chaos. Finally, we conclude by proposing some questions and ideas for future research work.« less

Feature extraction algorithm for space targets based on fractal theory

NASA Astrophysics Data System (ADS)

Tian, Balin; Yuan, Jianping; Yue, Xiaokui; Ning, Xin

2007-11-01

In order to offer a potential for extending the life of satellites and reducing the launch and operating costs, satellite servicing including conducting repairs, upgrading and refueling spacecraft on-orbit become much more frequently. Future space operations can be more economically and reliably executed using machine vision systems, which can meet real time and tracking reliability requirements for image tracking of space surveillance system. Machine vision was applied to the research of relative pose for spacecrafts, the feature extraction algorithm was the basis of relative pose. In this paper fractal geometry based edge extraction algorithm which can be used in determining and tracking the relative pose of an observed satellite during proximity operations in machine vision system was presented. The method gets the gray-level image distributed by fractal dimension used the Differential Box-Counting (DBC) approach of the fractal theory to restrain the noise. After this, we detect the consecutive edge using Mathematical Morphology. The validity of the proposed method is examined by processing and analyzing images of space targets. The edge extraction method not only extracts the outline of the target, but also keeps the inner details. Meanwhile, edge extraction is only processed in moving area to reduce computation greatly. Simulation results compared edge detection using the method which presented by us with other detection methods. The results indicate that the presented algorithm is a valid method to solve the problems of relative pose for spacecrafts.

The Fractal Self at Play

ERIC Educational Resources Information Center

Marks-Tarlow, Terry

2010-01-01

In this article, the author draws on contemporary science to illuminate the relationship between early play experiences, processes of self-development, and the later emergence of the fractal self. She argues that orientation within social space is a primary function of early play and developmentally a two-step process. With other people and with…

Characterization of microgravity effects on bone structure and strength using fractal analysis

NASA Technical Reports Server (NTRS)

Acharya, Raj S.; Shackelford, Linda

1995-01-01

The effect of micro-gravity on the musculoskeletal system has been well studied. Significant changes in bone and muscle have been shown after long term space flight. Similar changes have been demonstrated due to bed rest. Bone demineralization is particularly profound in weight bearing bones. Much of the current techniques to monitor bone condition use bone mass measurements. However, bone mass measurements are not reliable to distinguish Osteoporotic and Normal subjects. It has been shown that the overlap between normals and osteoporosis is found for all of the bone mass measurement technologies: single and dual photon absorptiometry, quantitative computed tomography and direct measurement of bone area/volume on biopsy as well as radiogrammetry. A similar discordance is noted in the fact that it has not been regularly possible to find the expected correlation between severity of osteoporosis and degree of bone loss. Structural parameters such as trabecular connectivity have been proposed as features for assessing bone conditions. In this report, we use fractal analysis to characterize bone structure. We show that the fractal dimension computed with MRI images and X-Ray images of the patella are the same. Preliminary experimental results show that the fractal dimension computed from MRI images of vertebrae of human subjects before bedrest is higher than during bedrest.

Fractals in the neurosciences, Part II: clinical applications and future perspectives.

PubMed

Di Ieva, Antonio; Esteban, Francisco J; Grizzi, Fabio; Klonowski, Wlodzimierz; Martín-Landrove, Miguel

2015-02-01

It has been ascertained that the human brain is a complex system studied at multiple scales, from neurons and microcircuits to macronetworks. The brain is characterized by a hierarchical organization that gives rise to its highly topological and functional complexity. Over the last decades, fractal geometry has been shown as a universal tool for the analysis and quantification of the geometric complexity of natural objects, including the brain. The fractal dimension has been identified as a quantitative parameter for the evaluation of the roughness of neural structures, the estimation of time series, and the description of patterns, thus able to discriminate different states of the brain in its entire physiopathological spectrum. Fractal-based computational analyses have been applied to the neurosciences, particularly in the field of clinical neurosciences including neuroimaging and neuroradiology, neurology and neurosurgery, psychiatry and psychology, and neuro-oncology and neuropathology. After a review of the basic concepts of fractal analysis and its main applications to the basic neurosciences in part I of this series, here, we review the main applications of fractals to the clinical neurosciences for a holistic approach towards a fractal geometry model of the brain. © The Author(s) 2013.

Fractal reaction kinetics.

PubMed

Kopelman, R

1988-09-23

Classical reaction kinetics has been found to be unsatisfactory when the reactants are spatially constrained on the microscopic level by either walls, phase boundaries, or force fields. Recently discovered theories of heterogeneous reaction kinetics have dramatic consequences, such as fractal orders for elementary reactions, self-ordering and self-unmixing of reactants, and rate coefficients with temporal "memories." The new theories were needed to explain the results of experiments and supercomputer simulations of reactions that were confined to low dimensions or fractal dimensions or both. Among the practical examples of "fractal-like kinetics" are chemical reactions in pores of membranes, excitation trapping in molecular aggregates, exciton fusion in composite materials, and charge recombination in colloids and clouds.

A Complex Story: Universal Preference vs. Individual Differences Shaping Aesthetic Response to Fractals Patterns

PubMed Central

Street, Nichola; Forsythe, Alexandra M.; Reilly, Ronan; Taylor, Richard; Helmy, Mai S.

2016-01-01

Fractal patterns offer one way to represent the rough complexity of the natural world. Whilst they dominate many of our visual experiences in nature, little large-scale perceptual research has been done to explore how we respond aesthetically to these patterns. Previous research (Taylor et al., 2011) suggests that the fractal patterns with mid-range fractal dimensions (FDs) have universal aesthetic appeal. Perceptual and aesthetic responses to visual complexity have been more varied with findings suggesting both linear (Forsythe et al., 2011) and curvilinear (Berlyne, 1970) relationships. Individual differences have been found to account for many of the differences we see in aesthetic responses but some, such as culture, have received little attention within the fractal and complexity research fields. This two-study article aims to test preference responses to FD and visual complexity, using a large cohort (N = 443) of participants from around the world to allow universality claims to be tested. It explores the extent to which age, culture and gender can predict our preferences for fractally complex patterns. Following exploratory analysis that found strong correlations between FD and visual complexity, a series of linear mixed-effect models were implemented to explore if each of the individual variables could predict preference. The first tested a linear complexity model (likelihood of selecting the more complex image from the pair of images) and the second a mid-range FD model (likelihood of selecting an image within mid-range). Results show that individual differences can reliably predict preferences for complexity across culture, gender and age. However, in fitting with current findings the mid-range models show greater consistency in preference not mediated by gender, age or culture. This article supports the established theory that the mid-range fractal patterns appear to be a universal construct underlying preference but also highlights the fragility of

A Complex Story: Universal Preference vs. Individual Differences Shaping Aesthetic Response to Fractals Patterns.

PubMed

Street, Nichola; Forsythe, Alexandra M; Reilly, Ronan; Taylor, Richard; Helmy, Mai S

2016-01-01

Fractal patterns offer one way to represent the rough complexity of the natural world. Whilst they dominate many of our visual experiences in nature, little large-scale perceptual research has been done to explore how we respond aesthetically to these patterns. Previous research (Taylor et al., 2011) suggests that the fractal patterns with mid-range fractal dimensions (FDs) have universal aesthetic appeal. Perceptual and aesthetic responses to visual complexity have been more varied with findings suggesting both linear (Forsythe et al., 2011) and curvilinear (Berlyne, 1970) relationships. Individual differences have been found to account for many of the differences we see in aesthetic responses but some, such as culture, have received little attention within the fractal and complexity research fields. This two-study article aims to test preference responses to FD and visual complexity, using a large cohort (N = 443) of participants from around the world to allow universality claims to be tested. It explores the extent to which age, culture and gender can predict our preferences for fractally complex patterns. Following exploratory analysis that found strong correlations between FD and visual complexity, a series of linear mixed-effect models were implemented to explore if each of the individual variables could predict preference. The first tested a linear complexity model (likelihood of selecting the more complex image from the pair of images) and the second a mid-range FD model (likelihood of selecting an image within mid-range). Results show that individual differences can reliably predict preferences for complexity across culture, gender and age. However, in fitting with current findings the mid-range models show greater consistency in preference not mediated by gender, age or culture. This article supports the established theory that the mid-range fractal patterns appear to be a universal construct underlying preference but also highlights the fragility of

A fractal model for nuclear organization: current evidence and biological implications

PubMed Central

Bancaud, Aurélien; Lavelle, Christophe; Huet, Sébastien; Ellenberg, Jan

2012-01-01

Chromatin is a multiscale structure on which transcription, replication, recombination and repair of the genome occur. To fully understand any of these processes at the molecular level under physiological conditions, a clear picture of the polymorphic and dynamic organization of chromatin in the eukaryotic nucleus is required. Recent studies indicate that a fractal model of chromatin architecture is consistent with both the reaction-diffusion properties of chromatin interacting proteins and with structural data on chromatin interminglement. In this study, we provide a critical overview of the experimental evidence that support a fractal organization of chromatin. On this basis, we discuss the functional implications of a fractal chromatin model for biological processes and propose future experiments to probe chromatin organization further that should allow to strongly support or invalidate the fractal hypothesis. PMID:22790985

A conservation law, entropy principle and quantization of fractal dimensions in hadron interactions

NASA Astrophysics Data System (ADS)

Zborovský, I.

2018-04-01

Fractal self-similarity of hadron interactions demonstrated by the z-scaling of inclusive spectra is studied. The scaling regularity reflects fractal structure of the colliding hadrons (or nuclei) and takes into account general features of fragmentation processes expressed by fractal dimensions. The self-similarity variable z is a function of the momentum fractions x1 and x2 of the colliding objects carried by the interacting hadron constituents and depends on the momentum fractions ya and yb of the scattered and recoil constituents carried by the inclusive particle and its recoil counterpart, respectively. Based on entropy principle, new properties of the z-scaling concept are found. They are conservation of fractal cumulativity in hadron interactions and quantization of fractal dimensions characterizing hadron structure and fragmentation processes at a constituent level.

Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis

NASA Astrophysics Data System (ADS)

Mantica, Giorgio; Perotti, Luca

2016-09-01

Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase-space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical rôle of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non-standard Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments. Dedicated to the memory of Joseph Ford, on the twentieth anniversary of his departure.

Chaos, Fractals, and Polynomials.

ERIC Educational Resources Information Center

Tylee, J. Louis; Tylee, Thomas B.

1996-01-01

Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)

Fractals in biology and medicine

NASA Technical Reports Server (NTRS)

Havlin, S.; Buldyrev, S. V.; Goldberger, A. L.; Mantegna, R. N.; Ossadnik, S. M.; Peng, C. K.; Simons, M.; Stanley, H. E.

1995-01-01

Our purpose is to describe some recent progress in applying fractal concepts to systems of relevance to biology and medicine. We review several biological systems characterized by fractal geometry, with a particular focus on the long-range power-law correlations found recently in DNA sequences containing noncoding material. Furthermore, we discuss the finding that the exponent alpha quantifying these long-range correlations ("fractal complexity") is smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the normal heart is characterized by long-range "anticorrelations" which are absent in the diseased heart.

Langevin Equation on Fractal Curves

NASA Astrophysics Data System (ADS)

Satin, Seema; Gangal, A. D.

2016-07-01

We analyze random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, plays an important role in this analysis. A Langevin equation with a particular model of noise is proposed and solved using techniques of the Fα-Calculus.

Fractal Patterns of Neural Activity Exist within the Suprachiasmatic Nucleus and Require Extrinsic Network Interactions

PubMed Central

Hu, Kun; Meijer, Johanna H.; Shea, Steven A.; vanderLeest, Henk Tjebbe; Pittman-Polletta, Benjamin; Houben, Thijs; van Oosterhout, Floor; Deboer, Tom; Scheer, Frank A. J. L.

2012-01-01

The mammalian central circadian pacemaker (the suprachiasmatic nucleus, SCN) contains thousands of neurons that are coupled through a complex network of interactions. In addition to the established role of the SCN in generating rhythms of ∼24 hours in many physiological functions, the SCN was recently shown to be necessary for normal self-similar/fractal organization of motor activity and heart rate over a wide range of time scales—from minutes to 24 hours. To test whether the neural network within the SCN is sufficient to generate such fractal patterns, we studied multi-unit neural activity of in vivo and in vitro SCNs in rodents. In vivo SCN-neural activity exhibited fractal patterns that are virtually identical in mice and rats and are similar to those in motor activity at time scales from minutes up to 10 hours. In addition, these patterns remained unchanged when the main afferent signal to the SCN, namely light, was removed. However, the fractal patterns of SCN-neural activity are not autonomous within the SCN as these patterns completely broke down in the isolated in vitro SCN despite persistence of circadian rhythmicity. Thus, SCN-neural activity is fractal in the intact organism and these fractal patterns require network interactions between the SCN and extra-SCN nodes. Such a fractal control network could underlie the fractal regulation observed in many physiological functions that involve the SCN, including motor control and heart rate regulation. PMID:23185285

The Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Fractal

NASA Astrophysics Data System (ADS)

Sadegh, Sanaz; Higgin, Jenny; Mannion, Patrick; Tamkun, Michael; Krapf, Diego

A broad range of membrane proteins display anomalous diffusion on the cell surface. Different methods provide evidence for obstructed subdiffusion and diffusion on a fractal space, but the underlying structure inducing anomalous diffusion has never been visualized due to experimental challenges. We addressed this problem by imaging the cortical actin at high resolution while simultaneously tracking individual membrane proteins in live mammalian cells. Our data show that actin introduces barriers leading to compartmentalization of the plasma membrane and that membrane proteins are transiently confined within actin fences. Furthermore, superresolution imaging shows that the cortical actin is organized into a self-similar fractal. These results present a hierarchical nanoscale picture of the plasma membrane and demonstrate direct interactions between the actin cortex and the cell surface.

Fuzzy membership functions for analysis of high-resolution CT images of diffuse pulmonary diseases.

PubMed

Almeida, Eliana; Rangayyan, Rangaraj M; Azevedo-Marques, Paulo M

2015-08-01

We propose the use of fuzzy membership functions to analyze images of diffuse pulmonary diseases (DPDs) based on fractal and texture features. The features were extracted from preprocessed regions of interest (ROIs) selected from high-resolution computed tomography images. The ROIs represent five different patterns of DPDs and normal lung tissue. A Gaussian mixture model (GMM) was constructed for each feature, with six Gaussians modeling the six patterns. Feature selection was performed and the GMMs of the five significant features were used. From the GMMs, fuzzy membership functions were obtained by a probability-possibility transformation and further statistical analysis was performed. An average classification accuracy of 63.5% was obtained for the six classes. For four of the six classes, the classification accuracy was superior to 65%, and the best classification accuracy was 75.5% for one class. The use of fuzzy membership functions to assist in pattern classification is an alternative to deterministic approaches to explore strategies for medical diagnosis.

Lévy processes on a generalized fractal comb

NASA Astrophysics Data System (ADS)

Sandev, Trifce; Iomin, Alexander; Méndez, Vicenç

2016-09-01

Comb geometry, constituted of a backbone and fingers, is one of the most simple paradigm of a two-dimensional structure, where anomalous diffusion can be realized in the framework of Markov processes. However, the intrinsic properties of the structure can destroy this Markovian transport. These effects can be described by the memory and spatial kernels. In particular, the fractal structure of the fingers, which is controlled by the spatial kernel in both the real and the Fourier spaces, leads to the Lévy processes (Lévy flights) and superdiffusion. This generalization of the fractional diffusion is described by the Riesz space fractional derivative. In the framework of this generalized fractal comb model, Lévy processes are considered, and exact solutions for the probability distribution functions are obtained in terms of the Fox H-function for a variety of the memory kernels, and the rate of the superdiffusive spreading is studied by calculating the fractional moments. For a special form of the memory kernels, we also observed a competition between long rests and long jumps. Finally, we considered the fractal structure of the fingers controlled by a Weierstrass function, which leads to the power-law kernel in the Fourier space. This is a special case, when the second moment exists for superdiffusion in this competition between long rests and long jumps.

Emergence of fractal scaling in complex networks

NASA Astrophysics Data System (ADS)

Wei, Zong-Wen; Wang, Bing-Hong

2016-09-01

Some real-world networks are shown to be fractal or self-similar. It is widespread that such a phenomenon originates from the repulsion between hubs or disassortativity. Here we show that this common belief fails to capture the causality. Our key insight to address it is to pinpoint links critical to fractality. Those links with small edge betweenness centrality (BC) constitute a special architecture called fractal reference system, which gives birth to the fractal structure of those reported networks. In contrast, a small amount of links with high BC enable small-world effects, hiding the intrinsic fractality. With enough of such links removed, fractal scaling spontaneously arises from nonfractal networks. Our results provide a multiple-scale view on the structure and dynamics and place fractality as a generic organizing principle of complex networks on a firmer ground.

Some problems in fractal differential equations

NASA Astrophysics Data System (ADS)

Su, Weiyi

2016-06-01

Based upon the fractal calculus on local fields, or p-type calculus, or Gibbs-Butzer calculus ([1],[2]), we suggest a constructive idea for "fractal differential equations", beginning from some special examples to a general theory. However, this is just an original idea, it needs lots of later work to support. In [3], we show example "two dimension wave equations with fractal boundaries", and in this note, other examples, as well as an idea to construct fractal differential equations are shown.

Thermodynamics of photons on fractals.

PubMed

Akkermans, Eric; Dunne, Gerald V; Teplyaev, Alexander

2010-12-03

A thermodynamical treatment of a massless scalar field (a photon) confined to a fractal spatial manifold leads to an equation of state relating pressure to internal energy, PV(s) = U/d(s), where d(s) is the spectral dimension and V(s) defines the "spectral volume." For regular manifolds, V(s) coincides with the usual geometric spatial volume, but on a fractal this is not necessarily the case. This is further evidence that on a fractal, momentum space can have a different dimension than position space. Our analysis also provides a natural definition of the vacuum (Casimir) energy of a fractal. We suggest ways that these unusual properties might be probed experimentally.

Universal characteristics of fractal fluctuations in prime number distribution

NASA Astrophysics Data System (ADS)

Selvam, A. M.

2014-11-01

The frequency of occurrence of prime numbers at unit number spacing intervals exhibits self-similar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows, stock market fluctuations and population dynamics. The physics of long-range correlations exhibited by fractals is not yet identified. A recently developed general systems theory visualizes the eddy continuum underlying fractals to result from the growth of large eddies as the integrated mean of enclosed small scale eddies, thereby generating a hierarchy of eddy circulations or an inter-connected network with associated long-range correlations. The model predictions are as follows: (1) The probability distribution and power spectrum of fractals follow the same inverse power law which is a function of the golden mean. The predicted inverse power law distribution is very close to the statistical normal distribution for fluctuations within two standard deviations from the mean of the distribution. (2) Fractals signify quantum-like chaos since variance spectrum represents probability density distribution, a characteristic of quantum systems such as electron or photon. (3) Fractal fluctuations of frequency distribution of prime numbers signify spontaneous organization of underlying continuum number field into the ordered pattern of the quasiperiodic Penrose tiling pattern. The model predictions are in agreement with the probability distributions and power spectra for different sets of frequency of occurrence of prime numbers at unit number interval for successive 1000 numbers. Prime numbers in the first 10 million numbers were used for the study.

Nonlinear dynamics, fractals, cardiac physiology and sudden death

NASA Technical Reports Server (NTRS)

Goldberger, Ary L.

1987-01-01

The authors propose a diametrically opposite viewpoint to the generally accepted tendency of equating healthy function with order and disease with chaos. With regard to the question of sudden cardiac death and chaos, it is suggested that certain features of dynamical chaos related to fractal structure and fractal dynamics may be important organizing principles in normal physiology and that certain pathologies, including ventricular fibrillation, represent a class of 'pathological periodicities'. Some laboratory work bearing on the relation of nonlinear analysis to physiological and pathophysiological data is briefly reviewed, with tentative theories and models described in reference to the mechanism of ventricular fibrillation.

Fractal dimension analysis of weight-bearing bones of rats during skeletal unloading

NASA Technical Reports Server (NTRS)

Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Sanhueza, A. I.; Yamauchi, M.

2001-01-01

Fractal analysis was used to quantify changes in trabecular bone induced through the use of a rat tail-suspension model to simulate microgravity-induced osteopenia. Fractal dimensions were estimated from digitized radiographs obtained from tail-suspended and ambulatory rats. Fifty 4-month-old male Sprague-Dawley rats were divided into groups of 24 ambulatory (control) and 26 suspended (test) animals. Rats of both groups were killed after periods of 1, 4, and 8 weeks. Femurs and tibiae were removed and radiographed with standard intraoral films and digitized using a flatbed scanner. Square regions of interest were cropped at proximal, middle, and distal areas of each bone. Fractal dimensions were estimated from slopes of regression lines fitted to circularly averaged plots of log power vs. log spatial frequency. The results showed that the computed fractal dimensions were significantly greater for images of trabecular bones from tail-suspended groups than for ambulatory groups (p < 0.01) at 1 week. Periods between 1 and 4 weeks likewise yielded significantly different estimates (p < 0.05), consistent with an increase in bone loss. In the tibiae, the proximal regions of the suspended group produced significantly greater fractal dimensions than other regions (p < 0.05), which suggests they were more susceptible to unloading. The data are consistent with other studies demonstrating osteopenia in microgravity environments and the regional response to skeletal unloading. Thus, fractal analysis could be a useful technique to evaluate the structural changes of bone.

Rheological and fractal characteristics of unconditioned and conditioned water treatment residuals.

PubMed

Dong, Y J; Wang, Y L; Feng, J

2011-07-01

The rheological and fractal characteristics of raw (unconditioned) and conditioned water treatment residuals (WTRs) were investigated in this study. Variations in morphology, size, and image fractal dimensions of the flocs/aggregates in these WTR systems with increasing polymer doses were analyzed. The results showed that when the raw WTRs were conditioned with the polymer CZ8688, the optimum polymer dosage was observed at 24 kg/ton dry sludge. The average diameter of irregularly shaped flocs/aggregates in the WTR suspensions increased from 42.54 μm to several hundred micrometers with increasing polymer doses. Furthermore, the aggregates in the conditioned WTR system displayed boundary/surface and mass fractals. At the optimum polymer dosage, the aggregates formed had a volumetric average diameter of about 820.7 μm, with a one-dimensional fractal dimension of 1.01 and a mass fractal dimension of 2.74 on the basis of the image analysis. Rheological tests indicated that the conditioned WTRs at the optimum polymer dosage showed higher levels of shear-thinning behavior than the raw WTRs. Variations in the limiting viscosity (η(∞)) of conditioned WTRs with sludge content could be described by a linear equation, which were different from the often-observed empirical exponential relationship for most municipal sludge. With increasing temperature, the η(∞) of the raw WTRs decreased more rapidly than that of the raw WTRs. Good fitting results for the relationships between lgη(∞)∼T using the Arrhenius equation indicate that the WTRs had a much higher activation energy for viscosity of about 17.86-26.91 J/mol compared with that of anaerobic granular sludge (2.51 J/mol) (Mu and Yu, 2006). In addition, the Bingham plastic model adequately described the rheological behavior of the conditioned WTRs, whereas the rheology of the raw WTRs fit the Herschel-Bulkley model well at only certain sludge contents. Considering the good power-law relationships between the

A fractal image analysis methodology for heat damage inspection in carbon fiber reinforced composites

NASA Astrophysics Data System (ADS)

Haridas, Aswin; Crivoi, Alexandru; Prabhathan, P.; Chan, Kelvin; Murukeshan, V. M.

2017-06-01

The use of carbon fiber-reinforced polymer (CFRP) composite materials in the aerospace industry have far improved the load carrying properties and the design flexibility of aircraft structures. A high strength to weight ratio, low thermal conductivity, and a low thermal expansion coefficient gives it an edge for applications demanding stringent loading conditions. Specifically, this paper focuses on the behavior of CFRP composites under stringent thermal loads. The properties of composites are largely affected by external thermal loads, especially when the loads are beyond the glass temperature, Tg, of the composite. Beyond this, the composites are subject to prominent changes in mechanical and thermal properties which may further lead to material decomposition. Furthermore, thermal damage formation being chaotic, a strict dimension cannot be associated with the formed damage. In this context, this paper focuses on comparing multiple speckle image analysis algorithms to effectively characterize the formed thermal damages on the CFRP specimen. This would provide us with a fast method for quantifying the extent of heat damage in carbon composites, thus reducing the required time for inspection. The image analysis methods used for the comparison include fractal dimensional analysis of the formed speckle pattern and analysis of number and size of various connecting elements in the binary image.

Electrical conductivity modeling in fractal non-saturated porous media

NASA Astrophysics Data System (ADS)

Wei, W.; Cai, J.; Hu, X.; Han, Q.

2016-12-01

The variety of electrical conductivity in non-saturated conditions is important to study electric conduction in natural sedimentary rocks. The electrical conductivity in completely saturated porous media is a porosity-function representing the complex connected behavior of single conducting phases (pore fluid). For partially saturated conditions, the electrical conductivity becomes even more complicated since the connectedness of pore. Archie's second law is an empirical electrical conductivity-porosity and -saturation model that has been used to predict the formation factor of non-saturated porous rock. However, the physical interpretation of its parameters, e.g., the cementation exponent m and the saturation exponent n, remains questionable. On basis of our previous work, we combine the pore-solid fractal (PSF) model to build an electrical conductivity model in non-saturated porous media. Our theoretical porosity- and saturation-dependent models contain endmember properties, such as fluid electrical conductivities, pore fractal dimension and tortuosity fractal dimension (representing the complex degree of electrical flowing path). We find the presented model with non-saturation-dependent electrical conductivity datasets indicate excellent match between theory and experiments. This means the value of pore fractal dimension and tortuosity fractal dimension change from medium to medium and depends not only on geometrical properties of pore structure but also characteristics of electrical current flowing in the non-saturated porous media.

Fractal aggregates in tennis ball systems

NASA Astrophysics Data System (ADS)

Sabin, J.; Bandín, M.; Prieto, G.; Sarmiento, F.

2009-09-01

We present a new practical exercise to explain the mechanisms of aggregation of some colloids which are otherwise not easy to understand. We have used tennis balls to simulate, in a visual way, the aggregation of colloids under reaction-limited colloid aggregation (RLCA) and diffusion-limited colloid aggregation (DLCA) regimes. We have used the images of the cluster of balls, following Forrest and Witten's pioneering studies on the aggregation of smoke particles, to estimate their fractal dimension.

Fractal analysis of radiologists' visual scanning pattern in screening mammography

NASA Astrophysics Data System (ADS)

Alamudun, Folami T.; Yoon, Hong-Jun; Hudson, Kathy; Morin-Ducote, Garnetta; Tourassi, Georgia

2015-03-01

Several researchers have investigated radiologists' visual scanning patterns with respect to features such as total time examining a case, time to initially hit true lesions, number of hits, etc. The purpose of this study was to examine the complexity of the radiologists' visual scanning pattern when viewing 4-view mammographic cases, as they typically do in clinical practice. Gaze data were collected from 10 readers (3 breast imaging experts and 7 radiology residents) while reviewing 100 screening mammograms (24 normal, 26 benign, 50 malignant). The radiologists' scanpaths across the 4 mammographic views were mapped to a single 2-D image plane. Then, fractal analysis was applied on the composite 4- view scanpaths. For each case, the complexity of each radiologist's scanpath was measured using fractal dimension estimated with the box counting method. The association between the fractal dimension of the radiologists' visual scanpath, case pathology, case density, and radiologist experience was evaluated using fixed effects ANOVA. ANOVA showed that the complexity of the radiologists' visual search pattern in screening mammography is dependent on case specific attributes (breast parenchyma density and case pathology) as well as on reader attributes, namely experience level. Visual scanning patterns are significantly different for benign and malignant cases than for normal cases. There is also substantial inter-observer variability which cannot be explained only by experience level.

A user-friendly modified pore-solid fractal model

PubMed Central

Ding, Dian-yuan; Zhao, Ying; Feng, Hao; Si, Bing-cheng; Hill, Robert Lee

2016-01-01

The primary objective of this study was to evaluate a range of calculation points on water retention curves (WRC) instead of the singularity point at air-entry suction in the pore-solid fractal (PSF) model, which additionally considered the hysteresis effect based on the PSF theory. The modified pore-solid fractal (M-PSF) model was tested using 26 soil samples from Yangling on the Loess Plateau in China and 54 soil samples from the Unsaturated Soil Hydraulic Database. The derivation results showed that the M-PSF model is user-friendly and flexible for a wide range of calculation point options. This model theoretically describes the primary differences between the soil moisture desorption and the adsorption processes by the fractal dimensions. The M-PSF model demonstrated good performance particularly at the calculation points corresponding to the suctions from 100 cm to 1000 cm. Furthermore, the M-PSF model, used the fractal dimension of the particle size distribution, exhibited an accepted performance of WRC predictions for different textured soils when the suction values were ≥100 cm. To fully understand the function of hysteresis in the PSF theory, the role of allowable and accessible pores must be examined. PMID:27996013

Fractal analysis of time varying data

DOEpatents

Vo-Dinh, Tuan; Sadana, Ajit

2002-01-01

Characteristics of time varying data, such as an electrical signal, are analyzed by converting the data from a temporal domain into a spatial domain pattern. Fractal analysis is performed on the spatial domain pattern, thereby producing a fractal dimension D.sub.F. The fractal dimension indicates the regularity of the time varying data.

Fractal electrodynamics via non-integer dimensional space approach

NASA Astrophysics Data System (ADS)

Tarasov, Vasily E.

2015-09-01

Using the recently suggested vector calculus for non-integer dimensional space, we consider electrodynamics problems in isotropic case. This calculus allows us to describe fractal media in the framework of continuum models with non-integer dimensional space. We consider electric and magnetic fields of fractal media with charges and currents in the framework of continuum models with non-integer dimensional spaces. An application of the fractal Gauss's law, the fractal Ampere's circuital law, the fractal Poisson equation for electric potential, and equation for fractal stream of charges are suggested. Lorentz invariance and speed of light in fractal electrodynamics are discussed. An expression for effective refractive index of non-integer dimensional space is suggested.

Fractal analysis: A new tool in transient volcanic ash plume characterization.

NASA Astrophysics Data System (ADS)

Tournigand, Pierre-Yves; Peña Fernandez, Juan Jose; Taddeucci, Jacopo; Perugini, Diego; Sesterhenn, Jörn

2017-04-01

Transient volcanic plumes are time-dependent features generated by unstable eruptive sources. They represent a threat to human health and infrastructures, and a challenge to characterize due to their intrinsic instability. Plumes have been investigated through physical (e.g. visible, thermal, UV, radar imagery), experimental and numerical studies in order to provide new insights about their dynamics and better anticipate their behavior. It has been shown experimentally that plume dynamics is strongly dependent to source conditions and that plume shape evolution holds key to retrieve these conditions. In this study, a shape evolution analysis is performed on thermal high-speed videos of volcanic plumes from three different volcanoes Sakurajima (Japan), Stromboli (Italy) and Fuego (Guatemala), recorded with a FLIR SC655 thermal camera during several field campaigns between 2012 and 2016. To complete this dataset, three numerical gas-jet simulations at different Reynolds number (2000, 5000 and 10000) have been used in order to set reference values to the natural cases. Turbulent flow shapes are well known to feature scale-invariant structures and a high degree of complexity. For this reason we characterized the bi-dimensional shape of natural and synthetic plumes by using a fractal descriptor. Such method has been applied in other studies on experimental turbulent jets as well as on atmospheric clouds and have shown promising results. At each time-step plume contour has been manually outlined and measured using the box-counting method. This method consists in covering the image with squares of variable sizes and counting the number of squares containing the plume outline. The negative slope of the number of squares in function of their size in a log-log plot gives the fractal dimension of the plume at a given time. Preliminary results show an increase over time of the fractal dimension for natural volcanic plume as well as for the numerically simulated ones, but at

Evolving random fractal Cantor superlattices for the infrared using a genetic algorithm

PubMed Central

Bossard, Jeremy A.; Lin, Lan; Werner, Douglas H.

2016-01-01

Ordered and chaotic superlattices have been identified in Nature that give rise to a variety of colours reflected by the skin of various organisms. In particular, organisms such as silvery fish possess superlattices that reflect a broad range of light from the visible to the UV. Such superlattices have previously been identified as ‘chaotic’, but we propose that apparent ‘chaotic’ natural structures, which have been previously modelled as completely random structures, should have an underlying fractal geometry. Fractal geometry, often described as the geometry of Nature, can be used to mimic structures found in Nature, but deterministic fractals produce structures that are too ‘perfect’ to appear natural. Introducing variability into fractals produces structures that appear more natural. We suggest that the ‘chaotic’ (purely random) superlattices identified in Nature are more accurately modelled by multi-generator fractals. Furthermore, we introduce fractal random Cantor bars as a candidate for generating both ordered and ‘chaotic’ superlattices, such as the ones found in silvery fish. A genetic algorithm is used to evolve optimal fractal random Cantor bars with multiple generators targeting several desired optical functions in the mid-infrared and the near-infrared. We present optimized superlattices demonstrating broadband reflection as well as single and multiple pass bands in the near-infrared regime. PMID:26763335

Deterministic chaos and fractal complexity in the dynamics of cardiovascular behavior: perspectives on a new frontier.

PubMed

Sharma, Vijay

2009-09-10

Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts.

Deterministic Chaos and Fractal Complexity in the Dynamics of Cardiovascular Behavior: Perspectives on a New Frontier

PubMed Central

Sharma, Vijay

2009-01-01

Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts. PMID:19812706

Fractal properties and denoising of lidar signals from cirrus clouds

NASA Astrophysics Data System (ADS)

van den Heuvel, J. C.; Driesenaar, M. L.; Lerou, R. J. L.

2000-02-01

Airborne lidar signals of cirrus clouds are analyzed to determine the cloud structure. Climate modeling and numerical weather prediction benefit from accurate modeling of cirrus clouds. Airborne lidar measurements of the European Lidar in Space Technology Experiment (ELITE) campaign were analyzed by combining shots to obtain the backscatter at constant altitude. The signal at high altitude was analyzed for horizontal structure of cirrus clouds. The power spectrum and the structure function show straight lines on a double logarithmic plot. This behavior is characteristic for a Brownian fractal. Wavelet analysis using the Haar wavelet confirms the fractal aspects. It is shown that the horizontal structure of cirrus can be described by a fractal with a dimension of 1.8 over length scales that vary 4 orders of magnitude. We use the fractal properties in a new denoising method. Denoising is required for future lidar measurements from space that have a low signal to noise ratio. Our wavelet denoising is based on the Haar wavelet and uses the statistical fractal properties of cirrus clouds in a method based on the maximum a posteriori (MAP) probability. This denoising based on wavelets is tested on airborne lidar signals from ELITE using added Gaussian noise. Superior results with respect to averaging are obtained.

Roughness Perception of Haptically Displayed Fractal Surfaces

NASA Technical Reports Server (NTRS)

Costa, Michael A.; Cutkosky, Mark R.; Lau, Sonie (Technical Monitor)

2000-01-01

Surface profiles were generated by a fractal algorithm and haptically rendered on a force feedback joystick, Subjects were asked to use the joystick to explore pairs of surfaces and report to the experimenter which of the surfaces they felt was rougher. Surfaces were characterized by their root mean square (RMS) amplitude and their fractal dimension. The most important factor affecting the perceived roughness of the fractal surfaces was the RMS amplitude of the surface. When comparing surfaces of fractal dimension 1.2-1.35 it was found that the fractal dimension was negatively correlated with perceived roughness.

Fractal planetary rings: Energy inequalities and random field model

NASA Astrophysics Data System (ADS)

Malyarenko, Anatoliy; Ostoja-Starzewski, Martin

2017-12-01

This study is motivated by a recent observation, based on photographs from the Cassini mission, that Saturn’s rings have a fractal structure in radial direction. Accordingly, two questions are considered: (1) What Newtonian mechanics argument in support of such a fractal structure of planetary rings is possible? (2) What kinematics model of such fractal rings can be formulated? Both challenges are based on taking planetary rings’ spatial structure as being statistically stationary in time and statistically isotropic in space, but statistically nonstationary in space. An answer to the first challenge is given through an energy analysis of circular rings having a self-generated, noninteger-dimensional mass distribution [V. E. Tarasov, Int. J. Mod Phys. B 19, 4103 (2005)]. The second issue is approached by taking the random field of angular velocity vector of a rotating particle of the ring as a random section of a special vector bundle. Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions defined on the Cartesian square F2 of the radial cross-section F of the rings, where F is a fat fractal.

Fractal analysis of narwhal space use patterns.

PubMed

Laidre, Kristin L; Heide-Jørgensen, Mads P; Logsdon, Miles L; Hobbs, Roderick C; Dietz, Rune; VanBlaricom, Glenn R

2004-01-01

Quantifying animal movement in response to a spatially and temporally heterogeneous environment is critical to understanding the structural and functional landscape influences on population viability. Generalities of landscape structure can easily be extended to the marine environment, as marine predators inhabit a patchy, dynamic system, which influences animal choice and behavior. An innovative use of the fractal measure of complexity, indexing the linearity of movement paths over replicate temporal scales, was applied to satellite tracking data collected from narwhals (Monodon monoceros) (n = 20) in West Greenland and the eastern Canadian high Arctic. Daily movements of individuals were obtained using polar orbiting satellites via the ARGOS data location and collection system. Geographic positions were filtered to obtain a daily good quality position for each whale. The length of total pathway was measured over seven different temporal length scales (step lengths), ranging from one day to one week, and a seasonal mean was calculated. Fractal dimension (D) was significantly different between seasons, highest during summer (D = 1.61, SE 0.04) and winter (D = 1.69, SE 0.06) when whales made convoluted movements in focal areas. Fractal dimension was lowest during fall (D = 1.34, SE 0.03) when whales were migrating south ahead of the forming sea ice. There were no significant effects of size category or sex on fractal dimension by season. The greater linearity of movement during the migration period suggests individuals do not intensively forage on patchy resources until they arrive at summer or winter sites. The highly convoluted movements observed during summer and winter suggest foraging or searching efforts in localized areas. Significant differences between the fractal dimensions on two separate wintering grounds in Baffin Bay suggest differential movement patterns in response to the dynamics of sea ice.

Skin cancer texture analysis of OCT images based on Haralick, fractal dimension, Markov random field features, and the complex directional field features

NASA Astrophysics Data System (ADS)

Raupov, Dmitry S.; Myakinin, Oleg O.; Bratchenko, Ivan A.; Zakharov, Valery P.; Khramov, Alexander G.

2016-10-01

In this paper, we propose a report about our examining of the validity of OCT in identifying changes using a skin cancer texture analysis compiled from Haralick texture features, fractal dimension, Markov random field method and the complex directional features from different tissues. Described features have been used to detect specific spatial characteristics, which can differentiate healthy tissue from diverse skin cancers in cross-section OCT images (B- and/or C-scans). In this work, we used an interval type-II fuzzy anisotropic diffusion algorithm for speckle noise reduction in OCT images. The Haralick texture features as contrast, correlation, energy, and homogeneity have been calculated in various directions. A box-counting method is performed to evaluate fractal dimension of skin probes. Markov random field have been used for the quality enhancing of the classifying. Additionally, we used the complex directional field calculated by the local gradient methodology to increase of the assessment quality of the diagnosis method. Our results demonstrate that these texture features may present helpful information to discriminate tumor from healthy tissue. The experimental data set contains 488 OCT-images with normal skin and tumors as Basal Cell Carcinoma (BCC), Malignant Melanoma (MM) and Nevus. All images were acquired from our laboratory SD-OCT setup based on broadband light source, delivering an output power of 20 mW at the central wavelength of 840 nm with a bandwidth of 25 nm. We obtained sensitivity about 97% and specificity about 73% for a task of discrimination between MM and Nevus.

Definition of fractal topography to essential understanding of scale-invariance

NASA Astrophysics Data System (ADS)

Jin, Yi; Wu, Ying; Li, Hui; Zhao, Mengyu; Pan, Jienan

2017-04-01

Fractal behavior is scale-invariant and widely characterized by fractal dimension. However, the cor-respondence between them is that fractal behavior uniquely determines a fractal dimension while a fractal dimension can be related to many possible fractal behaviors. Therefore, fractal behavior is independent of the fractal generator and its geometries, spatial pattern, and statistical properties in addition to scale. To mathematically describe fractal behavior, we propose a novel concept of fractal topography defined by two scale-invariant parameters, scaling lacunarity (P) and scaling coverage (F). The scaling lacunarity is defined as the scale ratio between two successive fractal generators, whereas the scaling coverage is defined as the number ratio between them. Consequently, a strictly scale-invariant definition for self-similar fractals can be derived as D = log F /log P. To reflect the direction-dependence of fractal behaviors, we introduce another parameter Hxy, a general Hurst exponent, which is analytically expressed by Hxy = log Px/log Py where Px and Py are the scaling lacunarities in the x and y directions, respectively. Thus, a unified definition of fractal dimension is proposed for arbitrary self-similar and self-affine fractals by averaging the fractal dimensions of all directions in a d-dimensional space, which . Our definitions provide a theoretical, mechanistic basis for understanding the essentials of the scale-invariant property that reduces the complexity of modeling fractals.

Integrated quantitative fractal polarimetric analysis of monolayer lung cancer cells

NASA Astrophysics Data System (ADS)

Shrestha, Suman; Zhang, Lin; Quang, Tri; Farrahi, Tannaz; Narayan, Chaya; Deshpande, Aditi; Na, Ying; Blinzler, Adam; Ma, Junyu; Liu, Bo; Giakos, George C.

2014-05-01

Digital diagnostic pathology has become one of the most valuable and convenient advancements in technology over the past years. It allows us to acquire, store and analyze pathological information from the images of histological and immunohistochemical glass slides which are scanned to create digital slides. In this study, efficient fractal, wavelet-based polarimetric techniques for histological analysis of monolayer lung cancer cells will be introduced and different monolayer cancer lines will be studied. The outcome of this study indicates that application of fractal, wavelet polarimetric principles towards the analysis of squamous carcinoma and adenocarcinoma cancer cell lines may be proved extremely useful in discriminating among healthy and lung cancer cells as well as differentiating among different lung cancer cells.

An Approach to Study Elastic Vibrations of Fractal Cylinders

NASA Astrophysics Data System (ADS)

Steinberg, Lev; Zepeda, Mario

2016-11-01

This paper presents our study of dynamics of fractal solids. Concepts of fractal continuum and time had been used in definitions of a fractal body deformation and motion, formulation of conservation of mass, balance of momentum, and constitutive relationships. A linearized model, which was written in terms of fractal time and spatial derivatives, has been employed to study the elastic vibrations of fractal circular cylinders. Fractal differential equations of torsional, longitudinal and transverse fractal wave equations have been obtained and solution properties such as size and time dependence have been revealed.

Fractal dimension of turbulent black holes

NASA Astrophysics Data System (ADS)

Westernacher-Schneider, John Ryan

2017-11-01

We present measurements of the fractal dimension of a turbulent asymptotically anti-de Sitter black brane reconstructed from simulated boundary fluid data at the perfect fluid order using the fluid-gravity duality. We argue that the boundary fluid energy spectrum scaling as E (k )˜k-2 is a more natural setting for the fluid-gravity duality than the Kraichnan-Kolmogorov scaling of E (k )˜k-5 /3, but we obtain fractal dimensions D for spatial sections of the horizon H ∩Σ in both cases: D =2.584 (1 ) and D =2.645 (4 ), respectively. These results are consistent with the upper bound of D =3 , thereby resolving the tension with the recent claim in Adams et al. [Phys. Rev. Lett. 112, 151602 (2014), 10.1103/PhysRevLett.112.151602] that D =3 +1 /3 . We offer a critical examination of the calculation which led to their result, and show that their proposed definition of the fractal dimension performs poorly as a fractal dimension estimator on one-dimensional curves with known fractal dimension. Finally, we describe how to define and in principle calculate the fractal dimension of spatial sections of the horizon H ∩Σ in a covariant manner, and we speculate on assigning a "bootstrapped" value of fractal dimension to the entire horizon H when it is in a statistically quasisteady turbulent state.

Fractal mechanisms in the electrophysiology of the heart

NASA Technical Reports Server (NTRS)

Goldberger, A. L.

1992-01-01

The mathematical concept of fractals provides insights into complex anatomic branching structures that lack a characteristic (single) length scale, and certain complex physiologic processes, such as heart rate regulation, that lack a single time scale. Heart rate control is perturbed by alterations in neuro-autonomic function in a number of important clinical syndromes, including sudden cardiac death, congestive failure, cocaine intoxication, fetal distress, space sickness and physiologic aging. These conditions are associated with a loss of the normal fractal complexity of interbeat interval dynamics. Such changes, which may not be detectable using conventional statistics, can be quantified using new methods derived from "chaos theory.".

Fractal dimension of microbead assemblies used for protein detection.

PubMed

Hecht, Ariel; Commiskey, Patrick; Lazaridis, Filippos; Argyrakis, Panos; Kopelman, Raoul

2014-11-10

We use fractal analysis to calculate the protein concentration in a rotating magnetic assembly of microbeads of size 1 μm, which has optimized parameters of sedimentation, binding sites and magnetic volume. We utilize the original Forrest-Witten method, but due to the relatively small number of bead particles, which is of the order of 500, we use a large number of origins and also a large number of algorithm iterations. We find a value of the fractal dimension in the range 1.70-1.90, as a function of the thrombin concentration, which plays the role of binding the microbeads together. This is in good agreement with previous results from magnetorotation studies. The calculation of the fractal dimension using multiple points of reference can be used for any assembly with a relatively small number of particles. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Fractal Analysis of Rock Joint Profiles

NASA Astrophysics Data System (ADS)

Audy, Ondřej; Ficker, Tomáš

2017-10-01

Surface reliefs of rock joints are analyzed in geotechnics when shear strength of rocky slopes is estimated. The rock joint profiles actually are self-affine fractal curves and computations of their fractal dimensions require special methods. Many papers devoted to the fractal properties of these profiles were published in the past but only a few of those papers employed a convenient computational method that would have guaranteed a sound value of that dimension. As a consequence, anomalously low dimensions were presented. This contribution deals with two computational modifications that lead to sound fractal dimensions of the self-affine rock joint profiles. These are the modified box-counting method and the modified yard-stick method sometimes called the compass method. Both these methods are frequently applied to self-similar fractal curves but the self-affine profile curves due to their self-affine nature require modified computational procedures implemented in computer programs.

Fractal Properties of Some Machined Surfaces

NASA Astrophysics Data System (ADS)

Thomas, T. R.; Rosén, B.-G.

Many surface profiles are self-affine fractals defined by fractal dimension D and topothesy Λ. Traditionally these parameters are derived laboriously from the slope and intercept of the profile's structure function. Recently a quicker and more convenient derivation from standard roughness parameters has been suggested. Based on this derivation, it is shown that D and Λ depend on two dimensionless numbers: the ratio of the mean peak spacing to the rms roughness, and the ratio of the mean local peak spacing to the sampling interval. Using this approach, values of D and Λ are calculated for 125 profiles produced by polishing, plateau honing and various single-point machining processes. Different processes are shown to occupy different regions in D-Λ space, and polished surfaces show a relationship between D and Λ which is independent of the surface material.

Aesthetic Responses to Exact Fractals Driven by Physical Complexity

PubMed Central

Bies, Alexander J.; Blanc-Goldhammer, Daryn R.; Boydston, Cooper R.; Taylor, Richard P.; Sereno, Margaret E.

2016-01-01

Fractals are physically complex due to their repetition of patterns at multiple size scales. Whereas the statistical characteristics of the patterns repeat for fractals found in natural objects, computers can generate patterns that repeat exactly. Are these exact fractals processed differently, visually and aesthetically, than their statistical counterparts? We investigated the human aesthetic response to the complexity of exact fractals by manipulating fractal dimensionality, symmetry, recursion, and the number of segments in the generator. Across two studies, a variety of fractal patterns were visually presented to human participants to determine the typical response to exact fractals. In the first study, we found that preference ratings for exact midpoint displacement fractals can be described by a linear trend with preference increasing as fractal dimension increases. For the majority of individuals, preference increased with dimension. We replicated these results for other exact fractal patterns in a second study. In the second study, we also tested the effects of symmetry and recursion by presenting asymmetric dragon fractals, symmetric dragon fractals, and Sierpinski carpets and Koch snowflakes, which have radial and mirror symmetry. We found a strong interaction among recursion, symmetry and fractal dimension. Specifically, at low levels of recursion, the presence of symmetry was enough to drive high preference ratings for patterns with moderate to high levels of fractal dimension. Most individuals required a much higher level of recursion to recover this level of preference in a pattern that lacked mirror or radial symmetry, while others were less discriminating. This suggests that exact fractals are processed differently than their statistical counterparts. We propose a set of four factors that influence complexity and preference judgments in fractals that may extend to other patterns: fractal dimension, recursion, symmetry and the number of segments in a

Diagnostics of multi-fractality of magnetized plasma inside coronal holes and quiet sun areas

NASA Astrophysics Data System (ADS)

Abramenko, Valentyna

Turbulent and multi-fractal properties of magnetized plasma in solar Coronal Holes (CHs) and Quiet Sun (QS) photosphere were explored using high-resolution magnetograms measured with the New Solar Telescope (NST) at the Big Bear Solar Observatory (BBSO, USA), Hinode/SOT and SDO/HMI instruments. Distribution functions of size and magnetic flux measured for small-scale magnetic elements follow the log-normal law, which implies multi-fractal organization of the magnetic field and the absence of a unique power law for all scales. The magnetograms show multi-fractality in CHs on scales 400 - 10000 km, which becomes better pronounced as the spatial resolution of data improves. Photospheric granulation measured with NST exhibits multi-fractal properties on very small scales of 50 - 600 km. While multi-fractal nature of solar active regions is well known, newly established multi-fractality of weakest magnetic fields on the solar surface, i.e., in CHs and QS, leads us to a conclusion that the entire variety of solar magnetic fields is generated by a unique nonlinear dynamical process.

Fractal analysis for assessing tumour grade in microscopic images of breast tissue

NASA Astrophysics Data System (ADS)

Tambasco, Mauro; Costello, Meghan; Newcomb, Chris; Magliocco, Anthony M.

2007-03-01

In 2006, breast cancer is expected to continue as the leading form of cancer diagnosed in women, and the second leading cause of cancer mortality in this group. A method that has proven useful for guiding the choice of treatment strategy is the assessment of histological tumor grade. The grading is based upon the mitosis count, nuclear pleomorphism, and tubular formation, and is known to be subject to inter-observer variability. Since cancer grade is one of the most significant predictors of prognosis, errors in grading can affect patient management and outcome. Hence, there is a need to develop a breast cancer-grading tool that is minimally operator dependent to reduce variability associated with the current grading system, and thereby reduce uncertainty that may impact patient outcome. In this work, we explored the potential of a computer-based approach using fractal analysis as a quantitative measure of cancer grade for breast specimens. More specifically, we developed and optimized computational tools to compute the fractal dimension of low- versus high-grade breast sections and found them to be significantly different, 1.3+/-0.10 versus 1.49+/-0.10, respectively (Kolmogorov-Smirnov test, p<0.001). These results indicate that fractal dimension (a measure of morphologic complexity) may be a useful tool for demarcating low- versus high-grade cancer specimens, and has potential as an objective measure of breast cancer grade. Such prognostic value could provide more sensitive and specific information that would reduce inter-observer variability by aiding the pathologist in grading cancers.

Fractal Patterns and Chaos Games

ERIC Educational Resources Information Center

Devaney, Robert L.

2004-01-01

Teachers incorporate the chaos game and the concept of a fractal into various areas of the algebra and geometry curriculum. The chaos game approach to fractals provides teachers with an opportunity to help students comprehend the geometry of affine transformations.

Band structures in fractal grading porous phononic crystals

NASA Astrophysics Data System (ADS)

Wang, Kai; Liu, Ying; Liang, Tianshu; Wang, Bin

2018-05-01

In this paper, a new grading porous structure is introduced based on a Sierpinski triangle routine, and wave propagation in this fractal grading porous phononic crystal is investigated. The influences of fractal hierarchy and porosity on the band structures in fractal graidng porous phononic crystals are clarified. Vibration modes of unit cell at absolute band gap edges are given to manifest formation mechanism of absolute band gaps. The results show that absolute band gaps are easy to form in fractal structures comparatively to the normal ones with the same porosity. Structures with higher fractal hierarchies benefit multiple wider absolute band gaps. This work provides useful guidance in design of fractal porous phononic crystals.

Fractal Theory for Permeability Prediction, Venezuelan and USA Wells

NASA Astrophysics Data System (ADS)

Aldana, Milagrosa; Altamiranda, Dignorah; Cabrera, Ana

2014-05-01

Inferring petrophysical parameters such as permeability, porosity, water saturation, capillary pressure, etc, from the analysis of well logs or other available core data has always been of critical importance in the oil industry. Permeability in particular, which is considered to be a complex parameter, has been inferred using both empirical and theoretical techniques. The main goal of this work is to predict permeability values on different wells using Fractal Theory, based on a method proposed by Pape et al. (1999). This approach uses the relationship between permeability and the geometric form of the pore space of the rock. This method is based on the modified equation of Kozeny-Carman and a fractal pattern, which allows determining permeability as a function of the cementation exponent, porosity and the fractal dimension. Data from wells located in Venezuela and the United States of America are analyzed. Employing data of porosity and permeability obtained from core samples, and applying the Fractal Theory method, we calculated the prediction equations for each well. At the beginning, this was achieved by training with 50% of the data available for each well. Afterwards, these equations were tested inferring over 100% of the data to analyze possible trends in their distribution. This procedure gave excellent results in all the wells in spite of their geographic distance, generating permeability models with the potential to accurately predict permeability logs in the remaining parts of the well for which there are no core samples, using even porority logs. Additionally, empirical models were used to determine permeability and the results were compared with those obtained by applying the fractal method. The results indicated that, although there are empirical equations that give a proper adjustment, the prediction results obtained using fractal theory give a better fit to the core reference data.

Function representation with circle inversion map systems

NASA Astrophysics Data System (ADS)

Boreland, Bryson; Kunze, Herb

2017-01-01

The fractals literature develops the now well-known concept of local iterated function systems (using affine maps) with grey-level maps (LIFSM) as an approach to function representation in terms of the associated fixed point of the so-called fractal transform. While originally explored as a method to achieve signal (and 2-D image) compression, more recent work has explored various aspects of signal and image processing using this machinery. In this paper, we develop a similar framework for function representation using circle inversion map systems. Given a circle C with centre õ and radius r, inversion with respect to C transforms the point p˜ to the point p˜', such that p˜ and p˜' lie on the same radial half-line from õ and d(õ, p˜)d(õ, p˜') = r2, where d is Euclidean distance. We demonstrate the results with an example.

Fractal analysis of INSAR and correlation with graph-cut based image registration for coastline deformation analysis: post seismic hazard assessment of the 2011 Tohoku earthquake region

NASA Astrophysics Data System (ADS)

Dutta, P. K.; Mishra, O. P.

2012-04-01

Satellite imagery for 2011 earthquake off the Pacific coast of Tohoku has provided an opportunity to conduct image transformation analyses by employing multi-temporal images retrieval techniques. In this study, we used a new image segmentation algorithm to image coastline deformation by adopting graph cut energy minimization framework. Comprehensive analysis of available INSAR images using coastline deformation analysis helped extract disaster information of the affected region of the 2011 Tohoku tsunamigenic earthquake source zone. We attempted to correlate fractal analysis of seismic clustering behavior with image processing analogies and our observations suggest that increase in fractal dimension distribution is associated with clustering of events that may determine the level of devastation of the region. The implementation of graph cut based image registration technique helps us to detect the devastation across the coastline of Tohoku through change of intensity of pixels that carries out regional segmentation for the change in coastal boundary after the tsunami. The study applies transformation parameters on remotely sensed images by manually segmenting the image to recovering translation parameter from two images that differ by rotation. Based on the satellite image analysis through image segmentation, it is found that the area of 0.997 sq km for the Honshu region was a maximum damage zone localized in the coastal belt of NE Japan forearc region. The analysis helps infer using matlab that the proposed graph cut algorithm is robust and more accurate than other image registration methods. The analysis shows that the method can give a realistic estimate for recovered deformation fields in pixels corresponding to coastline change which may help formulate the strategy for assessment during post disaster need assessment scenario for the coastal belts associated with damages due to strong shaking and tsunamis in the world under disaster risk mitigation programs.

Fractal Signals & Space-Time Cartoons

NASA Astrophysics Data System (ADS)

Oetama, H. C. Jakob; Maksoed, W. H.

2016-03-01

In ``Theory of Scale Relativity'', 1991- L. Nottale states whereas ``scale relativity is a geometrical & fractal space-time theory''. It took in comparisons to ``a unified, wavelet based framework for efficiently synthetizing, analyzing ∖7 processing several broad classes of fractal signals''-Gregory W. Wornell:``Signal Processing with Fractals'', 1995. Furthers, in Fig 1.1. a simple waveform from statistically scale-invariant random process [ibid.,h 3 ]. Accompanying RLE Technical Report 566 ``Synthesis, Analysis & Processing of Fractal Signals'' as well as from Wornell, Oct 1991 herewith intended to deducts =a Δt + (1 - β Δ t) ...in Petersen, et.al: ``Scale invariant properties of public debt growth'',2010 h. 38006p2 to [1/{1- (2 α (λ) /3 π) ln (λ/r)}depicts in Laurent Nottale,1991, h 24. Acknowledgment devotes to theLates HE. Mr. BrigadierGeneral-TNI[rtd].Prof. Ir. HANDOJO.

Exterior dimension of fat fractals

NASA Technical Reports Server (NTRS)

Grebogi, C.; Mcdonald, S. W.; Ott, E.; Yorke, J. A.

1985-01-01

Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which is called the exterior dimension. In addition, it is shown that the exterior dimension is related to the 'uncertainty exponent' previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.

Power dissipation in fractal AC circuits

NASA Astrophysics Data System (ADS)

Chen, Joe P.; Rogers, Luke G.; Anderson, Loren; Andrews, Ulysses; Brzoska, Antoni; Coffey, Aubrey; Davis, Hannah; Fisher, Lee; Hansalik, Madeline; Loew, Stephen; Teplyaev, Alexander

2017-08-01

We extend Feynman’s analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Using (weak) self-similarity of our fractal structures, we provide algorithms for studying the equilibrium distribution of energy on these circuits. This extends the analysis of self-similar resistance networks introduced by Fukushima, Kigami, Kusuoka, and more recently studied by Strichartz et al.

Synthetic Minority Oversampling Technique and Fractal Dimension for Identifying Multiple Sclerosis

NASA Astrophysics Data System (ADS)

Zhang, Yu-Dong; Zhang, Yin; Phillips, Preetha; Dong, Zhengchao; Wang, Shuihua

Multiple sclerosis (MS) is a severe brain disease. Early detection can provide timely treatment. Fractal dimension can provide statistical index of pattern changes with scale at a given brain image. In this study, our team used susceptibility weighted imaging technique to obtain 676 MS slices and 880 healthy slices. We used synthetic minority oversampling technique to process the unbalanced dataset. Then, we used Canny edge detector to extract distinguishing edges. The Minkowski-Bouligand dimension was a fractal dimension estimation method and used to extract features from edges. Single hidden layer neural network was used as the classifier. Finally, we proposed a three-segment representation biogeography-based optimization to train the classifier. Our method achieved a sensitivity of 97.78±1.29%, a specificity of 97.82±1.60% and an accuracy of 97.80±1.40%. The proposed method is superior to seven state-of-the-art methods in terms of sensitivity and accuracy.

Poiseuille equation for steady flow of fractal fluid

NASA Astrophysics Data System (ADS)

Tarasov, Vasily E.

2016-07-01

Fractal fluid is considered in the framework of continuous models with noninteger dimensional spaces (NIDS). A recently proposed vector calculus in NIDS is used to get a description of fractal fluid flow in pipes with circular cross-sections. The Navier-Stokes equations of fractal incompressible viscous fluids are used to derive a generalization of the Poiseuille equation of steady flow of fractal media in pipe.

Excitation of Terahertz Charge Transfer Plasmons in Metallic Fractal Structures

NASA Astrophysics Data System (ADS)

Ahmadivand, Arash; Gerislioglu, Burak; Sinha, Raju; Vabbina, Phani Kiran; Karabiyik, Mustafa; Pala, Nezih

2017-08-01

There have been extensive researches on terahertz (THz) plasmonic structures supporting resonant modes to demonstrate nano and microscale devices with high efficiency and responsivity as well as frequency selectivity. Here, using antisymmetric plasmonic fractal Y-shaped (FYS) structures as building blocks, we introduce a highly tunable four-member fractal assembly to support charge transfer plasmons (CTPs) and classical dipolar resonant modes with significant absorption cross section in the THz domain. We first present that the unique geometrical nature of the FYS system and corresponding spectral response allow for supporting intensified dipolar plasmonic modes under polarised light exposure in a standalone structure. In addition to classical dipolar mode, for the very first time, we demonstrated CTPs in the THz domain due to the direct shuttling of the charges across the metallic fractal microantenna which led to sharp resonant absorption peaks. Using both numerical and experimental studies, we have investigated and confirmed the excitation of the CTP modes and highly tunable spectral response of the proposed plasmonic fractal structure. This understanding opens new and promising horizons for tightly integrated THz devices with high efficiency and functionality.

Down syndrome's brain dynamics: analysis of fractality in resting state.

PubMed

Hemmati, Sahel; Ahmadlou, Mehran; Gharib, Masoud; Vameghi, Roshanak; Sajedi, Firoozeh

2013-08-01

To the best knowledge of the authors there is no study on nonlinear brain dynamics of down syndrome (DS) patients, whereas brain is a highly complex and nonlinear system. In this study, fractal dimension of EEG, as a key characteristic of brain dynamics, showing irregularity and complexity of brain dynamics, was used for evaluation of the dynamical changes in the DS brain. The results showed higher fractality of the DS brain in almost all regions compared to the normal brain, which indicates less centrality and higher irregular or random functioning of the DS brain regions. Also, laterality analysis of the frontal lobe showed that the normal brain had a right frontal laterality of complexity whereas the DS brain had an inverse pattern (left frontal laterality). Furthermore, the high accuracy of 95.8 % obtained by enhanced probabilistic neural network classifier showed the potential of nonlinear dynamic analysis of the brain for diagnosis of DS patients. Moreover, the results showed that the higher EEG fractality in DS is associated with the higher fractality in the low frequencies (delta and theta), in broad regions of the brain, and the high frequencies (beta and gamma), majorly in the frontal regions.

Fractal Interrelationships in Field and Seismic Data

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wilson, T.H.; Dominic, Jovita; Halverson, Joel

1997-10-01

Size scaling interrelationships are evaluated in this study using a fractal model. Fractal models of several geologic variables are examined and include fracture patterns, reflection travel times, structural relief, drainage, topographic relief and active fault patterns. The fractal properties of structural relief inferred from seismic data and structural cross sections provide a quantitative means to characterize and compare complex structural patterns. Studies were conducted using seismic data from the Granny Creek oil field in the Appalachian Plateau. Previous studies of the field reveal that subtle detached structures present on the limb of a larger structure are associated with enhanced productionmore » from the field. Vertical increases of fractal dimension across the zone of detachment provide a measure of the extent to which detachment has occurred. The increases of fractal dimension are greatest in the more productive areas of the field. A result with equally important ramifications is that fracture systems do not appear to be intrinsically fractal as is often suggested in the literature. While examples of nearly identical patterns can be found at different scales supporting the idea of self-similarity, these examples are often taken from different areas and from different lithologies. Examination of fracture systems at different scales in the Valley and Ridge Province suggest that their distribution become increasingly sparse with scale reduction, and therefore are dissimilar or non-fractal. Box counting data in all cases failed to yield a fractal regime. The results obtained from this analysis bring into question the general applicability of reservoir simulations employing fractal models of fracture distribution. The same conclusions were obtained from the analysis of 1D fracture patterns such as those that might appear in a horizontal well.« less

Fractal characterization of fracture surfaces in concrete

USGS Publications Warehouse

Saouma, V.E.; Barton, C.C.; Gamaleldin, N.A.

1990-01-01

Fractal geometry is used to characterize the roughness of cracked concrete surfaces through a specially built profilometer, and the fractal dimension is subsequently correlated to the fracture toughness and direction of crack propagation. Preliminary results indicate that the fracture surface is indeed fractal over two orders of magnitudes with a dimension of approximately 1.20. ?? 1990.

Fractal analysis of heart rate dynamics as a predictor of mortality in patients with depressed left ventricular function after acute myocardial infarction. TRACE Investigators. TRAndolapril Cardiac Evaluation

NASA Technical Reports Server (NTRS)

Makikallio, T. H.; Hoiber, S.; Kober, L.; Torp-Pedersen, C.; Peng, C. K.; Goldberger, A. L.; Huikuri, H. V.

1999-01-01

A number of new methods have been recently developed to quantify complex heart rate (HR) dynamics based on nonlinear and fractal analysis, but their value in risk stratification has not been evaluated. This study was designed to determine whether selected new dynamic analysis methods of HR variability predict mortality in patients with depressed left ventricular (LV) function after acute myocardial infarction (AMI). Traditional time- and frequency-domain HR variability indexes along with short-term fractal-like correlation properties of RR intervals (exponent alpha) and power-law scaling (exponent beta) were studied in 159 patients with depressed LV function (ejection fraction <35%) after an AMI. By the end of 4-year follow-up, 72 patients (45%) had died and 87 (55%) were still alive. Short-term scaling exponent alpha (1.07 +/- 0.26 vs 0.90 +/- 0.26, p <0.001) and power-law slope beta (-1.35 +/- 0.23 vs -1.44 +/- 0.25, p <0.05) differed between survivors and those who died, but none of the traditional HR variability measures differed between these groups. Among all analyzed variables, reduced scaling exponent alpha (<0.85) was the best univariable predictor of mortality (relative risk 3.17, 95% confidence interval 1.96 to 5.15, p <0.0001), with positive and negative predictive accuracies of 65% and 86%, respectively. In the multivariable Cox proportional hazards analysis, mortality was independently predicted by the reduced exponent alpha (p <0.001) after adjustment for several clinical variables and LV function. A short-term fractal-like scaling exponent was the most powerful HR variability index in predicting mortality in patients with depressed LV function. Reduction in fractal correlation properties implies more random short-term HR dynamics in patients with increased risk of death after AMI.

Fractal Dimension Analysis of Transient Visual Evoked Potentials: Optimisation and Applications.

PubMed

Boon, Mei Ying; Henry, Bruce Ian; Chu, Byoung Sun; Basahi, Nour; Suttle, Catherine May; Luu, Chi; Leung, Harry; Hing, Stephen

2016-01-01

The visual evoked potential (VEP) provides a time series signal response to an external visual stimulus at the location of the visual cortex. The major VEP signal components, peak latency and amplitude, may be affected by disease processes. Additionally, the VEP contains fine detailed and non-periodic structure, of presently unclear relevance to normal function, which may be quantified using the fractal dimension. The purpose of this study is to provide a systematic investigation of the key parameters in the measurement of the fractal dimension of VEPs, to develop an optimal analysis protocol for application. VEP time series were mathematically transformed using delay time, τ, and embedding dimension, m, parameters. The fractal dimension of the transformed data was obtained from a scaling analysis based on straight line fits to the numbers of pairs of points with separation less than r versus log(r) in the transformed space. Optimal τ, m, and scaling analysis were obtained by comparing the consistency of results using different sampling frequencies. The optimised method was then piloted on samples of normal and abnormal VEPs. Consistent fractal dimension estimates were obtained using τ = 4 ms, designating the fractal dimension = D2 of the time series based on embedding dimension m = 7 (for 3606 Hz and 5000 Hz), m = 6 (for 1803 Hz) and m = 5 (for 1000Hz), and estimating D2 for each embedding dimension as the steepest slope of the linear scaling region in the plot of log(C(r)) vs log(r) provided the scaling region occurred within the middle third of the plot. Piloting revealed that fractal dimensions were higher from the sampled abnormal than normal achromatic VEPs in adults (p = 0.02). Variances of fractal dimension were higher from the abnormal than normal chromatic VEPs in children (p = 0.01). A useful analysis protocol to assess the fractal dimension of transformed VEPs has been developed.

The generalized 20/80 law using probabilistic fractals applied to petroleum field size

USGS Publications Warehouse

Crovelli, R.A.

1995-01-01

Fractal properties of the Pareto probability distribution are used to generalize "the 20/80 law." The 20/80 law is a heuristic law that has evolved over the years into the following rule of thumb for many populations: 20 percent of the population accounts for 80 percent of the total value. The general p100/q100 law in probabilistic form is defined with q as a function of p, where p is the population proportion and q is the proportion of total value. Using the Pareto distribution, the p100/q100 law in fractal form is derived with the parameter q being a fractal, where q unexpectedly possesses the scale invariance property. The 20/80 law is a special case of the p100/q100 law in fractal form. The p100/q100 law in fractal form is applied to petroleum fieldsize data to obtain p and q such that p100% of the oil fields greater than any specified scale or size in a geologic play account for q100% of the total oil of the fields. The theoretical percentages of total resources of oil using the fractal q are extremely close to the empirical percentages from the data using the statistic q. Also, the empirical scale invariance property of the statistic q for the petroleum fieldsize data is in excellent agreement with the theoretical scale invariance property of the fractal q. ?? 1995 Oxford University Press.

Variability of fractal dimension of solar radio flux

NASA Astrophysics Data System (ADS)

Bhatt, Hitaishi; Sharma, Som Kumar; Trivedi, Rupal; Vats, Hari Om

2018-04-01

In the present communication, the variation of the fractal dimension of solar radio flux is reported. Solar radio flux observations on a day to day basis at 410, 1415, 2695, 4995, and 8800 MHz are used in this study. The data were recorded at Learmonth Solar Observatory, Australia from 1988 to 2009 covering an epoch of two solar activity cycles (22 yr). The fractal dimension is calculated for the listed frequencies for this period. The fractal dimension, being a measure of randomness, represents variability of solar radio flux at shorter time-scales. The contour plot of fractal dimension on a grid of years versus radio frequency suggests high correlation with solar activity. Fractal dimension increases with increasing frequency suggests randomness increases towards the inner corona. This study also shows that the low frequency is more affected by solar activity (at low frequency fractal dimension difference between solar maximum and solar minimum is 0.42) whereas, the higher frequency is less affected by solar activity (here fractal dimension difference between solar maximum and solar minimum is 0.07). A good positive correlation is found between fractal dimension averaged over all frequencies and yearly averaged sunspot number (Pearson's coefficient is 0.87).

Topological characterization of antireflective and hydrophobic rough surfaces: are random process theory and fractal modeling applicable?

NASA Astrophysics Data System (ADS)

Borri, Claudia; Paggi, Marco

2015-02-01

The random process theory (RPT) has been widely applied to predict the joint probability distribution functions (PDFs) of asperity heights and curvatures of rough surfaces. A check of the predictions of RPT against the actual statistics of numerically generated random fractal surfaces and of real rough surfaces has been only partially undertaken. The present experimental and numerical study provides a deep critical comparison on this matter, providing some insight into the capabilities and limitations in applying RPT and fractal modeling to antireflective and hydrophobic rough surfaces, two important types of textured surfaces. A multi-resolution experimental campaign using a confocal profilometer with different lenses is carried out and a comprehensive software for the statistical description of rough surfaces is developed. It is found that the topology of the analyzed textured surfaces cannot be fully described according to RPT and fractal modeling. The following complexities emerge: (i) the presence of cut-offs or bi-fractality in the power-law power-spectral density (PSD) functions; (ii) a more pronounced shift of the PSD by changing resolution as compared to what was expected from fractal modeling; (iii) inaccuracy of the RPT in describing the joint PDFs of asperity heights and curvatures of textured surfaces; (iv) lack of resolution-invariance of joint PDFs of textured surfaces in case of special surface treatments, not accounted for by fractal modeling.

Fractal fluctuations in gaze speed visual search.

PubMed

Stephen, Damian G; Anastas, Jason

2011-04-01

Visual search involves a subtle coordination of visual memory and lower-order perceptual mechanisms. Specifically, the fluctuations in gaze may provide support for visual search above and beyond what may be attributed to memory. Prior research indicates that gaze during search exhibits fractal fluctuations, which allow for a wide sampling of the field of view. Fractal fluctuations constitute a case of fast diffusion that may provide an advantage in exploration. We present reanalyses of eye-tracking data collected by Stephen and Mirman (Cognition, 115, 154-165, 2010) for single-feature and conjunction search tasks. Fluctuations in gaze during these search tasks were indeed fractal. Furthermore, the degree of fractality predicted decreases in reaction time on a trial-by-trial basis. We propose that fractality may play a key role in explaining the efficacy of perceptual exploration.

A system framework of inter-enterprise machining quality control based on fractal theory

NASA Astrophysics Data System (ADS)

Zhao, Liping; Qin, Yongtao; Yao, Yiyong; Yan, Peng

2014-03-01

In order to meet the quality control requirement of dynamic and complicated product machining processes among enterprises, a system framework of inter-enterprise machining quality control based on fractal was proposed. In this system framework, the fractal-specific characteristic of inter-enterprise machining quality control function was analysed, and the model of inter-enterprise machining quality control was constructed by the nature of fractal structures. Furthermore, the goal-driven strategy of inter-enterprise quality control and the dynamic organisation strategy of inter-enterprise quality improvement were constructed by the characteristic analysis on this model. In addition, the architecture of inter-enterprise machining quality control based on fractal was established by means of Web service. Finally, a case study for application was presented. The result showed that the proposed method was available, and could provide guidance for quality control and support for product reliability in inter-enterprise machining processes.

Interfacial contact stiffness of fractal rough surfaces.

PubMed

Zhang, Dayi; Xia, Ying; Scarpa, Fabrizio; Hong, Jie; Ma, Yanhong

2017-10-09

In this work we describe a theoretical model that predicts the interfacial contact stiffness of fractal rough surfaces by considering the effects of elastic and plastic deformations of the fractal asperities. We also develop an original test rig that simulates dovetail joints for turbo machinery blades, which can fine tune the normal contact load existing between the contacting surfaces of the blade root. The interfacial contact stiffness is obtained through an inverse identification method in which finite element simulations are fitted to the experimental results. Excellent agreement is observed between the contact stiffness predicted by the theoretical model and by the analogous experimental results. We demonstrate that the contact stiffness is a power law function of the normal contact load with an exponent α within the whole range of fractal dimension D(1 < D < 2). We also show that for 1 < D < 1.5 the Pohrt-Popov behavior (α = 1/(3 - D)) is valid, however for 1.5 < D < 2, the exponent α is different and equal to 2(D - 1)/D. The diversity between the model developed in the work and the Pohrt-Popov one is explained in detail.

A fractal nature for polymerized laminin.

PubMed

Hochman-Mendez, Camila; Cantini, Marco; Moratal, David; Salmeron-Sanchez, Manuel; Coelho-Sampaio, Tatiana

2014-01-01

Polylaminin (polyLM) is a non-covalent acid-induced nano- and micro-structured polymer of the protein laminin displaying distinguished biological properties. Polylaminin stimulates neuritogenesis beyond the levels achieved by ordinary laminin and has been shown to promote axonal regeneration in animal models of spinal cord injury. Here we used confocal fluorescence microscopy (CFM), scanning electron microscopy (SEM) and atomic force microscopy (AFM) to characterize its three-dimensional structure. Renderization of confocal optical slices of immunostained polyLM revealed the aspect of a loose flocculated meshwork, which was homogeneously stained by the antibody. On the other hand, an ordinary matrix obtained upon adsorption of laminin in neutral pH (LM) was constituted of bulky protein aggregates whose interior was not accessible to the same anti-laminin antibody. SEM and AFM analyses revealed that the seed unit of polyLM was a flat polygon formed in solution whereas the seed structure of LM was highly heterogeneous, intercalating rod-like, spherical and thin spread lamellar deposits. As polyLM was visualized at progressively increasing magnifications, we observed that the morphology of the polymer was alike independently of the magnification used for the observation. A search for the Hausdorff dimension in images of the two matrices showed that polyLM, but not LM, presented fractal dimensions of 1.55, 1.62 and 1.70 after 1, 8 and 12 hours of adsorption, respectively. Data in the present work suggest that the intrinsic fractal nature of polymerized laminin can be the structural basis for the fractal-like organization of basement membranes in the neurogenic niches of the central nervous system.

Verifying the Dependence of Fractal Coefficients on Different Spatial Distributions

NASA Astrophysics Data System (ADS)

Gospodinov, Dragomir; Marekova, Elisaveta; Marinov, Alexander

2010-01-01

A fractal distribution requires that the number of objects larger than a specific size r has a power-law dependence on the size N(r) = C/rD∝r-D where D is the fractal dimension. Usually the correlation integral is calculated to estimate the correlation fractal dimension of epicentres. A `box-counting' procedure could also be applied giving the `capacity' fractal dimension. The fractal dimension can be an integer and then it is equivalent to a Euclidean dimension (it is zero of a point, one of a segment, of a square is two and of a cube is three). In general the fractal dimension is not an integer but a fractional dimension and there comes the origin of the term `fractal'. The use of a power-law to statistically describe a set of events or phenomena reveals the lack of a characteristic length scale, that is fractal objects are scale invariant. Scaling invariance and chaotic behavior constitute the base of a lot of natural hazards phenomena. Many studies of earthquakes reveal that their occurrence exhibits scale-invariant properties, so the fractal dimension can characterize them. It has first been confirmed that both aftershock rate decay in time and earthquake size distribution follow a power law. Recently many other earthquake distributions have been found to be scale-invariant. The spatial distribution of both regional seismicity and aftershocks show some fractal features. Earthquake spatial distributions are considered fractal, but indirectly. There are two possible models, which result in fractal earthquake distributions. The first model considers that a fractal distribution of faults leads to a fractal distribution of earthquakes, because each earthquake is characteristic of the fault on which it occurs. The second assumes that each fault has a fractal distribution of earthquakes. Observations strongly favour the first hypothesis. The fractal coefficients analysis provides some important advantages in examining earthquake spatial distribution, which are

Fractal Geometry Enables Classification of Different Lung Morphologies in a Model of Experimental Asthma

NASA Astrophysics Data System (ADS)

Obert, Martin; Hagner, Stefanie; Krombach, Gabriele A.; Inan, Selcuk; Renz, Harald

2015-06-01

Animal models represent the basis of our current understanding of the pathophysiology of asthma and are of central importance in the preclinical development of drug therapies. The characterization of irregular lung shapes is a major issue in radiological imaging of mice in these models. The aim of this study was to find out whether differences in lung morphology can be described by fractal geometry. Healthy and asthmatic mouse groups, before and after an acute asthma attack induced by methacholine, were studied. In vivo flat-panel-based high-resolution Computed Tomography (CT) was used for mice's thorax imaging. The digital image data of the mice's lungs were segmented from the surrounding tissue. After that, the lungs were divided by image gray-level thresholds into two additional subsets. One subset contained basically the air transporting bronchial system. The other subset corresponds mainly to the blood vessel system. We estimated the fractal dimension of all sets of the different mouse groups using the mass radius relation (mrr). We found that the air transporting subset of the bronchial lung tissue enables a complete and significant differentiation between all four mouse groups (mean D of control mice before methacholine treatment: 2.64 ± 0.06; after treatment: 2.76 ± 0.03; asthma mice before methacholine treatment: 2.37 ± 0.16; after treatment: 2.71 ± 0.03; p < 0.05). We conclude that the concept of fractal geometry allows a well-defined, quantitative numerical and objective differentiation of lung shapes — applicable most likely also in human asthma diagnostics.

a Fractal Network Model for Fractured Porous Media

NASA Astrophysics Data System (ADS)

Xu, Peng; Li, Cuihong; Qiu, Shuxia; Sasmito, Agus Pulung

2016-04-01

The transport properties and mechanisms of fractured porous media are very important for oil and gas reservoir engineering, hydraulics, environmental science, chemical engineering, etc. In this paper, a fractal dual-porosity model is developed to estimate the equivalent hydraulic properties of fractured porous media, where a fractal tree-like network model is used to characterize the fracture system according to its fractal scaling laws and topological structures. The analytical expressions for the effective permeability of fracture system and fractured porous media, tortuosity, fracture density and fraction are derived. The proposed fractal model has been validated by comparisons with available experimental data and numerical simulation. It has been shown that fractal dimensions for fracture length and aperture have significant effect on the equivalent hydraulic properties of fractured porous media. The effective permeability of fracture system can be increased with the increase of fractal dimensions for fracture length and aperture, while it can be remarkably lowered by introducing tortuosity at large branching angle. Also, a scaling law between the fracture density and fractal dimension for fracture length has been found, where the scaling exponent depends on the fracture number. The present fractal dual-porosity model may shed light on the transport physics of fractured porous media and provide theoretical basis for oil and gas exploitation, underground water, nuclear waste disposal and geothermal energy extraction as well as chemical engineering, etc.

The fractal nature of vacuum arc cathode spots

DOE Office of Scientific and Technical Information (OSTI.GOV)

Anders, Andre

2005-05-27

Cathode spot phenomena show many features of fractals, for example self-similar patterns in the emitted light and arc erosion traces. Although there have been hints on the fractal nature of cathode spots in the literature, the fractal approach to spot interpretation is underutilized. In this work, a brief review of spot properties is given, touching the differences between spot type 1 (on cathodes surfaces with dielectric layers) and spot type 2 (on metallic, clean surfaces) as well as the known spot fragment or cell structure. The basic properties of self-similarity, power laws, random colored noise, and fractals are introduced. Severalmore » points of evidence for the fractal nature of spots are provided. Specifically power laws are identified as signature of fractal properties, such as spectral power of noisy arc parameters (ion current, arc voltage, etc) obtained by fast Fourier transform. It is shown that fractal properties can be observed down to the cutoff by measurement resolution or occurrence of elementary steps in physical processes. Random walk models of cathode spot motion are well established: they go asymptotically to Brownian motion for infinitesimal step width. The power spectrum of the arc voltage noise falls as 1/f {sup 2}, where f is frequency, supporting a fractal spot model associated with Brownian motion.« less

Fractal dimension and nonlinear dynamical processes

NASA Astrophysics Data System (ADS)

McCarty, Robert C.; Lindley, John P.

1993-11-01

Mandelbrot, Falconer and others have demonstrated the existence of dimensionally invariant geometrical properties of non-linear dynamical processes known as fractals. Barnsley defines fractal geometry as an extension of classical geometry. Such an extension, however, is not mathematically trivial Of specific interest to those engaged in signal processing is the potential use of fractal geometry to facilitate the analysis of non-linear signal processes often referred to as non-linear time series. Fractal geometry has been used in the modeling of non- linear time series represented by radar signals in the presence of ground clutter or interference generated by spatially distributed reflections around the target or a radar system. It was recognized by Mandelbrot that the fractal geometries represented by man-made objects had different dimensions than the geometries of the familiar objects that abound in nature such as leaves, clouds, ferns, trees, etc. The invariant dimensional property of non-linear processes suggests that in the case of acoustic signals (active or passive) generated within a dispersive medium such as the ocean environment, there exists much rich structure that will aid in the detection and classification of various objects, man-made or natural, within the medium.

Fractality à la carte: a general particle aggregation model.

PubMed

Nicolás-Carlock, J R; Carrillo-Estrada, J L; Dossetti, V

2016-01-19

In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters' fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.

On the question of fractal packing structure in metallic glasses

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ding, Jun; Asta, Mark; Ritchie, Robert O.

2017-07-25

This work addresses the long-standing debate over fractal models of packing structure in metallic glasses (MGs). Through detailed fractal and percolation analyses of MG structures, derived from simulations spanning a range of compositions and quenching rates, we conclude that there is no fractal atomic-level structure associated with the packing of all atoms or solute-centered clusters. The results are in contradiction with conclusions derived from previous studies based on analyses of shifts in radial distribution function and structure factor peaks associated with volume changes induced by pressure and compositional variations. Here in this paper, the interpretation of such shifts is shownmore » to be challenged by the heterogeneous nature of MG structure and deformation at the atomic scale. Moreover, our analysis in the present work illustrates clearly the percolation theory applied to MGs, for example, the percolation threshold and characteristics of percolation clusters formed by subsets of atoms, which can have important consequences for structure–property relationships in these amorphous materials.« less

Determination of Uniaxial Compressive Strength of Ankara Agglomerate Considering Fractal Geometry of Blocks

NASA Astrophysics Data System (ADS)

Coskun, Aycan; Sonmez, Harun; Ercin Kasapoglu, K.; Ozge Dinc, S.; Celal Tunusluoglu, M.

2010-05-01

The uniaxial compressive strength (UCS) of rock material is a crucial parameter to be used for design stages of slopes, tunnels and foundations to be constructed in/on geological medium. However, preparation of high quality cores from geological mixtures or fragmented rocks such as melanges, fault rocks, coarse pyroclastic rocks, breccias and sheared serpentinites is often extremely difficult. According to the studies performed in literature, this type of geological materials may be grouped as welded and unwelded birmocks. Success of preparation of core samples from welded bimrocks is slightly better than unwelded ones. Therefore, some studies performed on the welded bimrocks to understand the mechanical behavior of geological mixture materials composed of stronger and weaker components (Gokceoglu, 2002; Sonmez et al., 2004; Sonmez et al., 2006; Kahraman, et al., 2008). The overall strength of bimrocks are generally depends on strength contrast between blocks and matrix; types and strength of matrix; type, size, strength, shape and orientation of blocks and volumetric block proportion. In previously proposed prediction models, while UCS of unwelded bimrocks may be determined by decreasing the UCS of matrix considering the volumetric block proportion, the welded ones can be predicted by considering both UCS of matrix and blocks together (Lindquist, 1994; Lindquist and Goodman, 1994; Sonmez et al., 2006 and Sonmez et al., 2009). However, there is a few attempts were performed about the effect of blocks shape and orientation on the strength of bimrock (Linqduist, 1994 and Kahraman, et al., 2008). In this study, Ankara agglomerate, which is composed of andesite blocks and surrounded weak tuff matrix, was selected as study material. Image analyses were performed on bottom, top and side faces of cores to identify volumetric block portions. In addition to the image analyses, andesite blocks on bottom, top and side faces were digitized for determination of fractal

Pitfalls in Fractal Time Series Analysis: fMRI BOLD as an Exemplary Case

PubMed Central

Eke, Andras; Herman, Peter; Sanganahalli, Basavaraju G.; Hyder, Fahmeed; Mukli, Peter; Nagy, Zoltan

2012-01-01

This article will be positioned on our previous work demonstrating the importance of adhering to a carefully selected set of criteria when choosing the suitable method from those available ensuring its adequate performance when applied to real temporal signals, such as fMRI BOLD, to evaluate one important facet of their behavior, fractality. Earlier, we have reviewed on a range of monofractal tools and evaluated their performance. Given the advance in the fractal field, in this article we will discuss the most widely used implementations of multifractal analyses, too. Our recommended flowchart for the fractal characterization of spontaneous, low frequency fluctuations in fMRI BOLD will be used as the framework for this article to make certain that it will provide a hands-on experience for the reader in handling the perplexed issues of fractal analysis. The reason why this particular signal modality and its fractal analysis has been chosen was due to its high impact on today’s neuroscience given it had powerfully emerged as a new way of interpreting the complex functioning of the brain (see “intrinsic activity”). The reader will first be presented with the basic concepts of mono and multifractal time series analyses, followed by some of the most relevant implementations, characterization by numerical approaches. The notion of the dichotomy of fractional Gaussian noise and fractional Brownian motion signal classes and their impact on fractal time series analyses will be thoroughly discussed as the central theme of our application strategy. Sources of pitfalls and way how to avoid them will be identified followed by a demonstration on fractal studies of fMRI BOLD taken from the literature and that of our own in an attempt to consolidate the best practice in fractal analysis of empirical fMRI BOLD signals mapped throughout the brain as an exemplary case of potentially wide interest. PMID:23227008

Fractal analysis of polyferric chloride-humic acid (PFC-HA) flocs in different topological spaces.

PubMed

Wang, Yili; Lu, Jia; Baiyu, Du; Shi, Baoyou; Wang, Dongsheng

2009-01-01

The fractal dimensions in different topological spaces of polyferric chloride-humic acid (PFC-HA) flocs, formed in flocculating different kinds of humic acids (HA) water at different initial pH (9.0, 7.0, 5.0) and PFC dosages, were calculated by effective density-maximum diameter, image analysis, and N2 absorption-desorption methods, respectively. The mass fractal dimensions (Df) of PFC-HA flocs were calculated by bi-logarithm relation of effective density with maximum diameter and Logan empirical equation. The Df value was more than 2.0 at initial pH of 7.0, which was 11% and 13% higher than those at pH 9.0 and 5.0, respectively, indicating the most compact flocs formed in flocculated HA water at initial pH of 7.0. The image analysis for those flocs indicates that after flocculating the HA water at initial pH greater than 7.0 with PFC flocculant, the fractal dimensions of D2 (logA vs. logdL) and D3 (logVsphere VS. logdL) of PFC-HA flocs decreased with the increase of PFC dosages, and PFC-HA flocs showed a gradually looser structure. At the optimum dosage of PFC, the D2 (logA vs. logdL) values of the flocs show 14%-43% difference with their corresponding Df, and they even had different tendency with the change of initial pH values. However, the D2 values of the flocs formed at three different initial pH in HA solution had a same tendency with the corresponding Dr. Based on fractal Frenkel-Halsey-Hill (FHH) adsorption and desorption equations, the pore surface fractal dimensions (Ds) for dried powders of PFC-HA flocs formed in HA water with initial pH 9.0 and 7.0 were all close to 2.9421, and the Ds values of flocs formed at initial pH 5.0 were less than 2.3746. It indicated that the pore surface fractal dimensions of PFC-HA flocs dried powder mainly show the irregularity from the mesopore-size distribution and marcopore-size distribution.

Modeling Fractal Structure of City-Size Distributions Using Correlation Functions

PubMed Central

Chen, Yanguang

2011-01-01

Zipf's law is one the most conspicuous empirical facts for cities, however, there is no convincing explanation for the scaling relation between rank and size and its scaling exponent. Using the idea from general fractals and scaling, I propose a dual competition hypothesis of city development to explain the value intervals and the special value, 1, of the power exponent. Zipf's law and Pareto's law can be mathematically transformed into one another, but represent different processes of urban evolution, respectively. Based on the Pareto distribution, a frequency correlation function can be constructed. By scaling analysis and multifractals spectrum, the parameter interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf distribution, a size correlation function can be built, and it is opposite to the first one. By the second correlation function and multifractals notion, the Pareto exponent interval is derived as [1, 2). Thus the process of urban evolution falls into two effects: one is the Pareto effect indicating city number increase (external complexity), and the other the Zipf effect indicating city size growth (internal complexity). Because of struggle of the two effects, the scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium with each other, the scaling exponent approaches 1. A series of mathematical experiments on hierarchical correlation are employed to verify the models and a conclusion can be drawn that if cities in a given region follow Zipf's law, the frequency and size correlations will follow the scaling law. This theory can be generalized to interpret the inverse power-law distributions in various fields of physical and social sciences. PMID:21949753

Electro-chemical manifestation of nanoplasmonics in fractal media

NASA Astrophysics Data System (ADS)

Baskin, Emmanuel; Iomin, Alexander

2013-06-01

Electrodynamics of composite materials with fractal geometry is studied in the framework of fractional calculus. This consideration establishes a link between fractal geometry of the media and fractional integrodifferentiation. The photoconductivity in the vicinity of the electrode-electrolyte fractal interface is studied. The methods of fractional calculus are employed to obtain an analytical expression for the giant local enhancement of the optical electric field inside the fractal composite structure at the condition of the surface plasmon excitation. This approach makes it possible to explain experimental data on photoconductivity in the nano-electrochemistry.

Effective degrees of freedom of a random walk on a fractal.

PubMed

Balankin, Alexander S

2015-12-01

We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν-dimensional space F(ν) equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν) and fractal dimensionalities is deduced. The intrinsic time of random walk in F(ν) is inferred. The Laplacian operator in F(ν) is constructed. This allows us to map physical problems on fractals into the corresponding problems in F(ν). In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.

Effective degrees of freedom of a random walk on a fractal

NASA Astrophysics Data System (ADS)

Balankin, Alexander S.

2015-12-01

We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν -dimensional space Fν equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν ) and fractal dimensionalities is deduced. The intrinsic time of random walk in Fν is inferred. The Laplacian operator in Fν is constructed. This allows us to map physical problems on fractals into the corresponding problems in Fν. In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.

Fractal dust constrains the collisional history of comets

NASA Astrophysics Data System (ADS)

Fulle, M.; Blum, J.

2017-07-01

The fractal dust particles observed by Rosetta cannot form in the physical conditions observed today in comet 67P/Churyumov-Gerasimenko (67P hereinafter), being instead consistent with models of the pristine dust aggregates coagulated in the solar nebula. Since bouncing collisions in the protoplanetary disc restructure fractals into compact aggregates (pebbles), the only way to preserve fractals in a comet is the gentle gravitational collapse of a mixture of pebbles and fractals, which must occur before their mutual collision speeds overcome ≈1 m s-1. This condition fixes the pebble radius to ≲1 cm, as confirmed by Comet Nucleus Infrared and Visible Analyser onboard Philae. Here, we show that the flux of fractal particles measured by Rosetta constrains the 67P nucleus in a random packing of cm-sized pebbles, with all the voids among them filled by fractal particles. This structure is inconsistent with any catastrophic collision, which would have compacted or dispersed most fractals, thus leaving empty most voids in the reassembled nucleus. Comets are less numerous than current estimates, as confirmed by lacking small craters on Pluto and Charon. Bilobate comets accreted at speeds <1 m s-1 from cometesimals born in the same disc stream.

Perceptual and Physiological Responses to Jackson Pollock's Fractals

PubMed Central

Taylor, Richard P.; Spehar, Branka; Van Donkelaar, Paul; Hagerhall, Caroline M.

2011-01-01

Fractals have been very successful in quantifying the visual complexity exhibited by many natural patterns, and have captured the imagination of scientists and artists alike. Our research has shown that the poured patterns of the American abstract painter Jackson Pollock are also fractal. This discovery raises an intriguing possibility – are the visual characteristics of fractals responsible for the long-term appeal of Pollock's work? To address this question, we have conducted 10 years of scientific investigation of human response to fractals and here we present, for the first time, a review of this research that examines the inter-relationship between the various results. The investigations include eye tracking, visual preference, skin conductance, and EEG measurement techniques. We discuss the artistic implications of the positive perceptual and physiological responses to fractal patterns. PMID:21734876

A Fractal Dimension Survey of Active Region Complexity

NASA Technical Reports Server (NTRS)

McAteer, R. T. James; Gallagher, Peter; Ireland, Jack

2005-01-01

A new approach to quantifying the magnetic complexity of active regions using a fractal dimension measure is presented. This fully-automated approach uses full disc MDI magnetograms of active regions from a large data set (2742 days of the SoHO mission; 9342 active regions) to compare the calculated fractal dimension to both Mount Wilson classification and flare rate. The main Mount Wilson classes exhibit no distinct fractal dimension distribution, suggesting a self-similar nature of all active regions. Solar flare productivity exhibits an increase in both the frequency and GOES X-ray magnitude of flares from regions with higher fractal dimensions. Specifically a lower threshold fractal dimension of 1.2 and 1.25 exists as a necessary, but not sufficient, requirement for an active region to produce M- and X-class flares respectively .

Fractal Analysis of Visual Search Activity for Mass Detection During Mammographic Screening

DOE PAGES

Alamudun, Folami T.; Yoon, Hong-Jun; Hudson, Kathy; ...

2017-02-21

Purpose: The objective of this study was to assess the complexity of human visual search activity during mammographic screening using fractal analysis and to investigate its relationship with case and reader characteristics. Methods: The study was performed for the task of mammographic screening with simultaneous viewing of four coordinated breast views as typically done in clinical practice. Eye-tracking data and diagnostic decisions collected for 100 mammographic cases (25 normal, 25 benign, 50 malignant) and 10 readers (three board certified radiologists and seven radiology residents), formed the corpus data for this study. The fractal dimension of the readers’ visual scanning patternsmore » was computed with the Minkowski–Bouligand box-counting method and used as a measure of gaze complexity. Individual factor and group-based interaction ANOVA analysis was performed to study the association between fractal dimension, case pathology, breast density, and reader experience level. The consistency of the observed trends depending on gaze data representation was also examined. Results: Case pathology, breast density, reader experience level, and individual reader differences are all independent predictors of the visual scanning pattern complexity when screening for breast cancer. No higher order effects were found to be significant. Conclusions: Fractal characterization of visual search behavior during mammographic screening is dependent on case properties and image reader characteristics.« less

Connotations of pixel-based scale effect in remote sensing and the modified fractal-based analysis method

NASA Astrophysics Data System (ADS)

Feng, Guixiang; Ming, Dongping; Wang, Min; Yang, Jianyu

2017-06-01

Scale problems are a major source of concern in the field of remote sensing. Since the remote sensing is a complex technology system, there is a lack of enough cognition on the connotation of scale and scale effect in remote sensing. Thus, this paper first introduces the connotations of pixel-based scale and summarizes the general understanding of pixel-based scale effect. Pixel-based scale effect analysis is essentially important for choosing the appropriate remote sensing data and the proper processing parameters. Fractal dimension is a useful measurement to analysis pixel-based scale. However in traditional fractal dimension calculation, the impact of spatial resolution is not considered, which leads that the scale effect change with spatial resolution can't be clearly reflected. Therefore, this paper proposes to use spatial resolution as the modified scale parameter of two fractal methods to further analyze the pixel-based scale effect. To verify the results of two modified methods (MFBM (Modified Windowed Fractal Brownian Motion Based on the Surface Area) and MDBM (Modified Windowed Double Blanket Method)); the existing scale effect analysis method (information entropy method) is used to evaluate. And six sub-regions of building areas and farmland areas were cut out from QuickBird images to be used as the experimental data. The results of the experiment show that both the fractal dimension and information entropy present the same trend with the decrease of spatial resolution, and some inflection points appear at the same feature scales. Further analysis shows that these feature scales (corresponding to the inflection points) are related to the actual sizes of the geo-object, which results in fewer mixed pixels in the image, and these inflection points are significantly indicative of the observed features. Therefore, the experiment results indicate that the modified fractal methods are effective to reflect the pixel-based scale effect existing in remote sensing

Fractal Analysis of Radiologists Visual Scanning Pattern in Screening Mammography

DOE Office of Scientific and Technical Information (OSTI.GOV)

Alamudun, Folami T; Yoon, Hong-Jun; Hudson, Kathy

2015-01-01

Several investigators have investigated radiologists visual scanning patterns with respect to features such as total time examining a case, time to initially hit true lesions, number of hits, etc. The purpose of this study was to examine the complexity of the radiologists visual scanning pattern when viewing 4-view mammographic cases, as they typically do in clinical practice. Gaze data were collected from 10 readers (3 breast imaging experts and 7 radiology residents) while reviewing 100 screening mammograms (24 normal, 26 benign, 50 malignant). The radiologists scanpaths across the 4 mammographic views were mapped to a single 2-D image plane. Then,more » fractal analysis was applied on the derived scanpaths using the box counting method. For each case, the complexity of each radiologist s scanpath was estimated using fractal dimension. The association between gaze complexity, case pathology, case density, and radiologist experience was evaluated using 3 factor fixed effects ANOVA. ANOVA showed that case pathology, breast density, and experience level are all independent predictors of the visual scanning pattern complexity. Visual scanning patterns are significantly different for benign and malignant cases than for normal cases as well as when breast parenchyma density changes.« less

Transport properties of electrons in fractal magnetic-barrier structures

NASA Astrophysics Data System (ADS)

Sun, Lifeng; Fang, Chao; Guo, Yong

2010-09-01

Quantum transport properties in fractal magnetically modulated structures are studied by the transfer-matrix method. It is found that the transmission spectra depend sensitively not only on the incident energy and the direction of the wave vector but also on the stage of the fractal structures. Resonance splitting, enhancement, and position shift of the resonance peaks under different magnetic modulation are observed at four different fractal stages, and the relationship between the conductance in the fractal structure and magnetic modulation is also revealed. The results indicate the spectra of the transmission can be considered as fingerprints for the fractal structures, which show the subtle correspondence between magnetic structures and transport behaviors.

Characterization of branch complexity by fractal analyses

USGS Publications Warehouse

Alados, C.L.; Escos, J.; Emlen, J.M.; Freeman, D.C.

1999-01-01

The comparison between complexity in the sense of space occupancy (box-counting fractal dimension D(c) and information dimension D1) and heterogeneity in the sense of space distribution (average evenness index f and evenness variation coefficient J(cv)) were investigated in mathematical fractal objects and natural branch structures. In general, increased fractal dimension was paired with low heterogeneity. Comparisons between branch architecture in Anthyllis cytisoides under different slope exposure and grazing impact revealed that branches were more complex and more homogeneously distributed for plants on northern exposures than southern, while grazing had no impact during a wet year. Developmental instability was also investigated by the statistical noise of the allometric relation between internode length and node order. In conclusion, our study demonstrated that fractal dimension of branch structure can be used to analyze the structural organization of plants, especially if we consider not only fractal dimension but also shoot distribution within the canopy (lacunarity). These indexes together with developmental instability analyses are good indicators of growth responses to the environment.

Empirical Relationships Between Optical Properties and Equivalent Diameters of Fractal Soot Aggregates at 550 Nm Wavelength.

NASA Technical Reports Server (NTRS)

Pandey, Apoorva; Chakrabarty, Rajan K.; Liu, Li; Mishchenko, Michael I.

2015-01-01

Soot aggregates (SAs)-fractal clusters of small, spherical carbonaceous monomers-modulate the incoming visible solar radiation and contribute significantly to climate forcing. Experimentalists and climate modelers typically assume a spherical morphology for SAs when computing their optical properties, causing significant errors. Here, we calculate the optical properties of freshly-generated (fractal dimension Df = 1.8) and aged (Df = 2.6) SAs at 550 nm wavelength using the numericallyexact superposition T-Matrix method. These properties were expressed as functions of equivalent aerosol diameters as measured by contemporary aerosol instruments. This work improves upon previous efforts wherein SA optical properties were computed as a function of monomer number, rendering them unusable in practical applications. Future research will address the sensitivity of variation in refractive index, fractal prefactor, and monomer overlap of SAs on the reported empirical relationships.

Undergraduate Experiment with Fractal Diffraction Gratings

ERIC Educational Resources Information Center

Monsoriu, Juan A.; Furlan, Walter D.; Pons, Amparo; Barreiro, Juan C.; Gimenez, Marcos H.

2011-01-01

We present a simple diffraction experiment with fractal gratings based on the triadic Cantor set. Diffraction by fractals is proposed as a motivating strategy for students of optics in the potential applications of optical processing. Fraunhofer diffraction patterns are obtained using standard equipment present in most undergraduate physics…

The Application of Fractal and Multifractal Theory in Hydraulic-Flow-Unit Characterization and Permeability Estimation

NASA Astrophysics Data System (ADS)

Chen, X.; Yao, G.; Cai, J.

2017-12-01

Pore structure characteristics are important factors in influencing the fluid transport behavior of porous media, such as pore-throat ratio, pore connectivity and size distribution, moreover, wettability. To accurately characterize the diversity of pore structure among HFUs, five samples selected from different HFUs (porosities are approximately equal, however permeability varies widely) were chosen to conduct micro-computerized tomography test to acquire direct 3D images of pore geometries and to perform mercury injection experiments to obtain the pore volume-radii distribution. To characterize complex and high nonlinear pore structure of all samples, three classic fractal geometry models were applied. Results showed that each HFU has similar box-counting fractal dimension and generalized fractal dimension in the number-area model, but there are significant differences in multifractal spectrums. In the radius-volume model, there are three obvious linear segments, corresponding to three fractal dimension values, and the middle one is proved as the actual fractal dimension according to the maximum radius. In the number-radius model, the spherical-pore size distribution extracted by maximum ball algorithm exist a decrease in the number of small pores compared with the fractal power rate rather than the traditional linear law. Among the three models, only multifractal analysis can classify the HFUs accurately. Additionally, due to the tightness and low-permeability in reservoir rocks, connate water film existing in the inner surface of pore channels commonly forms bound water. The conventional model which is known as Yu-Cheng's model has been proved to be typically not applicable. Considering the effect of irreducible water saturation, an improved fractal permeability model was also deduced theoretically. The comparison results showed that the improved model can be applied to calculate permeability directly and accurately in such unconventional rocks.

Fractal dimension analysis of complexity in Ligeti piano pieces

NASA Astrophysics Data System (ADS)

Bader, Rolf

2005-04-01

Fractal correlation dimensional analysis has been performed with whole solo piano pieces by Gyrgy Ligeti at every 50ms interval of the pieces. The resulting curves of development of complexity represented by the fractal dimension showed up a very reasonable correlation with the perceptional density of events during these pieces. The seventh piece of Ligeti's ``Musica ricercata'' was used as a test case. Here, each new part of the piece was followed by an increase of the fractal dimension because of the increase of information at the part changes. The second piece ``Galamb borong,'' number seven of the piano Etudes was used, because Ligeti wrote these Etudes after studying fractal geometry. Although the piece is not fractal in the strict mathematical sense, the overall structure of the psychoacoustic event-density as well as the detailed event development is represented by the fractal dimension plot.

Quantitative assessment of early diabetic retinopathy using fractal analysis.

PubMed

Cheung, Ning; Donaghue, Kim C; Liew, Gerald; Rogers, Sophie L; Wang, Jie Jin; Lim, Shueh-Wen; Jenkins, Alicia J; Hsu, Wynne; Li Lee, Mong; Wong, Tien Y

2009-01-01

Fractal analysis can quantify the geometric complexity of the retinal vascular branching pattern and may therefore offer a new method to quantify early diabetic microvascular damage. In this study, we examined the relationship between retinal fractal dimension and retinopathy in young individuals with type 1 diabetes. We conducted a cross-sectional study of 729 patients with type 1 diabetes (aged 12-20 years) who had seven-field stereoscopic retinal photographs taken of both eyes. From these photographs, retinopathy was graded according to the modified Airlie House classification, and fractal dimension was quantified using a computer-based program following a standardized protocol. In this study, 137 patients (18.8%) had diabetic retinopathy signs; of these, 105 had mild retinopathy. Median (interquartile range) retinal fractal dimension was 1.46214 (1.45023-1.47217). After adjustment for age, sex, diabetes duration, A1C, blood pressure, and total cholesterol, increasing retinal vascular fractal dimension was significantly associated with increasing odds of retinopathy (odds ratio 3.92 [95% CI 2.02-7.61] for fourth versus first quartile of fractal dimension). In multivariate analysis, each 0.01 increase in retinal vascular fractal dimension was associated with a nearly 40% increased odds of retinopathy (1.37 [1.21-1.56]). This association remained after additional adjustment for retinal vascular caliber. Greater retinal fractal dimension, representing increased geometric complexity of the retinal vasculature, is independently associated with early diabetic retinopathy signs in type 1 diabetes. Fractal analysis of fundus photographs may allow quantitative measurement of early diabetic microvascular damage.

Using Peano Curves to Construct Laplacians on Fractals

NASA Astrophysics Data System (ADS)

Molitor, Denali; Ott, Nadia; Strichartz, Robert

2015-12-01

We describe a new method to construct Laplacians on fractals using a Peano curve from the circle onto the fractal, extending an idea that has been used in the case of certain Julia sets. The Peano curve allows us to visualize eigenfunctions of the Laplacian by graphing the pullback to the circle. We study in detail three fractals: the pentagasket, the octagasket and the magic carpet. We also use the method for two nonfractal self-similar sets, the torus and the equilateral triangle, obtaining appealing new visualizations of eigenfunctions on the triangle. In contrast to the many familiar pictures of approximations to standard Peano curves, that do no show self-intersections, our descriptions of approximations to the Peano curves have self-intersections that play a vital role in constructing graph approximations to the fractal with explicit graph Laplacians that give the fractal Laplacian in the limit.

ABC of multi-fractal spacetimes and fractional sea turtles

NASA Astrophysics Data System (ADS)

Calcagni, Gianluca

2016-04-01

We clarify what it means to have a spacetime fractal geometry in quantum gravity and show that its properties differ from those of usual fractals. A weak and a strong definition of multi-scale and multi-fractal spacetimes are given together with a sketch of the landscape of multi-scale theories of gravitation. Then, in the context of the fractional theory with q-derivatives, we explore the consequences of living in a multi-fractal spacetime. To illustrate the behavior of a non-relativistic body, we take the entertaining example of a sea turtle. We show that, when only the time direction is fractal, sea turtles swim at a faster speed than in an ordinary world, while they swim at a slower speed if only the spatial directions are fractal. The latter type of geometry is the one most commonly found in quantum gravity. For time-like fractals, relativistic objects can exceed the speed of light, but strongly so only if their size is smaller than the range of particle-physics interactions. We also find new results about log-oscillating measures, the measure presentation and their role in physical observations and in future extensions to nowhere-differentiable stochastic spacetimes.

Absorption and scattering by fractal aggregates and by their equivalent coated spheres

NASA Astrophysics Data System (ADS)

Kandilian, Razmig; Heng, Ri-Liang; Pilon, Laurent

2015-01-01

This paper demonstrates that the absorption and scattering cross-sections and the asymmetry factor of randomly oriented fractal aggregates of spherical monomers can be rapidly estimated as those of coated spheres with equivalent volume and average projected area. This was established for fractal aggregates with fractal dimension ranging from 2.0 to 3.0 and composed of up to 1000 monodisperse or polydisperse monomers with a wide range of size parameter and relative complex index of refraction. This equivalent coated sphere approximation was able to capture the effects of both multiple scattering and shading among constituent monomers on the integral radiation characteristics of the aggregates. It was shown to be superior to the Rayleigh-Debye-Gans approximation and to the equivalent coated sphere approximation proposed by Latimer. However, the scattering matrix element ratios of equivalent coated spheres featured large angular oscillations caused by internal reflection in the coating which were not observed in those of the corresponding fractal aggregates. Finally, the scattering phase function and the scattering matrix elements of aggregates with large monomer size parameter were found to have unique features that could be used in remote sensing applications.

Fractal Dynamics of Heartbeat Interval Fluctuations in Health and Disease

NASA Astrophysics Data System (ADS)

Meyer, M.; Marconi, C.; Rahmel, A.; Grassi, B.; Ferretti, G.; Skinner, J. E.; Cerretelli, P.

The dynamics of heartbeat interval time series were studied by a modified random walk analysis recently introduced as Detrended Fluctuation Analysis. In this analysis, the intrinsic fractal long-range power-law correlation properties of beat-to-beat fluctuations generated by the dynamical system (i.e. cardiac rhythm generator), after decomposition from extrinsic uncorrelated sources, can be quantified by the scaling exponent which, in healthy subjects, is about 1.0. The finding of a scaling coefficient of 1.0, indicating scale-invariant long-range power-law correlations (1/ƒnoise) of heartbeat fluctuations, would reflect a genuinely self-similar fractal process that typically generates fluctuations on a wide range of time scales. Lack of a characteristic time scale suggests that the neuroautonomic system underlying the control of heart rate dynamics helps prevent excessive mode-locking (error tolerance) that would restrict its functional responsiveness (plasticity) to environmental stimuli. The 1/ƒ dynamics of heartbeat interval fluctuations are unaffected by exposure to chronic hypoxia suggesting that the neuroautonomic cardiac control system is preadapted to hypoxia. Functional (hypothermia, cardiac disease) and/or structural (cardiac transplantation, early cardiac development) inactivation of neuroautonomic control is associated with the breakdown or absence of fractal complexity reflected by anticorrelated random walk-like dynamics, indicating that in these conditions the heart is unadapted to its environment.

Hierarchical socioeconomic fractality: The rich, the poor, and the middle-class

NASA Astrophysics Data System (ADS)

Eliazar, Iddo; Cohen, Morrel H.

2014-05-01

Since the seminal work of the Italian economist Vilfredo Pareto, the study of wealth and income has been a topic of active scientific exploration engaging researches ranging from economics and political science to econophysics and complex systems. This paper investigates the intrinsic fractality of wealth and income. To that end we introduce and characterize three forms of socioeconomic scale-invariance-poor fractality, rich fractality, and middle-class fractality-and construct hierarchical fractal approximations of general wealth and income distributions, based on the stitching of these three forms of fractality. Intertwining the theoretical results with real-world empirical data we then establish that the three forms of socioeconomic fractality-amalgamated into a composite hierarchical structure-underlie the distributions of wealth and income in human societies. We further establish that the hierarchical socioeconomic fractality of wealth and income is also displayed by empirical rank distributions observed across the sciences.

Analysis of Resting-State fMRI Topological Graph Theory Properties in Methamphetamine Drug Users Applying Box-Counting Fractal Dimension.

PubMed

Siyah Mansoory, Meysam; Oghabian, Mohammad Ali; Jafari, Amir Homayoun; Shahbabaie, Alireza

2017-01-01

Graph theoretical analysis of functional Magnetic Resonance Imaging (fMRI) data has provided new measures of mapping human brain in vivo. Of all methods to measure the functional connectivity between regions, Linear Correlation (LC) calculation of activity time series of the brain regions as a linear measure is considered the most ubiquitous one. The strength of the dependence obligatory for graph construction and analysis is consistently underestimated by LC, because not all the bivariate distributions, but only the marginals are Gaussian. In a number of studies, Mutual Information (MI) has been employed, as a similarity measure between each two time series of the brain regions, a pure nonlinear measure. Owing to the complex fractal organization of the brain indicating self-similarity, more information on the brain can be revealed by fMRI Fractal Dimension (FD) analysis. In the present paper, Box-Counting Fractal Dimension (BCFD) is introduced for graph theoretical analysis of fMRI data in 17 methamphetamine drug users and 18 normal controls. Then, BCFD performance was evaluated compared to those of LC and MI methods. Moreover, the global topological graph properties of the brain networks inclusive of global efficiency, clustering coefficient and characteristic path length in addict subjects were investigated too. Compared to normal subjects by using statistical tests (P<0.05), topological graph properties were postulated to be disrupted significantly during the resting-state fMRI. Based on the results, analyzing the graph topological properties (representing the brain networks) based on BCFD is a more reliable method than LC and MI.

Polarimetric Wavelet Fractal Remote Sensing Principles for Space Materials (Preprint)

DTIC Science & Technology

2012-06-04

previously introduced 9-10, 28. The combination of polarimetry and wavelet-fractal analysis yields enhanced knowledge of the spatial-temporal-frequency...applications in situations that require analysis over very short time durations or where information is localized, and have been combined with polarimetry ...and D.B. Chenault, “Near Infrared Imaging Polarimetry ”, Proc. SPIE 4481, pp. 30-31, 2001. [8] A.B. Mahler, P. Smith, R. Chipman, G. Smith, N

Three-dimensional fractal analysis of 99mTc-MAA SPECT images in chronic thromboembolic pulmonary hypertension for evaluation of response to balloon pulmonary angioplasty: association with pulmonary arterial pressure.

PubMed

Maruoka, Yasuhiro; Nagao, Michinobu; Baba, Shingo; Isoda, Takuro; Kitamura, Yoshiyuki; Yamazaki, Yuzo; Abe, Koichiro; Sasaki, Masayuki; Abe, Kohtaro; Honda, Hiroshi

2017-06-01

Balloon pulmonary angioplasty (BPA) is used for inoperable chronic thromboembolic pulmonary hypertension (CTEPH), but its effect cannot be evaluated noninvasively. We devised a noninvasive quantitative index of response to BPA using three-dimensional fractal analysis (3D-FA) of technetium-99m-macroaggregated albumin (Tc-MAA) single-photon emission computed tomography (SPECT). Forty CTEPH patients who underwent pulmonary perfusion scintigraphy and mean pulmonary arterial pressure (mPAP) measurement by right heart catheterization before and after BPA were studied. The total uptake volume (TUV) in bilateral lungs was determined from maximum intensity projection Tc-MAA SPECT images. Fractal dimension was assessed by 3D-FA. Parameters were compared before and after BPA, and between patients with post-BPA mPAP more than 30 mmHg and less than or equal to 30 mmHg. Receiver operating characteristic analysis was carried out. BPA significantly improved TUV (595±204-885±214 ml, P<0.001) and reduced the laterality of uptake (238±147-135±131 ml, P<0.001). Patients with poor therapeutic response (post-BPA mPAP≥30 mmHg, n=16) showed a significantly smaller TUV increase (P=0.044) and a significantly greater post-BPA fractal dimension (P<0.001) than the low-mPAP group. Fractal dimension correlated with mPAP values before and after BPA (P=0.013 and 0.001, respectively). A post-BPA fractal dimension threshold of 2.4 distinguished between BPA success and failure with 75% sensitivity, 79% specificity, 78% accuracy, and area under the curve of 0.85. 3D-FA using Tc-MAA SPECT pulmonary perfusion scintigraphy enables a noninvasive evaluation of the response of CTEPH patients to BPA.

Investigation of changes in fractal dimension from layered retinal structures of healthy and diabetic eyes with optical coherence tomography

NASA Astrophysics Data System (ADS)

Gao, Wei; Zakharov, Valery P.; Myakinin, Oleg O.; Bratchenko, Ivan A.; Artemyev, Dmitry N.; Kornilin, Dmitry V.

2015-07-01

Optical coherence tomography (OCT) is usually employed for the measurement of retinal thickness characterizing the structural changes of tissue. However, fractal dimension (FD) could also character the structural changes of tissue. Therefore, fractal dimension changes may provide further information regarding cellular layers and early damage in ocular diseases. We investigated the possibility of OCT in detecting changes in fractal dimension from layered retinal structures. OCT images were obtained from diabetic patients without retinopathy (DM, n = 38 eyes) or mild diabetic retinopathy (MDR, n = 43 eyes) and normal healthy subjects (Controls, n = 74 eyes). Fractal dimension was calculated using the differentiate box counting methodology. We evaluated the usefulness of quantifying fractal dimension of layered structures in the detection of retinal damage. Generalized estimating equations considering within-subject intereye relations were used to test for differences between the groups. A modified p value of <0.001 was considered statistically significant. Receiver operating characteristic (ROC) curves were constructed to describe the ability of fractal dimension to discriminate between the eyes of DM, MDR and healthy eyes. Significant decreases of fractal dimension were observed in all layers in the MDR eyes compared with controls except in the inner nuclear layer (INL). Significant decreases of fractal dimension were also observed in all layers in the MDR eyes compared with DM eyes. The highest area under receiver operating characteristic curve (AUROC) values estimated for fractal dimension were observed for the outer plexiform layer (OPL) and outer segment photoreceptors (OS) when comparing MDR eyes with controls. The highest AUROC value estimated for fractal dimension were also observed for the retinal nerve fiber layer (RNFL) and OS when comparing MDR eyes with DM eyes. Our results suggest that fractal dimension of the intraretinal layers may provide useful

Association between stride time fractality and gait adaptability during unperturbed and asymmetric walking.

PubMed

Ducharme, Scott W; Liddy, Joshua J; Haddad, Jeffrey M; Busa, Michael A; Claxton, Laura J; van Emmerik, Richard E A

2018-04-01

speed, fractal dynamics increased closer to 1/f when participants were exposed to asymmetric walking. These findings suggest there may not be a relationship between unperturbed preferred or slow speed walking fractal dynamics and gait adaptability. However, the emergent relationship between asymmetric walking fractal dynamics and limb phase adaptation may represent a functional reorganization of the locomotor system (i.e., improved interactivity between degrees of freedom within the system) to be better suited to attenuate externally generated perturbations at various spatiotemporal scales. Copyright © 2018 Elsevier B.V. All rights reserved.

Fractal patterns formed by growth of radial viscous fingers*

NASA Astrophysics Data System (ADS)

Praud, Olivier

2004-03-01

We examine fractal patterns formed by the injection of air into oil in a thin (0.13 mm) layer contained between two cylindrical glass plates of 288 mm diameter (a Hele-Shaw cell) [1]. The resultant radially grown patterns are similar to those formed in Diffusion Limited Aggregation (DLA), but the relation between the continuum limit of DLA and continuum (Laplacian) growth remains an open question. Our viscous fingering patterns in the limit of very high pressure difference reach an asymptotic state in which they exhibit a fractal dimension of 1.70± 0.02, in good agreement with a calculation of the fractal dimension of a DLA cluster, 1.713± 0.003 [2]. The generalized dimensions are also computed and show that the observed pattern is self-similar with Dq = 1.70 for all q. Further, the probability density function of shielding angles suggests the existence of a critical angle close to 75 degrees. This result is in accord with numerical and analytical evidence of a critical angle in DLA [3]. Thus fractal viscous fingering patterns and Diffusion Limited Aggregation clusters have a similar geometrical structure. *Work conducted in collaboration with H.L. Swinney, M.G. Moore and Eran Sharon [1] E. Sharon, M. G. Moore, W. D. McCormick, and H. L. Swinney, Phys. Rev. Lett. 91, 205504 (2003). [2] B.Davidovitch et A. Levermann and I. Procaccia, Phys. Rev. E 62, 5919 (2000). [3] D. A. Kessler et al., Phys. Rev. E 57, 6913 (1998).

Crack image segmentation based on improved DBC method

NASA Astrophysics Data System (ADS)

Cao, Ting; Yang, Nan; Wang, Fengping; Gao, Ting; Wang, Weixing

2017-11-01

With the development of computer vision technology, crack detection based on digital image segmentation method arouses global attentions among researchers and transportation ministries. Since the crack always exhibits the random shape and complex texture, it is still a challenge to accomplish reliable crack detection results. Therefore, a novel crack image segmentation method based on fractal DBC (differential box counting) is introduced in this paper. The proposed method can estimate every pixel fractal feature based on neighborhood information which can consider the contribution from all possible direction in the related block. The block moves just one pixel every time so that it could cover all the pixels in the crack image. Unlike the classic DBC method which only describes fractal feature for the related region, this novel method can effectively achieve crack image segmentation according to the fractal feature of every pixel. The experiment proves the proposed method can achieve satisfactory results in crack detection.

Bridging Three Orders of Magnitude: Multiple Scattered Waves Sense Fractal Microscopic Structures via Dispersion

NASA Astrophysics Data System (ADS)

Lambert, Simon A.; Näsholm, Sven Peter; Nordsletten, David; Michler, Christian; Juge, Lauriane; Serfaty, Jean-Michel; Bilston, Lynne; Guzina, Bojan; Holm, Sverre; Sinkus, Ralph

2015-08-01

Wave scattering provides profound insight into the structure of matter. Typically, the ability to sense microstructure is determined by the ratio of scatterer size to probing wavelength. Here, we address the question of whether macroscopic waves can report back the presence and distribution of microscopic scatterers despite several orders of magnitude difference in scale between wavelength and scatterer size. In our analysis, monosized hard scatterers 5 μ m in radius are immersed in lossless gelatin phantoms to investigate the effect of multiple reflections on the propagation of shear waves with millimeter wavelength. Steady-state monochromatic waves are imaged in situ via magnetic resonance imaging, enabling quantification of the phase velocity at a voxel size big enough to contain thousands of individual scatterers, but small enough to resolve the wavelength. We show in theory, experiments, and simulations that the resulting coherent superposition of multiple reflections gives rise to power-law dispersion at the macroscopic scale if the scatterer distribution exhibits apparent fractality over an effective length scale that is comparable to the probing wavelength. Since apparent fractality is naturally present in any random medium, microstructure can thereby leave its fingerprint on the macroscopically quantifiable power-law exponent. Our results are generic to wave phenomena and carry great potential for sensing microstructure that exhibits intrinsic fractality, such as, for instance, vasculature.

Electromagnetic backscattering from one-dimensional drifting fractal sea surface II: Electromagnetic backscattering model

NASA Astrophysics Data System (ADS)

Tao, Xie; William, Perrie; Shang-Zhuo, Zhao; He, Fang; Wen-Jin, Yu; Yi-Jun, He

2016-07-01

Sea surface current has a significant influence on electromagnetic (EM) backscattering signals and may constitute a dominant synthetic aperture radar (SAR) imaging mechanism. An effective EM backscattering model for a one-dimensional drifting fractal sea surface is presented in this paper. This model is used to simulate EM backscattering signals from the drifting sea surface. Numerical results show that ocean currents have a significant influence on EM backscattering signals from the sea surface. The normalized radar cross section (NRCS) discrepancies between the model for a coupled wave-current fractal sea surface and the model for an uncoupled fractal sea surface increase with the increase of incidence angle, as well as with increasing ocean currents. Ocean currents that are parallel to the direction of the wave can weaken the EM backscattering signal intensity, while the EM backscattering signal is intensified by ocean currents propagating oppositely to the wave direction. The model presented in this paper can be used to study the SAR imaging mechanism for a drifting sea surface. Project supported by the National Natural Science Foundation of China (Grant No. 41276187), the Global Change Research Program of China (Grant No. 2015CB953901), the Priority Academic Program Development of Jiangsu Higher Education Institutions, China, the Program for the Innovation Research and Entrepreneurship Team in Jiangsu Province, China, the Canadian Program on Energy Research and Development, and the Canadian World Class Tanker Safety Service Program.

Flow field topology of submerged jets with fractal generated turbulence

NASA Astrophysics Data System (ADS)

Cafiero, Gioacchino; Discetti, Stefano; Astarita, Tommaso

2015-11-01

Fractal grids (FGs) have been recently an object of numerous investigations due to the interesting capability of generating turbulence at multiple scales, thus paving the way to tune mixing and scalar transport. The flow field topology of a turbulent air jet equipped with a square FG is investigated by means of planar and volumetric particle image velocimetry. The comparison with the well-known features of a round jet without turbulence generators is also presented. The Reynolds number based on the nozzle exit section diameter for all the experiments is set to about 15 000. It is demonstrated that the presence of the grid enhances the entrainment rate and, as a consequence, the scalar transfer of the jet. Moreover, due to the effect of the jet external shear layer on the wake shed by the grid bars, the turbulence production region past the grid is significantly shortened with respect to the documented behavior of fractal grids in free-shear conditions. The organization of the large coherent structures in the FG case is also analyzed and discussed. Differently from the well-known generation of toroidal vortices due to the growth of azimuthal disturbances within the jet shear layer, the fractal grid introduces cross-wise disturbs which produce streamwise vortices; these structures, although characterized by a lower energy content, have a deeper streamwise penetration than the ring vortices, thus enhancing the entrainment process.

Pulse regime in formation of fractal fibers

DOE Office of Scientific and Technical Information (OSTI.GOV)

Smirnov, B. M., E-mail: bmsmirnov@gmail.com

The pulse regime of vaporization of a bulk metal located in a buffer gas is analyzed as a method of generation of metal atoms under the action of a plasma torch or a laser beam. Subsequently these atoms are transformed into solid nanoclusters, fractal aggregates and then into fractal fibers if the growth process proceeds in an external electric field. We are guided by metals in which transitions between s and d-electrons of their atoms are possible, since these metals are used as catalysts and filters in interaction with gas flows. The resistance of metal fractal structures to a gasmore » flow is evaluated that allows one to find optimal parameters of a fractal structure for gas flow propagation through it. The thermal regime of interaction between a plasma pulse or a laser beam and a metal surface is analyzed. It is shown that the basic energy from an external source is consumed on a bulk metal heating, and the efficiency of atom evaporation from the metal surface, that is the ratio of energy fluxes for vaporization and heating, is 10{sup –3}–10{sup –4} for transient metals under consideration. A typical energy flux (~10{sup 6} W/cm{sup 2}), a typical surface temperature (~3000 K), and a typical pulse duration (~1 μs) provide a sufficient amount of evaporated atoms to generate fractal fibers such that each molecule of a gas flow collides with the skeleton of fractal fibers many times.« less

Fractal Dimensionality of Pore and Grain Volume of a Siliciclastic Marine Sand

NASA Astrophysics Data System (ADS)

Reed, A. H.; Pandey, R. B.; Lavoie, D. L.

Three-dimensional (3D) spatial distributions of pore and grain volumes were determined from high-resolution computer tomography (CT) images of resin-impregnated marine sands. Using a linear gradient extrapolation method, cubic three-dimensional samples were constructed from two-dimensional CT images. Image porosity (0.37) was found to be consistent with the estimate of porosity by water weight loss technique (0.36). Scaling of the pore volume (Vp) with the linear size (L), V~LD provides the fractal dimensionalities of the pore volume (D=2.74+/-0.02) and grain volume (D=2.90+/-0.02) typical for sedimentary materials.

Fractal tomography and its application in 3D vision

NASA Astrophysics Data System (ADS)

Trubochkina, N.

2018-01-01

A three-dimensional artistic fractal tomography method that implements a non-glasses 3D visualization of fractal worlds in layered media is proposed. It is designed for the glasses-free 3D vision of digital art objects and films containing fractal content. Prospects for the development of this method in art galleries and the film industry are considered.

Wetting characteristics of 3-dimensional nanostructured fractal surfaces

NASA Astrophysics Data System (ADS)

Davis, Ethan; Liu, Ying; Jiang, Lijia; Lu, Yongfeng; Ndao, Sidy

2017-01-01

This article reports the fabrication and wetting characteristics of 3-dimensional nanostructured fractal surfaces (3DNFS). Three distinct 3DNFS surfaces, namely cubic, Romanesco broccoli, and sphereflake were fabricated using two-photon direct laser writing. Contact angle measurements were performed on the multiscale fractal surfaces to characterize their wetting properties. Average contact angles ranged from 66.8° for the smooth control surface to 0° for one of the fractal surfaces. The change in wetting behavior was attributed to modification of the interfacial surface properties due to the inclusion of 3-dimensional hierarchical fractal nanostructures. However, this behavior does not exactly obey existing surface wetting models in the literature. Potential applications for these types of surfaces in physical and biological sciences are also discussed.

Teaching about Fractals.

ERIC Educational Resources Information Center

Willson, Stephen J.

1991-01-01

Described is a course designed to teach students about fractals using various teaching methods including the computer. Discussed are why the course drew students, prerequisites, clientele, textbook, grading, computer usage, and the syllabus. (KR)

Two-Dimensional Animal-Like Fractals in Thin Films

NASA Astrophysics Data System (ADS)

Gao, Hong-jun; Xue, Zeng-quan; Wu, Quan-de; Pang, Shi-jin

1996-02-01

We present a few unique animal-like fractal patterns in ionized-cluster-beam deposited fullerene-tetracyanoquinodimethane thin films. The fractal patterns consisting of animal-like aggregates such as "fishes" and "quasi-seahorses" have been characterized by transmission electron microscopy. The results indicate that the small aggregates of the animal-like body are composed of many single crystals whose crystalline directions are generally different. The formation of the fractal patterns can be attributed to the cluster-diffusion-limited aggregation.

Evolution of Fractal Parameters through Development Stage of Soil Crust

NASA Astrophysics Data System (ADS)

Ospina, Abelardo; Florentino, Adriana; Tarquis, Ana Maria

2016-04-01

Soil surface characteristics are subjected to changes driven by several interactions between water, air, biotic and abiotic components. One of the examples of such interactions is provided through biological soil crusts (BSC) in arid and semi-arid environments. BSC are communities composed of cyanobacteria, fungi, mosses, lichens, algae and liverworts covering the soil surface and play an important role in ecosystem functioning. The characteristics and formation of these BSC influence the soil hydrological balance, control the mass of eroded sediment, increase stability of soil surface, and influence plant productivity through the modification of nitrogen and carbon cycle. The site of this work is located at Quibor and Ojo de Agua (Lara state, Venezuela). The Quibor Depression in Venezuela is a major agricultural area being at semi-arid conditions and limited drainage favor the natural process of salinization. Additionally, the extension and intensification of agriculture has led to over-exploitation of groundwater in the past 30 years (Méndoza et al., 2013). The soil microbial crust develops initially on physical crusts which are mainly generated since wetting and drying, being a recurrent feature in the Quíbor arid zone. The microbiotic crust is organic, composed of macro organisms (bryophytes and lichens) and microorganisms (cyanobacteria, fungi algae, etc.); growing on the ground, forming a thickness no greater than 3 mm. For further details see Toledo and Florentino (2009). This study focus on characterize the development stage of the BSC based on image analysis. To this end, grayscale images of different types of biological soil crust at different stages where taken, each image corresponding to an area of 12.96 cm2 with a resolution of 1024x1024 pixels (Ospina et al., 2015). For each image lacunarity and fractal dimension through the differential box counting method were calculated. These were made with the software ImageJ/Fraclac (Karperien, 2013

Iterons, fractals and computations of automata

NASA Astrophysics Data System (ADS)

Siwak, Paweł

1999-03-01

Processing of strings by some automata, when viewed on space-time (ST) diagrams, reveals characteristic soliton-like coherent periodic objects. They are inherently associated with iterations of automata mappings thus we call them the iterons. In the paper we present two classes of one-dimensional iterons: particles and filtrons. The particles are typical for parallel (cellular) processing, while filtrons, introduced in (32) are specific for serial processing of strings. In general, the images of iterated automata mappings exhibit not only coherent entities but also the fractals, and quasi-periodic and chaotic dynamics. We show typical images of such computations: fractals, multiplication by a number, and addition of binary numbers defined by a Turing machine. Then, the particles are presented as iterons generated by cellular automata in three computations: B/U code conversion (13, 29), majority classification (9), and in discrete version of the FPU (Fermi-Pasta-Ulam) dynamics (7, 23). We disclose particles by a technique of combinational recoding of ST diagrams (as opposed to sequential recoding). Subsequently, we recall the recursive filters based on FCA (filter cellular automata) window operators, and considered by Park (26), Ablowitz (1), Fokas (11), Fuchssteiner (12), Bruschi (5) and Jiang (20). We present the automata equivalents to these filters (33). Some of them belong to the class of filter automata introduced in (30). We also define and illustrate some properties of filtrons. Contrary to particles, the filtrons interact nonlocally in the sense that distant symbols may influence one another. Thus their interactions are very unusual. Some examples have been given in (32). Here we show new examples of filtron phenomena: multifiltron solitonic collisions, attracting and repelling filtrons, trapped bouncing filtrons (which behave like a resonance cavity) and quasi filtrons.

A fractal analysis of protein to DNA binding kinetics using biosensors.

PubMed

Sadana, Ajit

2003-08-01

A fractal analysis of a confirmative nature only is presented for the binding of estrogen receptor (ER) in solution to its corresponding DNA (estrogen response element, ERE) immobilized on a sensor chip surface [J. Biol. Chem. 272 (1997) 11384], and for the cooperative binding of human 1,25-dihydroxyvitamin D(3) receptor (VDR) to DNA with the 9-cis-retinoic acid receptor (RXR) [Biochemistry 35 (1996) 3309]. Ligands were also used to modulate the first reaction. Data taken from the literature may be modeled by using a single- or a dual-fractal analysis. Relationships are presented for the binding rate coefficient as a function of either the analyte concentration in solution or the fractal dimension that exists on the biosensor surface. The binding rate expressions developed exhibit a wide range of dependence on the degree of heterogeneity that exists on the surface, ranging from sensitive (order of dependence equal to 1.202) to very sensitive (order of dependence equal to 12.239). In general, the binding rate coefficient increases as the degree of heterogeneity or the fractal dimension of the surface increases. The predictive relationships presented provide further physical insights into the reactions occurring on the biosensor surface. Even though these reactions are occurring on the biosensor surface, the relationships presented should assist in understanding and in possibly manipulating the reactions occurring on cellular surfaces.

Fractal binding and dissociation kinetics of lecithin cholesterol acyl transferase (LCAT), a heart-related compound, on biosensor surfaces

NASA Astrophysics Data System (ADS)

Doke, Atul M.; Sadana, Ajit

2006-05-01

A fractal analysis is presented for the binding and dissociation of different heart-related compounds in solution to receptors immobilized on biosensor surfaces. The data analyzed include LCAT (lecithin cholesterol acyl transferase) concentrations in solution to egg-white apoA-I rHDL immobilized on a biosensor chip surface.1 Single- and dual- fractal models were employed to fit the data. Values of the binding and the dissociation rate coefficient(s), affinity values, and the fractal dimensions were obtained from the regression analysis provided by Corel Quattro Pro 8.0 (Corel Corporation Limited).2 The binding rate coefficients are quite sensitive to the degree of heterogeneity on the sensor chip surface. Predictive equations are developed for the binding rate coefficient as a function of the degree of heterogeneity present on the sensor chip surface and on the LCAT concentration in solution, and for the affinity as a function of the ratio of fractal dimensions present in the binding and the dissociation phases. The analysis presented provided physical insights into these analyte-receptor reactions occurring on different biosensor surfaces.

Fractals and Transformations.

ERIC Educational Resources Information Center

Bannon, Thomas J.

1991-01-01

Discussed are several different transformations based on the generation of fractals including self-similar designs, the chaos game, the koch curve, and the Sierpinski Triangle. Three computer programs which illustrate these concepts are provided. (CW)

The Fractal Behavior of Crystal Distribution of la Gloria Pluton, Chile

NASA Astrophysics Data System (ADS)

Gutiérrez, F. J.; Payacán, I. J.; Pasten, D.; Aravena, A.; Gelman, S. E.; Bachmann, O.; Parada, M. A.

2013-12-01

We utilize fractal analysis to study the spatial distributions of crystals in a 10 Ma granitic intrusion (La Gloria pluton) located in the central Chilean Andes. Previous work determined the crystal size distribution (CSD) and anisotropy of magnetic susceptibility (AMS) tensors throughout this pluton. Using orthogonal thin sections oriented along the AMS tensor axes, we have applied fractal analysis in three magmatic crystal families: plagioclase, ferromagnesian minerals (biotite and amphibole), and Fe-Ti oxides (magnetite with minor ilmenite). We find that plagioclase and ferromagnesian minerals have a Semi-logarithmic CSD (S-CSD), given by: log(n/n0)= -L/C (1) where n [mm-4], n0 [mm-4], L [mm] and C [mm] are crystal density, intercept (nucleation density; L=0), size of crystals (three axes) and characteristic length, respectively. In contrast, Fe-Ti oxides have a Fractal CSD (F-CSD, power law size distribution), given by: log(n)= - Dn log(L) + n1 (2) where Dn and n1 [log(mm-4)] are a non-dimensional proportionality constant and the logarithm of the initial crystallization density (n1 = log(n(L=1 mm))), respectively. Finally, we calculate the fractal dimension (D0) by applying the box-counting method on each crystal thin section image, using: log(N) = -D0 log(ɛ) (3) where N and ɛ are the number of boxes occupied by minerals and the length of the square box, respectively. Results indicate that D0 values (eq. 3) are well defined for all minerals, and are higher for plagioclase than for ferromagnesian minerals and lowest for Fe-Ti oxides. D0 values are correlated with n0 and -1/C for S-CSD (eq. 1), and with n1 values for F-CSD (eq. 2). These correlations between fractal dimensions with CSD parameters suggest crystal growth follows a fractal behaviour in magmatic systems. Fractal behaviour of CSD means that the spatial distribution of crystals follows an all-scale pattern as part of a self-organized magmatic system. We interpret S-CSD of plagioclase and

Fractal morphology, imaging and mass spectrometry of single aerosol particles in flight.

PubMed

Loh, N D; Hampton, C Y; Martin, A V; Starodub, D; Sierra, R G; Barty, A; Aquila, A; Schulz, J; Lomb, L; Steinbrener, J; Shoeman, R L; Kassemeyer, S; Bostedt, C; Bozek, J; Epp, S W; Erk, B; Hartmann, R; Rolles, D; Rudenko, A; Rudek, B; Foucar, L; Kimmel, N; Weidenspointner, G; Hauser, G; Holl, P; Pedersoli, E; Liang, M; Hunter, M S; Hunter, M M; Gumprecht, L; Coppola, N; Wunderer, C; Graafsma, H; Maia, F R N C; Ekeberg, T; Hantke, M; Fleckenstein, H; Hirsemann, H; Nass, K; White, T A; Tobias, H J; Farquar, G R; Benner, W H; Hau-Riege, S P; Reich, C; Hartmann, A; Soltau, H; Marchesini, S; Bajt, S; Barthelmess, M; Bucksbaum, P; Hodgson, K O; Strüder, L; Ullrich, J; Frank, M; Schlichting, I; Chapman, H N; Bogan, M J

2012-06-27

The morphology of micrometre-size particulate matter is of critical importance in fields ranging from toxicology to climate science, yet these properties are surprisingly difficult to measure in the particles' native environment. Electron microscopy requires collection of particles on a substrate; visible light scattering provides insufficient resolution; and X-ray synchrotron studies have been limited to ensembles of particles. Here we demonstrate an in situ method for imaging individual sub-micrometre particles to nanometre resolution in their native environment, using intense, coherent X-ray pulses from the Linac Coherent Light Source free-electron laser. We introduced individual aerosol particles into the pulsed X-ray beam, which is sufficiently intense that diffraction from individual particles can be measured for morphological analysis. At the same time, ion fragments ejected from the beam were analysed using mass spectrometry, to determine the composition of single aerosol particles. Our results show the extent of internal dilation symmetry of individual soot particles subject to non-equilibrium aggregation, and the surprisingly large variability in their fractal dimensions. More broadly, our methods can be extended to resolve both static and dynamic morphology of general ensembles of disordered particles. Such general morphology has implications in topics such as solvent accessibilities in proteins, vibrational energy transfer by the hydrodynamic interaction of amino acids, and large-scale production of nanoscale structures by flame synthesis.

Elasticity of fractal materials using the continuum model with non-integer dimensional space

NASA Astrophysics Data System (ADS)

Tarasov, Vasily E.

2015-01-01

Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for the equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are solved.

Fractal diffusion in high temperature polymer electrolyte fuel cell membranes

NASA Astrophysics Data System (ADS)

Hopfenmüller, Bernhard; Zorn, Reiner; Holderer, Olaf; Ivanova, Oxana; Lehnert, Werner; Lüke, Wiebke; Ehlers, Georg; Jalarvo, Niina; Schneider, Gerald J.; Monkenbusch, Michael; Richter, Dieter

2018-05-01

The performance of fuel cells depends largely on the proton diffusion in the proton conducting membrane, the core of a fuel cell. High temperature polymer electrolyte fuel cells are based on a polymer membrane swollen with phosphoric acid as the electrolyte, where proton conduction takes place. We studied the proton diffusion in such membranes with neutron scattering techniques which are especially sensitive to the proton contribution. Time of flight spectroscopy and backscattering spectroscopy have been combined to cover a broad dynamic range. In order to selectively observe the diffusion of protons potentially contributing to the ion conductivity, two samples were prepared, where in one of the samples the phosphoric acid was used with hydrogen replaced by deuterium. The scattering data from the two samples were subtracted in a suitable way after measurement. Thereby subdiffusive behavior of the proton diffusion has been observed and interpreted in terms of a model of fractal diffusion. For this purpose, a scattering function for fractal diffusion has been developed. The fractal diffusion dimension dw and the Hausdorff dimension df have been determined on the length scales covered in the neutron scattering experiments.

Fractal diffusion in high temperature polymer electrolyte fuel cell membranes

DOE PAGES

Hopfenmuller, Bernhard; Zorn, Reiner; Holderer, Olaf; ...

2018-05-29

In this paper, the performance of fuel cells depends largely on the proton diffusion in the proton conducting membrane, the core of a fuel cell. High temperature polymer electrolyte fuel cells are based on a polymer membrane swollen with phosphoric acid as the electrolyte, where proton conduction takes place. We studied the proton diffusion in such membranes with neutron scattering techniques which are especially sensitive to the proton contribution. Time of flight spectroscopy and backscattering spectroscopy have been combined to cover a broad dynamic range. In order to selectively observe the diffusion of protons potentially contributing to the ion conductivity,more » two samples were prepared, where in one of the samples the phosphoric acid was used with hydrogen replaced by deuterium. The scattering data from the two samples were subtracted in a suitable way after measurement. Thereby subdiffusive behavior of the proton diffusion has been observed and interpreted in terms of a model of fractal diffusion. For this purpose, a scattering function for fractal diffusion has been developed. The fractal diffusion dimension d w and the Hausdorff dimension d f have been determined on the length scales covered in the neutron scattering experiments.« less

Fractal diffusion in high temperature polymer electrolyte fuel cell membranes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hopfenmuller, Bernhard; Zorn, Reiner; Holderer, Olaf

In this paper, the performance of fuel cells depends largely on the proton diffusion in the proton conducting membrane, the core of a fuel cell. High temperature polymer electrolyte fuel cells are based on a polymer membrane swollen with phosphoric acid as the electrolyte, where proton conduction takes place. We studied the proton diffusion in such membranes with neutron scattering techniques which are especially sensitive to the proton contribution. Time of flight spectroscopy and backscattering spectroscopy have been combined to cover a broad dynamic range. In order to selectively observe the diffusion of protons potentially contributing to the ion conductivity,more » two samples were prepared, where in one of the samples the phosphoric acid was used with hydrogen replaced by deuterium. The scattering data from the two samples were subtracted in a suitable way after measurement. Thereby subdiffusive behavior of the proton diffusion has been observed and interpreted in terms of a model of fractal diffusion. For this purpose, a scattering function for fractal diffusion has been developed. The fractal diffusion dimension d w and the Hausdorff dimension d f have been determined on the length scales covered in the neutron scattering experiments.« less

a New Method for Calculating the Fractal Dimension of Surface Topography

NASA Astrophysics Data System (ADS)

Zuo, Xue; Zhu, Hua; Zhou, Yuankai; Li, Yan

2015-06-01

A new method termed as three-dimensional root-mean-square (3D-RMS) method, is proposed to calculate the fractal dimension (FD) of machined surfaces. The measure of this method is the root-mean-square value of surface data, and the scale is the side length of square in the projection plane. In order to evaluate the calculation accuracy of the proposed method, the isotropic surfaces with deterministic FD are generated based on the fractional Brownian function and Weierstrass-Mandelbrot (WM) fractal function, and two kinds of anisotropic surfaces are generated by stretching or rotating a WM fractal curve. Their FDs are estimated by the proposed method, as well as differential boxing-counting (DBC) method, triangular prism surface area (TPSA) method and variation method (VM). The results show that the 3D-RMS method performs better than the other methods with a lower relative error for both isotropic and anisotropic surfaces, especially for the surfaces with dimensions higher than 2.5, since the relative error between the estimated value and its theoretical value decreases with theoretical FD. Finally, the electrodeposited surface, end-turning surface and grinding surface are chosen as examples to illustrate the application of 3D-RMS method on the real machined surfaces. This method gives a new way to accurately calculate the FD from the surface topographic data.

Fractal Viscous Fingering in Fracture Networks

NASA Astrophysics Data System (ADS)

Boyle, E.; Sams, W.; Ferer, M.; Smith, D. H.

2007-12-01

We have used two very different physical models and computer codes to study miscible injection of a low- viscosity fluid into a simple fracture network, where it displaces a much-more viscous "defending" fluid through "rock" that is otherwise impermeable. The one code (NETfLow) is a standard pore level model, originally intended to treat laboratory-scale experiments; it assumes negligible mixing of the two fluids. The other code (NFFLOW) was written to treat reservoir-scale engineering problems; It explicitly treats the flow through the fractures and allows for significant mixing of the fluids at the interface. Both codes treat the fractures as parallel plates, of different effective apertures. Results are presented for the composition profiles from both codes. Independent of the degree of fluid-mixing, the profiles from both models have a functional form identical to that for fractal viscous fingering (i.e., diffusion limited aggregation, DLA). The two codes that solve the equations for different models gave similar results; together they suggest that the injection of a low-viscosity fluid into large- scale fracture networks may be much more significantly affected by fractal fingering than previously illustrated.

Surface and mass fractals in vapor-phase aggregates

NASA Astrophysics Data System (ADS)

Hurd, Alan J.; Schaefer, Dale W.; Martin, James E.

1987-03-01

Several types of fumed-silica aggregates with differing surface areas were studied over a wide range of spatial resolution by employing both light and neutron scattering. At intermediate length scales, between 100 and 1000 Å, the aggregates are mass fractals with Dm~=1.7-2.0, in basic agreement with simulations of aggregating clusters. At short length scales below 100 Å where the nature of the surfaces of the primary particles dominates the scattering, some of the samples appear to be fractally rough. In particular, a higher surface area seems to be correlated not with smaller primary particles in the aggregates, as previously assumed, but with fractally rough surfaces having Ds as high as 2.5. These may be the first materials discovered to have both mass and surface fractal structure.

Fractals and the irreducibility of consciousness in plants and animals.

PubMed

Gardiner, John

2013-08-01

In both plants and animals consciousness is fractal. Since fractals can only pass information in one direction it is impossible to extrapolate backward to find the rule that governs the fractal. Thus, similarly, it will be impossible to completely determine the rule or rules that govern consciousness.

A new numerical approximation of the fractal ordinary differential equation

NASA Astrophysics Data System (ADS)

Atangana, Abdon; Jain, Sonal

2018-02-01

The concept of fractal medium is present in several real-world problems, for instance, in the geological formation that constitutes the well-known subsurface water called aquifers. However, attention has not been quite devoted to modeling for instance, the flow of a fluid within these media. We deem it important to remind the reader that the concept of fractal derivative is not to represent the fractal sharps but to describe the movement of the fluid within these media. Since this class of ordinary differential equations is highly complex to solve analytically, we present a novel numerical scheme that allows to solve fractal ordinary differential equations. Error analysis of the method is also presented. Application of the method and numerical approximation are presented for fractal order differential equation. The stability and the convergence of the numerical schemes are investigated in detail. Also some exact solutions of fractal order differential equations are presented and finally some numerical simulations are presented.

Robust estimation of fractal measures for characterizing the structural complexity of the human brain: optimization and reproducibility

PubMed Central

Goñi, Joaquín; Sporns, Olaf; Cheng, Hu; Aznárez-Sanado, Maite; Wang, Yang; Josa, Santiago; Arrondo, Gonzalo; Mathews, Vincent P; Hummer, Tom A; Kronenberger, William G; Avena-Koenigsberger, Andrea; Saykin, Andrew J.; Pastor, María A.

2013-01-01

High-resolution isotropic three-dimensional reconstructions of human brain gray and white matter structures can be characterized to quantify aspects of their shape, volume and topological complexity. In particular, methods based on fractal analysis have been applied in neuroimaging studies to quantify the structural complexity of the brain in both healthy and impaired conditions. The usefulness of such measures for characterizing individual differences in brain structure critically depends on their within-subject reproducibility in order to allow the robust detection of between-subject differences. This study analyzes key analytic parameters of three fractal-based methods that rely on the box-counting algorithm with the aim to maximize within-subject reproducibility of the fractal characterizations of different brain objects, including the pial surface, the cortical ribbon volume, the white matter volume and the grey matter/white matter boundary. Two separate datasets originating from different imaging centers were analyzed, comprising, 50 subjects with three and 24 subjects with four successive scanning sessions per subject, respectively. The reproducibility of fractal measures was statistically assessed by computing their intra-class correlations. Results reveal differences between different fractal estimators and allow the identification of several parameters that are critical for high reproducibility. Highest reproducibility with intra-class correlations in the range of 0.9–0.95 is achieved with the correlation dimension. Further analyses of the fractal dimensions of parcellated cortical and subcortical gray matter regions suggest robustly estimated and region-specific patterns of individual variability. These results are valuable for defining appropriate parameter configurations when studying changes in fractal descriptors of human brain structure, for instance in studies of neurological diseases that do not allow repeated measurements or for disease

Fractality of eroded coastlines of correlated landscapes.

PubMed

Morais, P A; Oliveira, E A; Araújo, N A M; Herrmann, H J; Andrade, J S

2011-07-01

Using numerical simulations of a simple sea-coast mechanical erosion model, we investigate the effect of spatial long-range correlations in the lithology of coastal landscapes on the fractal behavior of the corresponding coastlines. In the model, the resistance of a coast section to erosion depends on the local lithology configuration as well as on the number of neighboring sea sides. For weak sea forces, the sea is trapped by the coastline and the eroding process stops after some time. For strong sea forces erosion is perpetual. The transition between these two regimes takes place at a critical sea force, characterized by a fractal coastline front. For uncorrelated landscapes, we obtain, at the critical value, a fractal dimension D=1.33, which is consistent with the dimension of the accessible external perimeter of the spanning cluster in two-dimensional percolation. For sea forces above the critical value, our results indicate that the coastline is self-affine and belongs to the Kardar-Parisi-Zhang universality class. In the case of landscapes generated with power-law spatial long-range correlations, the coastline fractal dimension changes continuously with the Hurst exponent H, decreasing from D=1.34 to 1.04, for H=0 and 1, respectively. This nonuniversal behavior is compatible with the multitude of fractal dimensions found for real coastlines.

Fractals and the irreducibility of consciousness in plants and animals

PubMed Central

Gardiner, John

2013-01-01

In both plants and animals consciousness is fractal. Since fractals can only pass information in one direction it is impossible to extrapolate backward to find the rule that governs the fractal. Thus, similarly, it will be impossible to completely determine the rule or rules that govern consciousness. PMID:23759545

Persistent fluctuations in stride intervals under fractal auditory stimulation.

PubMed

Marmelat, Vivien; Torre, Kjerstin; Beek, Peter J; Daffertshofer, Andreas

2014-01-01

Stride sequences of healthy gait are characterized by persistent long-range correlations, which become anti-persistent in the presence of an isochronous metronome. The latter phenomenon is of particular interest because auditory cueing is generally considered to reduce stride variability and may hence be beneficial for stabilizing gait. Complex systems tend to match their correlation structure when synchronizing. In gait training, can one capitalize on this tendency by using a fractal metronome rather than an isochronous one? We examined whether auditory cues with fractal variations in inter-beat intervals yield similar fractal inter-stride interval variability as isochronous auditory cueing in two complementary experiments. In Experiment 1, participants walked on a treadmill while being paced by either an isochronous or a fractal metronome with different variation strengths between beats in order to test whether participants managed to synchronize with a fractal metronome and to determine the necessary amount of variability for participants to switch from anti-persistent to persistent inter-stride intervals. Participants did synchronize with the metronome despite its fractal randomness. The corresponding coefficient of variation of inter-beat intervals was fixed in Experiment 2, in which participants walked on a treadmill while being paced by non-isochronous metronomes with different scaling exponents. As expected, inter-stride intervals showed persistent correlations similar to self-paced walking only when cueing contained persistent correlations. Our results open up a new window to optimize rhythmic auditory cueing for gait stabilization by integrating fractal fluctuations in the inter-beat intervals.

CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY: The Fractal Dimensions of Complex Networks

NASA Astrophysics Data System (ADS)

Guo, Long; Cai, XU

2009-08-01

It is shown that many real complex networks share distinctive features, such as the small-world effect and the heterogeneous property of connectivity of vertices, which are different from random networks and regular lattices. Although these features capture the important characteristics of complex networks, their applicability depends on the style of networks. To unravel the universal characteristics many complex networks have in common, we study the fractal dimensions of complex networks using the method introduced by Shanker. We find that the average 'density' (ρ(r)) of complex networks follows a better power-law function as a function of distance r with the exponent df, which is defined as the fractal dimension, in some real complex networks. Furthermore, we study the relation between df and the shortcuts Nadd in small-world networks and the size N in regular lattices. Our present work provides a new perspective to understand the dependence of the fractal dimension df on the complex network structure.

Methods of nanoassembly of a fractal polymer and materials formed thereby

DOE Office of Scientific and Technical Information (OSTI.GOV)

Newkome, George R; Moorefield, Charles N

2012-07-24

The invention relates to the formation of synthesized fractal constructs and the methods of chemical self-assembly for the preparation of a non-dendritic, nano-scale, fractal constructs or molecules. More particularly, the invention relates to fractal constructs formed by molecular self-assembly, to create synthetic, nanometer-scale fractal shapes. In an embodiment, a nanoscale Sierpinski hexagonal gasket is formed. This non-dendritic, perfectly self-similar fractal macromolecule is comprised of bisterpyridine building blocks that are bound together by coordination to 36 Ru and 6 Fe ions to form a nearly planar array of increasingly larger hexagons around a hollow center.

Methods of nanoassembly of a fractal polymer and materials formed thereby

DOE Office of Scientific and Technical Information (OSTI.GOV)

Newkome, George R; Moorefield, Charles N

2014-09-23

The invention relates to the formation of synthesized fractal constructs and the methods of chemical self-assembly for the preparation of a non-dendritic, nano-scale, fractal constructs or molecules. More particularly, the invention relates to fractal constructs formed by molecular self-assembly, to create synthetic, nanometer-scale fractal shapes. In an embodiment, a nanoscale Sierpinski hexagonal gasket is formed. This non-dendritic, perfectly self-similar fractal macromolecule is comprised of bisterpyridine building blocks that are bound together by coordination to (36) Ru and (6) Fe ions to form a nearly planar array of increasingly larger hexagons around a hollow center.

Fractal rigidity in migraine

NASA Astrophysics Data System (ADS)

Latka, Miroslaw; Glaubic-Latka, Marta; Latka, Dariusz; West, Bruce J.

2004-04-01

We study the middle cerebral artery blood flow velocity (MCAfv) in humans using transcranial Doppler ultrasonography (TCD). Scaling properties of time series of the axial flow velocity averaged over a cardiac beat interval may be characterized by two exponents. The short time scaling exponent (STSE) determines the statistical properties of fluctuations of blood flow velocities in short-time intervals while the Hurst exponent describes the long-term fractal properties. In many migraineurs the value of the STSE is significantly reduced and may approach that of the Hurst exponent. This change in dynamical properties reflects the significant loss of short-term adaptability and the overall hyperexcitability of the underlying cerebral blood flow control system. We call this effect fractal rigidity.

Is volcanic phenomena of fractal nature?

NASA Astrophysics Data System (ADS)

Quevedo, R.; Lopez, D. A. L.; Alparone, S.; Hernandez Perez, P. A.; Sagiya, T.; Barrancos, J.; Rodriguez-Santana, A. A.; Ramos, A.; Calvari, S.; Perez, N. M.

2016-12-01

A particular resonance waveform pattern has been detected beneath different physical volcano manifestations from recent 2011-2012 period of volcanic unrest at El Hierro Island, Canary Islands, and also from other worldwide volcanoes with different volcanic typology. This mentioned pattern appears to be a fractal time dependent waveform repeated in different time scales (periods of time). This time dependent feature suggests this resonance as a new approach to volcano phenomena for predicting such interesting matters as earthquakes, gas emission, deformation etc. as this fractal signal has been discovered hidden in a wide typical volcanic parameters measurements. It is known that the resonance phenomenon occurring in nature usually denote a structure, symmetry or a subjacent law (Fermi et al., 1952; and later -about enhanced cross-sections symmetry in protons collisions), which, in this particular case, may be indicative of some physical interactions showing a sequence not completely chaotic but cyclic provided with symmetries. The resonance and fractal model mentioned allowed the authors to make predictions in cycles from a few weeks to months. In this work an equation for this waveform has been described and also correlations with volcanic parameters and fractal behavior demonstration have been performed, including also some suggestive possible explanations of this signal origin.

Seeing shapes in seemingly random spatial patterns: Fractal analysis of Rorschach inkblots

PubMed Central

Taylor, R. P.; Martin, T. P.; Montgomery, R. D.; Smith, J. H.; Micolich, A. P.; Boydston, C.; Scannell, B. C.; Fairbanks, M. S.; Spehar, B.

2017-01-01

Rorschach inkblots have had a striking impact on the worlds of art and science because of the remarkable variety of associations with recognizable and namable objects they induce. Originally adopted as a projective psychological tool to probe mental health, psychologists and artists have more recently interpreted the variety of induced images simply as a signature of the observers’ creativity. Here we analyze the relationship between the spatial scaling parameters of the inkblot patterns and the number of induced associations, and suggest that the perceived images are induced by the fractal characteristics of the blot edges. We discuss how this relationship explains the frequent observation of images in natural scenery. PMID:28196082

Fractal Music: The Mathematics Behind "Techno" Music

ERIC Educational Resources Information Center

Padula, Janice

2005-01-01

This article describes sound waves, their basis in the sine curve, Fourier's theorem of infinite series, the fractal equation and its application to the composition of music, together with algorithms (such as those employed by meteorologist Edward Lorenz in his discovery of chaos theory) that are now being used to compose fractal music on…

Enhanced Graphene Photodetector with Fractal Metasurface.

PubMed

Fang, Jieran; Wang, Di; DeVault, Clayton T; Chung, Ting-Fung; Chen, Yong P; Boltasseva, Alexandra; Shalaev, Vladimir M; Kildishev, Alexander V

2017-01-11

Graphene has been demonstrated to be a promising photodetection material because of its ultrabroadband optical absorption, compatibility with CMOS technology, and dynamic tunability in optical and electrical properties. However, being a single atomic layer thick, graphene has intrinsically small optical absorption, which hinders its incorporation with modern photodetecting systems. In this work, we propose a gold snowflake-like fractal metasurface design to realize broadband and polarization-insensitive plasmonic enhancement in graphene photodetector. We experimentally obtain an enhanced photovoltage from the fractal metasurface that is an order of magnitude greater than that generated at a plain gold-graphene edge and such an enhancement in the photovoltage sustains over the entire visible spectrum. We also observed a relatively constant photoresponse with respect to polarization angles of incident light, as a result of the combination of two orthogonally oriented concentric hexagonal fractal geometries in one metasurface.

A Fractal Permeability Model for Shale Oil Reservoir

NASA Astrophysics Data System (ADS)

Zhang, Tao; Dong, Mingzhe; Li, Yajun

2018-01-01

In this work, a fractal analytical model is proposed to predict the permeability of shale reservoir. The proposed model explicitly relates the permeability to the micro-structural parameters (tortuosity, pore area fractal dimensions, porosity and slip velocity coefficient) of shale.

Dynamic fractals in spatial evolutionary games

NASA Astrophysics Data System (ADS)

Kolotev, Sergei; Malyutin, Aleksandr; Burovski, Evgeni; Krashakov, Sergei; Shchur, Lev

2018-06-01

We investigate critical properties of a spatial evolutionary game based on the Prisoner's Dilemma. Simulations demonstrate a jump in the component densities accompanied by drastic changes in average sizes of the component clusters. We argue that the cluster boundary is a random fractal. Our simulations are consistent with the fractal dimension of the boundary being equal to 2, and the cluster boundaries are hence asymptotically space filling as the system size increases.

Is Fractal 1/f Scaling in Stream Chemistry Universal?

NASA Astrophysics Data System (ADS)

Hrachowitz, M.

2016-12-01

Stream water chemistry data from catchments worldwide suggest that catchments act as filters that transform white noise, i.e. random input signals such as in precipitation, into 1/fαnoise whose slope in a power spectrum typically ranges between -0.5>α> -1.5. This previously lead to the hypothesis that catchments act as fractal filters, i.e. a slope of α=-1 may be a universal and intrinsic property of catchments. That would have considerable implications on the predictability of stream water chemistry, as both, temporal short- and long-range interdependence control the system response. While short memories and thus flatter slopes with α closer to 0 indicate poor short term but good long-term predictability, steeper slopes (α <<-1) indicate the opposite. In fractal systems, i.e. α=-1, this therefore leads to inherent problems of predicting both, short and long-term response patterns. The hypothesis of catchments acting as fractal filters remains to be tested more profoundly. It is not yet clear, if observed inter-catchment variations in α need to be interpreted as noise in the signal or if the variations underlie a systematic pattern and can be explained by some characteristic of catchment function. Here we will test the hypothesis that the spectral slope of stream water chemistry is not necessarily α=-1 and that catchments therefore do not inherently act as fractal filters. Further, it will be tested if closer links between the variations in spectral slope and hydrological function of catchments can be identified. The combined data-analysis and modelling study uses hydrochemical data (i.e. Cl-) from a wide range of catchments worldwide. The study catchments are physically contrasting, from distinct climate zones, and with distinct landscapes and vegetation. To identify patterns in the variations of α, firstly the power spectra of observed stream chemistry are compared with physical catchment characteristics using methods such as cluster analysis. In a

Fractal continuum model for tracer transport in a porous medium.

PubMed

Herrera-Hernández, E C; Coronado, M; Hernández-Coronado, H

2013-12-01

A model based on the fractal continuum approach is proposed to describe tracer transport in fractal porous media. The original approach has been extended to treat tracer transport and to include systems with radial and uniform flow, which are cases of interest in geoscience. The models involve advection due to the fluid motion in the fractal continuum and dispersion whose mathematical expression is taken from percolation theory. The resulting advective-dispersive equations are numerically solved for continuous and for pulse tracer injection. The tracer profile and the tracer breakthrough curve are evaluated and analyzed in terms of the fractal parameters. It has been found in this work that anomalous transport frequently appears, and a condition on the fractal parameter values to predict when sub- or superdiffusion might be expected has been obtained. The fingerprints of fractality on the tracer breakthrough curve in the explored parameter window consist of an early tracer breakthrough and long tail curves for the spherical and uniform flow cases, and symmetric short tailed curves for the radial flow case.

Chimera states in brain networks: Empirical neural vs. modular fractal connectivity

NASA Astrophysics Data System (ADS)

Chouzouris, Teresa; Omelchenko, Iryna; Zakharova, Anna; Hlinka, Jaroslav; Jiruska, Premysl; Schöll, Eckehard

2018-04-01

Complex spatiotemporal patterns, called chimera states, consist of coexisting coherent and incoherent domains and can be observed in networks of coupled oscillators. The interplay of synchrony and asynchrony in complex brain networks is an important aspect in studies of both the brain function and disease. We analyse the collective dynamics of FitzHugh-Nagumo neurons in complex networks motivated by its potential application to epileptology and epilepsy surgery. We compare two topologies: an empirical structural neural connectivity derived from diffusion-weighted magnetic resonance imaging and a mathematically constructed network with modular fractal connectivity. We analyse the properties of chimeras and partially synchronized states and obtain regions of their stability in the parameter planes. Furthermore, we qualitatively simulate the dynamics of epileptic seizures and study the influence of the removal of nodes on the network synchronizability, which can be useful for applications to epileptic surgery.

Persistent Fluctuations in Stride Intervals under Fractal Auditory Stimulation

PubMed Central

Marmelat, Vivien; Torre, Kjerstin; Beek, Peter J.; Daffertshofer, Andreas

2014-01-01

Stride sequences of healthy gait are characterized by persistent long-range correlations, which become anti-persistent in the presence of an isochronous metronome. The latter phenomenon is of particular interest because auditory cueing is generally considered to reduce stride variability and may hence be beneficial for stabilizing gait. Complex systems tend to match their correlation structure when synchronizing. In gait training, can one capitalize on this tendency by using a fractal metronome rather than an isochronous one? We examined whether auditory cues with fractal variations in inter-beat intervals yield similar fractal inter-stride interval variability as isochronous auditory cueing in two complementary experiments. In Experiment 1, participants walked on a treadmill while being paced by either an isochronous or a fractal metronome with different variation strengths between beats in order to test whether participants managed to synchronize with a fractal metronome and to determine the necessary amount of variability for participants to switch from anti-persistent to persistent inter-stride intervals. Participants did synchronize with the metronome despite its fractal randomness. The corresponding coefficient of variation of inter-beat intervals was fixed in Experiment 2, in which participants walked on a treadmill while being paced by non-isochronous metronomes with different scaling exponents. As expected, inter-stride intervals showed persistent correlations similar to self-paced walking only when cueing contained persistent correlations. Our results open up a new window to optimize rhythmic auditory cueing for gait stabilization by integrating fractal fluctuations in the inter-beat intervals. PMID:24651455

Fractal characterization of a fractured chalk reservoir - The Laegerdorf case

DOE Office of Scientific and Technical Information (OSTI.GOV)

Stoelum, H.H.; Koestler, A.G.; Feder, J.

1991-03-01

What is the matrix block size distribution of a fractured reservoir In order to answer this question and assess the potential of fractal geometry as a method of characterization of fracture networks, a pilot study has been done of the fractured chalk quarry in Laegerdorf. The fractures seen on the quarry walls were traced in the field for a total area of {approximately}200 {times} 45 m. The digitized pictures have been analyzed by a standard box-counting method. This analysis gave a fractal dimension of similarity varying from 1.33 for fractured areas between faults, to 1.43 for the fault zone, andmore » 1.53 for the highly deformed fault gouge. The amplitude showed a similar trend. The fractal dimension for the whole system of fractures is {approximately}1.55. In other words, fracture networks in chalk have a nonlinear, fractal geometry, and so matrix block size is a scaling property of chalk reservoirs. In terms of rock mechanics, the authors interpret the variation of the fractal dimension as follows: A small fractal dimension and amplitude are associated with brittle deformation in the elastic regime, while a large fractal dimension and amplitude are associated with predominantly ductile, strain softening deformation in the plastic regime. The interaction between the two regimes of deformation in the rock body is a key element of successful characterization and may be approached by seeing the rock as a non-Newtonian viscoelastic medium. The fractal dimension for the whole is close to a material independent limit that constrains the development of fractures.« less

Fractal and multifractal analyses of bipartite networks

NASA Astrophysics Data System (ADS)

Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua

2017-03-01

Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions.

Fractal and multifractal analyses of bipartite networks.

PubMed

Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua

2017-03-31

Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions.

Fractal and multifractal analyses of bipartite networks

PubMed Central

Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua

2017-01-01

Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions. PMID:28361962

Regularized Laplacian determinants of self-similar fractals

NASA Astrophysics Data System (ADS)

Chen, Joe P.; Teplyaev, Alexander; Tsougkas, Konstantinos

2018-06-01

We study the spectral zeta functions of the Laplacian on fractal sets which are locally self-similar fractafolds, in the sense of Strichartz. These functions are known to meromorphically extend to the entire complex plane, and the locations of their poles, sometimes referred to as complex dimensions, are of special interest. We give examples of locally self-similar sets such that their complex dimensions are not on the imaginary axis, which allows us to interpret their Laplacian determinant as the regularized product of their eigenvalues. We then investigate a connection between the logarithm of the determinant of the discrete graph Laplacian and the regularized one.

Percolation Laws of a Fractal Fracture-Pore Double Medium

NASA Astrophysics Data System (ADS)

Zhao, Yangsheng; Feng, Zengchao; Lv, Zhaoxing; Zhao, Dong; Liang, Weiguo

2016-12-01

The fracture-pore double porosity medium is one of the most common media in nature, for example, rock mass in strata. Fracture has a more significant effect on fluid flow than a pore in a fracture-pore double porosity medium. Hence, the fracture effect on percolation should be considered when studying the percolation phenomenon in porous media. In this paper, based on the fractal distribution law, three-dimensional (3D) fracture surfaces, and two-dimensional (2D) fracture traces in rock mass, the locations of fracture surfaces or traces are determined using a random function of uniform distribution. Pores are superimposed to build a fractal fracture-pore double medium. Numerical experiments were performed to show percolation phenomena in the fracture-pore double medium. The percolation threshold can be determined from three independent variables (porosity n, fracture fractal dimension D, and initial value of fracture number N0). Once any two are determined, the percolation probability exists at a critical point with the remaining parameter changing. When the initial value of the fracture number is greater than zero, the percolation threshold in the fracture-pore medium is much smaller than that in a pore medium. When the fracture number equals zero, the fracture-pore medium degenerates to a pore medium, and both percolation thresholds are the same.

Passenger flow analysis of Beijing urban rail transit network using fractal approach

NASA Astrophysics Data System (ADS)

Li, Xiaohong; Chen, Peiwen; Chen, Feng; Wang, Zijia

2018-04-01

To quantify the spatiotemporal distribution of passenger flow and the characteristics of an urban rail transit network, we introduce four radius fractal dimensions and two branch fractal dimensions by combining a fractal approach with passenger flow assignment model. These fractal dimensions can numerically describe the complexity of passenger flow in the urban rail transit network and its change characteristics. Based on it, we establish a fractal quantification method to measure the fractal characteristics of passenger follow in the rail transit network. Finally, we validate the reasonability of our proposed method by using the actual data of Beijing subway network. It has been shown that our proposed method can effectively measure the scale-free range of the urban rail transit network, network development and the fractal characteristics of time-varying passenger flow, which further provides a reference for network planning and analysis of passenger flow.

Fractal active contour model for segmenting the boundary of man-made target in nature scenes

NASA Astrophysics Data System (ADS)

Li, Min; Tang, Yandong; Wang, Lidi; Shi, Zelin

2006-02-01

In this paper, a novel geometric active contour model based on the fractal dimension feature to extract the boundary of man-made target in nature scenes is presented. In order to suppress the nature clutters, an adaptive weighting function is defined using the fractal dimension feature. Then the weighting function is introduced into the geodesic active contour model to detect the boundary of man-made target. Curve driven by our proposed model can evolve gradually from the initial position to the boundary of man-made target without being disturbed by nature clutters, even if the initial curve is far away from the true boundary. Experimental results validate the effectiveness and feasibility of our model.

Ulam method and fractal Weyl law for Perron-Frobenius operators

NASA Astrophysics Data System (ADS)

Ermann, L.; Shepelyansky, D. L.

2010-06-01

We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show numerically that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent ν = d-1, where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we numerically find the Weyl exponent ν = d/2 where d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation ν = d0/2 where d0 is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.

Resource Letter FR-1: Fractals

NASA Astrophysics Data System (ADS)

Hurd, Alan J.

1988-11-01

This Resource Letter provides a guide to the literature on fractals. Although ``fractal'' is a relatively new term in science, unifying many new ideas with established ones, its wide application and general popularity have made it one of the fastest growing fields in statistical physics. The letter E after an item indicates elementary level or material of general interest to persons becoming informed in the field; the letter I, for intermediate level, indicates material of somewhat more specialized nature; and the letter A indicates rather specialized or advanced material. An asterisk (*) indicates those articles to be included in an accompanying Reprint Book.

Fractal universe and quantum gravity.

PubMed

Calcagni, Gianluca

2010-06-25

We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energy momentum with the bulk with integer topological dimension. However, the total energy momentum is conserved. We consider the dynamics and the propagator of a scalar field. Implications for quantum gravity, cosmology, and the cosmological constant are discussed.

Surface areas of fractally rough particles studied by scattering

NASA Astrophysics Data System (ADS)

Hurd, Alan J.; Schaefer, Dale W.; Smith, Douglas M.; Ross, Steven B.; Le Méhauté, Alain; Spooner, Steven

1989-05-01

The small-angle scattering from fractally rough surfaces has the potential to give information on the surface area at a given resolution. By use of quantitative neutron and x-ray scattering, a direct comparison of surface areas of fractally rough powders was made between scattering and adsorption techniques. This study supports a recently proposed correction to the theory for scattering from fractal surfaces. In addition, the scattering data provide an independent calibration of molecular adsorbate areas.

Process for applying control variables having fractal structures

DOEpatents

Bullock, IV, Jonathan S.; Lawson, Roger L.

1996-01-01

A process and apparatus for the application of a control variable having a fractal structure to a body or process. The process of the present invention comprises the steps of generating a control variable having a fractal structure and applying the control variable to a body or process reacting in accordance with the control variable. The process is applicable to electroforming where first, second and successive pulsed-currents are applied to cause the deposition of material onto a substrate, such that the first pulsed-current, the second pulsed-current, and successive pulsed currents form a fractal pulsed-current waveform.

Process for applying control variables having fractal structures

DOEpatents

Bullock, J.S. IV; Lawson, R.L.

1996-01-23

A process and apparatus are disclosed for the application of a control variable having a fractal structure to a body or process. The process of the present invention comprises the steps of generating a control variable having a fractal structure and applying the control variable to a body or process reacting in accordance with the control variable. The process is applicable to electroforming where first, second and successive pulsed-currents are applied to cause the deposition of material onto a substrate, such that the first pulsed-current, the second pulsed-current, and successive pulsed currents form a fractal pulsed-current waveform. 3 figs.

A modified abstraction of Sierpiński fractals towards enhanced sensitivity of a cross-coupled bow-tie nanostructure

NASA Astrophysics Data System (ADS)

Hasan, Dihan; Lee, Chengkuo

2018-06-01

We experimentally demonstrate a modified abstraction of a fractal geometry (up to order M = 2), namely the Sierpiński fractal, with intrinsic self-similarity for a multitude of infrared sensing applications. The modification particularly strengthens the dipolar resonance and enables optical magnetism at longer wavelengths on a relatively miniaturized footprint. In contrast to the conventional resonant sensing, we harness the broadband electric field enhancement of the modified fractal patterns originating from the lightning rod effect in the non-resonant regime. We demonstrate strong enhancement of molecular absorption at mid-IR by the fractal patterns in the non-resonant regime even under extreme thermal broadening. Finally, we extend the work towards the functional study of the molecular fingerprint of ultra-thin film (∼5 nm) on a non-complementary metamaterial platform in the non-resonant regime. With the help of the solid state chemical dewetting of the monolayer, we also successfully demonstrate a new type of cross-coupling mediated sensitivity of the multispectral and mutually coupled fractal patterns. The research clearly indicates the usefulness of broadband electric field enhancement by the second order fractal pattern for on chip, complete profiling of mid-IR fingerprints of biological elements, i.e. cell, and protein monolayer on a limited footprint and under versatile morphological states.

Comprehensive Fractal Description of Porosity of Coal of Different Ranks

PubMed Central

Ren, Jiangang; Zhang, Guocheng; Song, Zhimin; Liu, Gaofeng; Li, Bing

2014-01-01

We selected, as the objects of our research, lignite from the Beizao Mine, gas coal from the Caiyuan Mine, coking coal from the Xiqu Mine, and anthracite from the Guhanshan Mine. We used the mercury intrusion method and the low-temperature liquid nitrogen adsorption method to analyze the structure and shape of the coal pores and calculated the fractal dimensions of different aperture segments in the coal. The experimental results show that the fractal dimension of the aperture segment of lignite, gas coal, and coking coal with an aperture of greater than or equal to 10 nm, as well as the fractal dimension of the aperture segment of anthracite with an aperture of greater than or equal to 100 nm, can be calculated using the mercury intrusion method; the fractal dimension of the coal pore, with an aperture range between 2.03 nm and 361.14 nm, can be calculated using the liquid nitrogen adsorption method, of which the fractal dimensions bounded by apertures of 10 nm and 100 nm are different. Based on these findings, we defined and calculated the comprehensive fractal dimensions of the coal pores and achieved the unity of fractal dimensions for full apertures of coal pores, thereby facilitating, overall characterization for the heterogeneity of the coal pore structure. PMID:24955407

Fractal density modeling of crustal heterogeneity from the KTB deep hole

NASA Astrophysics Data System (ADS)

Chen, Guoxiong; Cheng, Qiuming

2017-03-01

Fractal or multifractal concepts have significantly enlightened our understanding of crustal heterogeneity. Much attention has focused on 1/f scaling natures of physicochemical heterogeneity of Earth crust from fractal increment perspective. In this study, fractal density model from fractal clustering point of view is used to characterize the scaling behaviors of heterogeneous sources recorded at German Continental Deep Drilling Program (KTB) main hole, and of special contribution is the local and global multifractal analysis revisited by using Haar wavelet transform (HWT). Fractal density modeling of mass accumulation generalizes the unit of rock density from integer (e.g., g/cm3) to real numbers (e.g., g/cmα), so that crustal heterogeneities with respect to source accumulation are quantified by singularity strength of fractal density in α-dimensional space. From that perspective, we found that the bulk densities of metamorphic rocks exhibit fractal properties but have a weak multifractality, decreasing with the depth. The multiscaling natures of chemical logs also have been evidenced, and the observed distinct fractal laws for mineral contents are related to their different geochemical behaviors within complex lithological context. Accordingly, scaling distributions of mineral contents have been recognized as a main contributor to the multifractal natures of heterogeneous density for low-porosity crystalline rocks. This finally allows us to use de Wijs cascade process to explain the mechanism of fractal density. In practice, the proposed local singularity analysis based on HWT is suggested as an attractive high-pass filtering to amplify weak signatures of well logs as well as to delineate microlithological changes.

H-fractal seismic metamaterial with broadband low-frequency bandgaps

NASA Astrophysics Data System (ADS)

Du, Qiujiao; Zeng, Yi; Xu, Yang; Yang, Hongwu; Zeng, Zuoxun

2018-03-01

The application of metamaterial in civil engineering to achieve isolation of a building by controlling the propagation of seismic waves is a substantial challenge because seismic waves, a superposition of longitudinal and shear waves, are more complex than electromagnetic and acoustic waves. In this paper, we design a broadband seismic metamaterial based on H-shaped fractal pillars and report numerical simulation of band structures for seismic surface waves propagating. Comparative study on the band structures of H-fractal seismic metamaterials with different levels shows that a new level of fractal structure creates new band gap, widens the total band gaps and shifts the same band gap towards lower frequencies. Moreover, the vibration modes for H-fractal seismic metamaterials are computed and analyzed to clarify the mechanism of widening band gaps. A numerical investigation of seismic surface waves propagation on a 2D array of fractal unit cells on the surface of semi-infinite substrate is proposed to show the efficiency of earthquake shielding in multiple complete band gaps.

Human physiological benefits of viewing nature: EEG responses to exact and statistical fractal patterns.

PubMed

Hagerhall, C M; Laike, T; Küller, M; Marcheschi, E; Boydston, C; Taylor, R P

2015-01-01

Psychological and physiological benefits of viewing nature have been extensively studied for some time. More recently it has been suggested that some of these positive effects can be explained by nature's fractal properties. Virtually all studies on human responses to fractals have used stimuli that represent the specific form of fractal geometry found in nature, i.e. statistical fractals, as opposed to fractal patterns which repeat exactly at different scales. This raises the question of whether human responses like preference and relaxation are being driven by fractal geometry in general or by the specific form of fractal geometry found in nature. In this study we consider both types of fractals (statistical and exact) and morph one type into the other. Based on the Koch curve, nine visual stimuli were produced in which curves of three different fractal dimensions evolve gradually from an exact to a statistical fractal. The patterns were shown for one minute each to thirty-five subjects while qEEG was continuously recorded. The results showed that the responses to statistical and exact fractals differ, and that the natural form of the fractal is important for inducing alpha responses, an indicator of a wakefully relaxed state and internalized attention.

Anisotropic fractal media by vector calculus in non-integer dimensional space

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tarasov, Vasily E., E-mail: tarasov@theory.sinp.msu.ru

2014-08-15

A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensionalmore » space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.« less

Anisotropic fractal media by vector calculus in non-integer dimensional space

NASA Astrophysics Data System (ADS)

Tarasov, Vasily E.

2014-08-01

A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.

Turbulent premixed flames on fractal-grid-generated turbulence

NASA Astrophysics Data System (ADS)

Soulopoulos, N.; Kerl, J.; Sponfeldner, T.; Beyrau, F.; Hardalupas, Y.; Taylor, A. M. K. P.; Vassilicos, J. C.

2013-12-01

A space-filling, low blockage fractal grid is used as a novel turbulence generator in a premixed turbulent flame stabilized by a rod. The study compares the flame behaviour with a fractal grid to the behaviour when a standard square mesh grid with the same effective mesh size and solidity as the fractal grid is used. The isothermal gas flow turbulence characteristics, including mean flow velocity and rms of velocity fluctuations and Taylor length, were evaluated from hot-wire measurements. The behaviour of the flames was assessed with direct chemiluminescence emission from the flame and high-speed OH-laser-induced fluorescence. The characteristics of the two flames are considered in terms of turbulent flame thickness, local flame curvature and turbulent flame speed. It is found that, for the same flow rate and stoichiometry and at the same distance downstream of the location of the grid, fractal-grid-generated turbulence leads to a more turbulent flame with enhanced burning rate and increased flame surface area.

Self-stabilized Fractality of Sea-coasts Through Damped Erosion

NASA Astrophysics Data System (ADS)

Sapoval, B.; Baldassari, A.; Gabrielli, A.

2004-05-01

Coastline morphology is of current interest in geophysical research and coastline erosion has important economic consequences. At the same time, although the geometry of seacoasts is often used as an introductory archetype of fractal morphology in nature there has been no explanation about which physical mechanism could justify that empirical observation. The present work propose a minimal, but robust, model of evolution of rocky coasts towards fractality. The model describes how a stationary fractal geometry arises spontaneously from the mutual self-stabilization of a rocky coast morphology and sea eroding power. If, on one hand, erosion generally increases the geometrical irregularity of the coast, on the other hand this increase creates a stronger damping of the sea and a consequent diminution of its eroding power. The increased damping argument relies on the studies of fractal acoustical cavities, which have shown that viscous damping is augmented on a longer, irregular, surface. A minimal two-dimensional model of erosion is introduced which leads to the through a complex dynamics of the earth-sea interface, to the appearance of a stationary fractal seacoast with dimension close to 4/3. Fractal geometry plays here the role of a morphological attractor directly related to percolation geometry. The model reproduces at least qualitatively some of the features of real coasts using only simple ingredients: the randomness of the lithology and the decrease of the erosion power of the sea. B. Sapoval, Fractals (Aditech, Paris, 1989). B. Sapoval, O. Haeberlé, and S.Russ, J. Acoust. Soc. Am., 2014 (1997). B. Hébert B., B. Sapoval, and S.Russ, J. Acoust. Soc. Am., 1567 (1999).

Fractal Geometry in the Arts: AN Overview across the Different Cultures

NASA Astrophysics Data System (ADS)

Sala, Nicoletta

Fractal, in mathematics, is a geometric shape that is complex and detailed in structure at any level of magnification. The word "fractal" was coined less than thirty years ago by one of history's most creative and mathematicians, Benoit Mandelbrot, whose work, The Fractal Geometry of Nature, first introduced and explained concepts underlying this new vision of the geometry. Although other mathematical thinkers like Georg Cantor (1845-1918), Felix Hausdorff (1868-1942), Gaston Julia (1893-1978), Helge von Koch (1870-1924), Giuseppe Peano (1858-1932), Lewis Richardson (1891-1953), Waclaw Sierpinski (1882-1969) and others had attained isolated insights of fractal understanding, such ideas were largely ignored until Mandelbrot's genius forged them at a single blow into a gorgeously coherent and fascinating discipline. Fractal geometry is applied in different field now: engineering, physics, chemistry, biology, and architecture. The aim of this paper is to introduce an approach where the arts are analysed using a fractal point of view.

Influence of Turbulent Flow and Fractal Scaling on Effective Permeability of Fracture Network

NASA Astrophysics Data System (ADS)

Zhu, J.

2017-12-01

A new approach is developed to calculate hydraulic gradient dependent effective permeability of a fractal fracture network where both laminar and turbulent flows may occur in individual fractures. A critical fracture length is used to distinguish flow characteristics in individual fractures. The developed new solutions can be used for the case of a general scaling relationship, an extension to the linear scaling. We examine the impact on the effective permeability of the network of fractal fracture network characteristics, which include the fractal scaling coefficient and exponent, fractal dimension, ratio of minimum over maximum fracture lengths. Results demonstrate that the developed solution can explain more variations of the effective permeability in relation to the fractal dimensions estimated from the field observations. At high hydraulic gradient the effective permeability decreases with the fractal scaling exponent, but increases with the fractal scaling exponent at low gradient. The effective permeability increases with the scaling coefficient, fractal dimension, fracture length ratio and maximum fracture length.

Fractal based curves in musical creativity: A critical annotation

NASA Astrophysics Data System (ADS)

Georgaki, Anastasia; Tsolakis, Christos

In this article we examine fractal curves and synthesis algorithms in musical composition and research. First we trace the evolution of different approaches for the use of fractals in music since the 80's by a literature review. Furthermore, we review representative fractal algorithms and platforms that implement them. Properties such as self-similarity (pink noise), correlation, memory (related to the notion of Brownian motion) or non correlation at multiple levels (white noise), can be used to develop hierarchy of criteria for analyzing different layers of musical structure. L-systems can be applied in the modelling of melody in different musical cultures as well as in the investigation of musical perception principles. Finally, we propose a critical investigation approach for the use of artificial or natural fractal curves in systematic musicology.

Fractal 1/f Dynamics Suggest Entanglement of Measurement and Human Performance

ERIC Educational Resources Information Center

Holden, John G.; Choi, Inhyun; Amazeen, Polemnia G.; Van Orden, Guy

2011-01-01

Variability of repeated measurements in human performances exhibits fractal 1/f noise. Yet the relative strength of this fractal pattern varies widely across conditions, tasks, and individuals. Four experiments illustrate how subtle details of the conditions of measurement change the fractal patterns observed across task conditions. The results…

Edge detection of optical subaperture image based on improved differential box-counting method

NASA Astrophysics Data System (ADS)

Li, Yi; Hui, Mei; Liu, Ming; Dong, Liquan; Kong, Lingqin; Zhao, Yuejin

2018-01-01

Optical synthetic aperture imaging technology is an effective approach to improve imaging resolution. Compared with monolithic mirror system, the image of optical synthetic aperture system is often more complex at the edge, and as a result of the existence of gap between segments, which makes stitching becomes a difficult problem. So it is necessary to extract the edge of subaperture image for achieving effective stitching. Fractal dimension as a measure feature can describe image surface texture characteristics, which provides a new approach for edge detection. In our research, an improved differential box-counting method is used to calculate fractal dimension of image, then the obtained fractal dimension is mapped to grayscale image to detect edges. Compared with original differential box-counting method, this method has two improvements as follows: by modifying the box-counting mechanism, a box with a fixed height is replaced by a box with adaptive height, which solves the problem of over-counting the number of boxes covering image intensity surface; an image reconstruction method based on super-resolution convolutional neural network is used to enlarge small size image, which can solve the problem that fractal dimension can't be calculated accurately under the small size image, and this method may well maintain scale invariability of fractal dimension. The experimental results show that the proposed algorithm can effectively eliminate noise and has a lower false detection rate compared with the traditional edge detection algorithms. In addition, this algorithm can maintain the integrity and continuity of image edge in the case of retaining important edge information.

Study on Conversion Between Momentum and Contrarian Based on Fractal Game

NASA Astrophysics Data System (ADS)

Wu, Xu; Song, Guanghui; Deng, Yan; Xu, Lin

2015-06-01

Based on the fractal game which is performed by the majority and the minority, the fractal market theory (FMT) is employed to describe the features of investors' decision-making. Accordingly, the process of fractal games is formed in order to analyze the statistical features of conversion between momentum and contrarian. The result shows that among three fractal game mechanisms, the statistical feature of simulated return rate series is much more similar to log returns on actual series. In addition, the conversion between momentum and contrarian is also extremely similar to real situation, which can reflect the effectiveness of using fractal game in analyzing the conversion between momentum and contrarian. Moreover, it also provides decision-making reference which helps investors develop effective investment strategy.

Statistical Analysis of the Fractal Gating Motions of the Enzyme Acetylcholinesterase

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shen, T Y.; Tai, Kaihsu; Mccammon, Andy

The enzyme acetylcholinesterase has an active site that is accessible only by a gorge or main channel from the surface, and perhaps by secondary channels such as the back door. Molecular-dynamics simulations show that these channels are too narrow most of the time to admit substrate or other small molecules. Binding of substrates is therefore gated by structural fluctuations of the enzyme. Here, we analyze the fluctuations of these possible channels, as observed in the 10.8-ns trajectory of the simulation. The probability density function of the gorge proper radius (defined in the text) was calculated. A double-peak feature of themore » function was discovered and therefore two states with a threshold were identified. The relaxation (transition probability) functions of these two states were also calculated. The results revealed a power-law decay trend and an oscillation around it, which show properties of fractal dynamics with a complex exponent. The cross correlation of potential energy versus proper radius was also investigated. We discuss possible physical models behind the fractal protein dynamics; the dynamic hierarchical model for glassy systems is evaluated in detail.« less

Is fractal 1/f scaling in stream chemistry universal?

NASA Astrophysics Data System (ADS)

Hrachowitz, Markus

2016-04-01

Stream water chemistry data from catchments worldwide suggest that catchments act as filters that transform white noise, i.e. random, input signals such as in precipitation, into 1/f^α noise whose slope in a power spectrum typically ranges between -0.5>α>-1.5. This previously lead to the hypothesis that catchments act as fractal filters. In other words, it was posed that considering uncertainty, a slope of α=-1 may be a universal and intrinsic property of catchments. Such fractal scaling characteristics would have considerable implications on the predictability of stream water chemistry, as both, temporal short- and long-range interdependence and memory control the system response. While short memories and thus flatter slopes with α closer to 0 indicate poor short term but good long-term predictability, steeper slopes with values of α <<-1 indicate the opposite. In fractal systems, i.e. where α=-1, this therefore leads to inherent problems of robustly predicting both, short and long-term response patterns. The hypothesis of catchments acting as fractal filters (α=-1), however, remains to be tested more profoundly. It is, for example, not yet clear, if the observed inter-catchment variations in α indeed need to be interpreted as uncertainty and noise in the signal or if the variations underlie a systematic pattern and can be explained by some characteristic of catchment function, as was recently suggested in a modelling study based two experimental catchments (Hrachowitz et al., 2015). Here we will therefore further test the hypothesis that the spectral slope of stream water chemistry is not necessarily α=-1 and that catchments therefore do not inherently act as fractal filters. Further, it will be tested if closer links between the variations in spectral slope and hydrological function of catchments can be identified. The combined data-analysis and modelling study uses hydrochemical data (i.e. Cl- and O-18) from a wide range of catchments worldwide to

Fractal Analyses of High-Resolution Cloud Droplet Measurements.

NASA Astrophysics Data System (ADS)

Malinowski, Szymon P.; Leclerc, Monique Y.; Baumgardner, Darrel G.

1994-02-01

Fractal analyses of individual cloud droplet distributions using aircraft measurements along one-dimensional horizontal cross sections through clouds are performed. Box counting and cluster analyses are used to determine spatial scales of inhomogeneity of cloud droplet spacing. These analyses reveal that droplet spatial distributions do not exhibit a fractal behavior. A high variability in local droplet concentration in cloud volumes undergoing mixing was found. In these regions, thin filaments of cloudy air with droplet concentration close to those observed in cloud cores were found. Results suggest that these filaments may be anisotropic. Additional box counting analyses performed for various classes of cloud droplet diameters indicate that large and small droplets are similarly distributed, except for the larger characteristic spacing of large droplets.A cloud-clear air interface defined by a certain threshold of total droplet count (TDC) was investigated. There are indications that this interface is a convoluted surface of a fractal nature, at least in actively developing cumuliform clouds. In contrast, TDC in the cloud interior does not have fractal or multifractal properties. Finally a random Cantor set (RCS) was introduced as a model of a fractal process with an ill-defined internal scale. A uniform measure associated with the RCS after several generations was introduced to simulate the TDC records. Comparison of the model with real TDC records indicates similar properties of both types of data series.

Fractal and twin SVM-based handgrip recognition for healthy subjects and trans-radial amputees using myoelectric signal.

PubMed

Arjunan, Sridhar Poosapadi; Kumar, Dinesh Kant; Jayadeva J

2016-02-01

Identifying functional handgrip patterns using surface electromygram (sEMG) signal recorded from amputee residual muscle is required for controlling the myoelectric prosthetic hand. In this study, we have computed the signal fractal dimension (FD) and maximum fractal length (MFL) during different grip patterns performed by healthy and transradial amputee subjects. The FD and MFL of the sEMG, referred to as the fractal features, were classified using twin support vector machines (TSVM) to recognize the handgrips. TSVM requires fewer support vectors, is suitable for data sets with unbalanced distributions, and can simultaneously be trained for improving both sensitivity and specificity. When compared with other methods, this technique resulted in improved grip recognition accuracy, sensitivity, and specificity, and this improvement was significant (κ=0.91).

Fractal markets: Liquidity and investors on different time horizons

NASA Astrophysics Data System (ADS)

Li, Da-Ye; Nishimura, Yusaku; Men, Ming

2014-08-01

In this paper, we propose a new agent-based model to study the source of liquidity and the “emergent” phenomenon in financial market with fractal structure. The model rests on fractal market hypothesis and agents with different time horizons of investments. What is interesting is that though the agent-based model reveals that the interaction between these heterogeneous agents affects the stability and liquidity of the financial market the real world market lacks detailed data to bring it to light since it is difficult to identify and distinguish the investors with different time horizons in the empirical approach. results show that in a relatively short period of time fractal market provides liquidity from investors with different horizons and the market gains stability when the market structure changes from uniformity to diversification. In the real world the fractal structure with the finite of horizons can only stabilize the market within limits. With the finite maximum horizons, the greater diversity of the investors and the fractal structure will not necessarily bring more stability to the market which might come with greater fluctuation in large time scale.

A Fractal Excursion.

ERIC Educational Resources Information Center

Camp, Dane R.

1991-01-01

After introducing the two-dimensional Koch curve, which is generated by simple recursions on an equilateral triangle, the process is extended to three dimensions with simple recursions on a regular tetrahedron. Included, for both fractal sequences, are iterative formulae, illustrations of the first several iterations, and a sample PASCAL program.…

Interactions between a fractal tree-like object and hydrodynamic turbulence: flow structure and characteristic mixing length

NASA Astrophysics Data System (ADS)

Meneveau, C. V.; Bai, K.; Katz, J.

2011-12-01

The vegetation canopy has a significant impact on various physical and biological processes such as forest microclimate, rainfall evaporation distribution and climate change. Most scaled laboratory experimental studies have used canopy element models that consist of rigid vertical strips or cylindrical rods that can be typically represented through only one or a few characteristic length scales, for example the diameter and height for cylindrical rods. However, most natural canopies and vegetation are highly multi-scale with branches and sub-branches, covering a wide range of length scales. Fractals provide a convenient idealization of multi-scale objects, since their multi-scale properties can be described in simple ways (Mandelbrot 1982). While fractal aspects of turbulence have been studied in several works in the past decades, research on turbulence generated by fractal objects started more recently. We present an experimental study of boundary layer flow over fractal tree-like objects. Detailed Particle-Image-Velocimetry (PIV) measurements are carried out in the near-wake of a fractal-like tree. The tree is a pre-fractal with five generations, with three branches and a scale reduction factor 1/2 at each generation. Its similarity fractal dimension (Mandelbrot 1982) is D ~ 1.58. Detailed mean velocity and turbulence stress profiles are documented, as well as their downstream development. We then turn attention to the turbulence mixing properties of the flow, specifically to the question whether a mixing length-scale can be identified in this flow, and if so, how it relates to the geometric length-scales in the pre-fractal object. Scatter plots of mean velocity gradient (shear) and Reynolds shear stress exhibit good linear relation at all locations in the flow. Therefore, in the transverse direction of the wake evolution, the Boussinesq eddy viscosity concept is appropriate to describe the mixing. We find that the measured mixing length increases with increasing

Fusion of multiscale wavelet-based fractal analysis on retina image for stroke prediction.

PubMed

Che Azemin, M Z; Kumar, Dinesh K; Wong, T Y; Wang, J J; Kawasaki, R; Mitchell, P; Arjunan, Sridhar P

2010-01-01

In this paper, we present a novel method of analyzing retinal vasculature using Fourier Fractal Dimension to extract the complexity of the retinal vasculature enhanced at different wavelet scales. Logistic regression was used as a fusion method to model the classifier for 5-year stroke prediction. The efficacy of this technique has been tested using standard pattern recognition performance evaluation, Receivers Operating Characteristics (ROC) analysis and medical prediction statistics, odds ratio. Stroke prediction model was developed using the proposed system.

A fractal growth model: Exploring the connection pattern of hubs in complex networks

NASA Astrophysics Data System (ADS)

Li, Dongyan; Wang, Xingyuan; Huang, Penghe

2017-04-01

Fractal is ubiquitous in many real-world networks. Previous researches showed that the strong disassortativity between the hub-nodes on all length scales was the key principle that gave rise to the fractal architecture of networks. Although fractal property emerged in some models, there were few researches about the fractal growth model and quantitative analyses about the strength of the disassortativity for fractal model. In this paper, we proposed a novel inverse renormalization method, named Box-based Preferential Attachment (BPA), to build the fractal growth models in which the Preferential Attachment was performed at box level. The proposed models provided a new framework that demonstrated small-world-fractal transition. Also, we firstly demonstrated the statistical characteristic of connection patterns of the hubs in fractal networks. The experimental results showed that, given proper growing scale and added edges, the proposed models could clearly show pure small-world or pure fractal or both of them. It also showed that the hub connection ratio showed normal distribution in many real-world networks. At last, the comparisons of connection pattern between the proposed models and the biological and technical networks were performed. The results gave useful reference for exploring the growth principle and for modeling the connection patterns for real-world networks.

A fractal analysis of quaternary, Cenozoic-Mesozoic, and Late Pennsylvanian sea level changes

NASA Technical Reports Server (NTRS)

Hsui, Albert T.; Rust, Kelly A.; Klein, George D.

1993-01-01

Sea level changes are related to both climatic variations and tectonic movements. The fractal dimensions of several sea level curves were compared to a modern climatic fractal dimension of 1.26 established for annual precipitation records. A similar fractal dimension (1.22) based on delta(O-18/O-16) in deep-sea sediments has been suggested to characterize climatic change during the past 2 m.y. Our analysis indicates that sea level changes over the past 150,000 to 250,000 years also exhibit comparable fractal dimensions. Sea level changes for periods longer than about 30 m.y. are found to produce fractal dimensions closer to unity and Missourian (Late Pennsylvanian) sea level changes yield a fractal dimension of 1.41. The fact that these sea level curves all possess fractal dimensions less than 1.5 indicates that sea level changes exhibit nonperiodic, long-run persistence. The different fractal dimensions calculated for the various time periods could be the result of a characteristic overprinting of the sediment recored by prevailing processes during deposition. For example, during the Quaternary, glacio-eustatic sea level changes correlate well with the present climatic signature. During the Missourian, however, mechanisms such as plate reorganization may have dominated, resulting in a significantly different fractal dimension.

Fractal Clustering and Knowledge-driven Validation Assessment for Gene Expression Profiling.

PubMed

Wang, Lu-Yong; Balasubramanian, Ammaiappan; Chakraborty, Amit; Comaniciu, Dorin

2005-01-01

DNA microarray experiments generate a substantial amount of information about the global gene expression. Gene expression profiles can be represented as points in multi-dimensional space. It is essential to identify relevant groups of genes in biomedical research. Clustering is helpful in pattern recognition in gene expression profiles. A number of clustering techniques have been introduced. However, these traditional methods mainly utilize shape-based assumption or some distance metric to cluster the points in multi-dimension linear Euclidean space. Their results shows poor consistence with the functional annotation of genes in previous validation study. From a novel different perspective, we propose fractal clustering method to cluster genes using intrinsic (fractal) dimension from modern geometry. This method clusters points in such a way that points in the same clusters are more self-affine among themselves than to the points in other clusters. We assess this method using annotation-based validation assessment for gene clusters. It shows that this method is superior in identifying functional related gene groups than other traditional methods.

Percolation thresholds and fractal dimensions for square and cubic lattices with long-range correlated defects

NASA Astrophysics Data System (ADS)

Zierenberg, Johannes; Fricke, Niklas; Marenz, Martin; Spitzner, F. P.; Blavatska, Viktoria; Janke, Wolfhard

2017-12-01

We study long-range power-law correlated disorder on square and cubic lattices. In particular, we present high-precision results for the percolation thresholds and the fractal dimension of the largest clusters as a function of the correlation strength. The correlations are generated using a discrete version of the Fourier filtering method. We consider two different metrics to set the length scales over which the correlations decay, showing that the percolation thresholds are highly sensitive to such system details. By contrast, we verify that the fractal dimension df is a universal quantity and unaffected by the choice of metric. We also show that for weak correlations, its value coincides with that for the uncorrelated system. In two dimensions we observe a clear increase of the fractal dimension with increasing correlation strength, approaching df→2 . The onset of this change does not seem to be determined by the extended Harris criterion.

Morphology and Fractal Characterization of Multiscale Pore Structures for Organic-Rich Lacustrine Shale Reservoirs

NASA Astrophysics Data System (ADS)

Wang, Yang; Wu, Caifang; Zhu, Yanming; Chen, Shangbin; Liu, Shimin; Zhang, Rui

Lacustrine shale gas has received considerable attention and has been playing an important role in unconventional natural gas production in China. In this study, multiple techniques, including total organic carbon (TOC) analysis, X-ray diffraction (XRD) analysis, field emission scanning electron microscopy (FE-SEM), helium pycnometry and low-pressure N2 adsorption have been applied to characterize the pore structure of lacustrine shale of Upper Triassic Yanchang Formation from the Ordos Basin. The results show that organic matter (OM) pores are the most important type dominating the pore system, while interparticle (interP) pores, intraparticle (intraP) and microfractures are also usually observed between or within different minerals. The shapes of OM pores are less complex compared with the other two pore types based on the Image-Pro Plus software analysis. In addition, the specific surface area ranges from 2.76m2/g to 10.26m2/g and the pore volume varies between 0.52m3/100g and 1.31m3/100g. Two fractal dimensions D1 and D2 were calculated using Frenkel-Halsey-Hill (FHH) method, with D1 varying between 2.510 and 2.632, and D2 varying between 2.617 and 2.814. Further investigation indicates that the fractal dimensions exhibit positive correlations with TOC contents, whereas there is no definite relationship observed between fractal dimensions and clay minerals. Meanwhile, the fractal dimensions increase with the increase in specific surface area, and is negatively correlated with the pore size.

Exhaled Aerosol Pattern Discloses Lung Structural Abnormality: A Sensitivity Study Using Computational Modeling and Fractal Analysis

PubMed Central

Xi, Jinxiang; Si, Xiuhua A.; Kim, JongWon; Mckee, Edward; Lin, En-Bing

2014-01-01

Background Exhaled aerosol patterns, also called aerosol fingerprints, provide clues to the health of the lung and can be used to detect disease-modified airway structures. The key is how to decode the exhaled aerosol fingerprints and retrieve the lung structural information for a non-invasive identification of respiratory diseases. Objective and Methods In this study, a CFD-fractal analysis method was developed to quantify exhaled aerosol fingerprints and applied it to one benign and three malign conditions: a tracheal carina tumor, a bronchial tumor, and asthma. Respirations of tracer aerosols of 1 µm at a flow rate of 30 L/min were simulated, with exhaled distributions recorded at the mouth. Large eddy simulations and a Lagrangian tracking approach were used to simulate respiratory airflows and aerosol dynamics. Aerosol morphometric measures such as concentration disparity, spatial distributions, and fractal analysis were applied to distinguish various exhaled aerosol patterns. Findings Utilizing physiology-based modeling, we demonstrated substantial differences in exhaled aerosol distributions among normal and pathological airways, which were suggestive of the disease location and extent. With fractal analysis, we also demonstrated that exhaled aerosol patterns exhibited fractal behavior in both the entire image and selected regions of interest. Each exhaled aerosol fingerprint exhibited distinct pattern parameters such as spatial probability, fractal dimension, lacunarity, and multifractal spectrum. Furthermore, a correlation of the diseased location and exhaled aerosol spatial distribution was established for asthma. Conclusion Aerosol-fingerprint-based breath tests disclose clues about the site and severity of lung diseases and appear to be sensitive enough to be a practical tool for diagnosis and prognosis of respiratory diseases with structural abnormalities. PMID:25105680

Numerical study of electromagnetic scattering from one-dimensional nonlinear fractal sea surface

NASA Astrophysics Data System (ADS)

Xie, Tao; He, Chao; William, Perrie; Kuang, Hai-Lan; Zou, Guang-Hui; Chen, Wei

2010-02-01

In recent years, linear fractal sea surface models have been developed for the sea surface in order to establish an electromagnetic backscattering model. Unfortunately, the sea surface is always nonlinear, particularly at high sea states. We present a nonlinear fractal sea surface model and derive an electromagnetic backscattering model. Using this model, we numerically calculate the normalized radar cross section (NRCS) of a nonlinear sea surface. Comparing the averaged NRCS between linear and nonlinear fractal models, we show that the NRCS of a linear fractal sea surface underestimates the NRCS of the real sea surface, especially for sea states with high fractal dimensions, and for dominant ocean surface gravity waves that are either very short or extremely long.

Fractal Analysis of Brain Blood Oxygenation Level Dependent (BOLD) Signals from Children with Mild Traumatic Brain Injury (mTBI).

PubMed

Dona, Olga; Noseworthy, Michael D; DeMatteo, Carol; Connolly, John F

2017-01-01

Conventional imaging techniques are unable to detect abnormalities in the brain following mild traumatic brain injury (mTBI). Yet patients with mTBI typically show delayed response on neuropsychological evaluation. Because fractal geometry represents complexity, we explored its utility in measuring temporal fluctuations of brain resting state blood oxygen level dependent (rs-BOLD) signal. We hypothesized that there could be a detectable difference in rs-BOLD signal complexity between healthy subjects and mTBI patients based on previous studies that associated reduction in signal complexity with disease. Fifteen subjects (13.4 ± 2.3 y/o) and 56 age-matched (13.5 ± 2.34 y/o) healthy controls were scanned using a GE Discovery MR750 3T MRI and 32-channel RF-coil. Axial FSPGR-3D images were used to prescribe rs-BOLD (TE/TR = 35/2000ms), acquired over 6 minutes. Motion correction was performed and anatomical and functional images were aligned and spatially warped to the N27 standard atlas. Fractal analysis, performed on grey matter, was done by estimating the Hurst exponent using de-trended fluctuation analysis and signal summation conversion methods. Voxel-wise fractal dimension (FD) was calculated for every subject in the control group to generate mean and standard deviation maps for regional Z-score analysis. Voxel-wise validation of FD normality across controls was confirmed, and non-Gaussian voxels (3.05% over the brain) were eliminated from subsequent analysis. For each mTBI patient, regions where Z-score values were at least 2 standard deviations away from the mean (i.e. where |Z| > 2.0) were identified. In individual patients the frequently affected regions were amygdala (p = 0.02), vermis(p = 0.03), caudate head (p = 0.04), hippocampus(p = 0.03), and hypothalamus(p = 0.04), all previously reported as dysfunctional after mTBI, but based on group analysis. It is well known that the brain is best modeled as a complex system. Therefore a measure of complexity

Temporal fractals in seabird foraging behaviour: diving through the scales of time

PubMed Central

MacIntosh, Andrew J. J.; Pelletier, Laure; Chiaradia, Andre; Kato, Akiko; Ropert-Coudert, Yan

2013-01-01

Animal behaviour exhibits fractal structure in space and time. Fractal properties in animal space-use have been explored extensively under the Lévy flight foraging hypothesis, but studies of behaviour change itself through time are rarer, have typically used shorter sequences generated in the laboratory, and generally lack critical assessment of their results. We thus performed an in-depth analysis of fractal time in binary dive sequences collected via bio-logging from free-ranging little penguins (Eudyptula minor) across full-day foraging trips (216 data points; 4 orders of temporal magnitude). Results from 4 fractal methods show that dive sequences are long-range dependent and persistent across ca. 2 orders of magnitude. This fractal structure correlated with trip length and time spent underwater, but individual traits had little effect. Fractal time is a fundamental characteristic of penguin foraging behaviour, and its investigation is thus a promising avenue for research on interactions between animals and their environments. PMID:23703258

Fractal symmetry of protein interior: what have we learned?

PubMed

Banerji, Anirban; Ghosh, Indira

2011-08-01

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially, there has been a startling upsurge of research that innovatively stretches the limits of fractal-based studies to present an array of unexpected results on the biophysical properties of protein interior. In this article, we introduce readers to the fundamentals of fractals, reviewing the commonality (and the lack of it) between these approaches before exploring the patterns in the results that they produced. Clustering the approaches in major schools of protein self-similarity studies, we describe the evolution of fractal dimension-based methodologies. The genealogy of approaches (and results) presented here portrays a clear picture of the contemporary state of fractal-based studies in the context of the protein interior. To underline the utility of fractal dimension-based measures further, we have performed a correlation dimension analysis on all of the available non-redundant protein structures, both at the level of an individual protein and at the level of structural domains. In this investigation, we were able to separately quantify the self-similar symmetries in spatial correlation patterns amongst peptide-dipole units, charged amino acids, residues with the π-electron cloud and hydrophobic amino acids. The results revealed that electrostatic environments in the interiors of proteins belonging to 'α/α toroid' (all-α class) and 'PLP-dependent transferase-like' domains (α/β class) are highly conducive. In contrast, the interiors of 'zinc finger design' ('designed proteins') and 'knottins' ('small proteins') were identified as folds with the least conducive electrostatic

EUS elastography (strain ratio) and fractal-based quantitative analysis for the diagnosis of solid pancreatic lesions.

PubMed

Carrara, Silvia; Di Leo, Milena; Grizzi, Fabio; Correale, Loredana; Rahal, Daoud; Anderloni, Andrea; Auriemma, Francesco; Fugazza, Alessandro; Preatoni, Paoletta; Maselli, Roberta; Hassan, Cesare; Finati, Elena; Mangiavillano, Benedetto; Repici, Alessandro

2018-06-01

EUS elastography is useful in characterizing solid pancreatic lesions (SPLs), and fractal analysis-based technology has been used to evaluate geometric complexity in oncology. The aim of this study was to evaluate EUS elastography (strain ratio) and fractal analysis for the characterization of SPLs. Consecutive patients with SPLs were prospectively enrolled between December 2015 and February 2017. Elastographic evaluation included parenchymal strain ratio (pSR) and wall strain ratio (wSR) and was performed with a new compact US processor. Elastographic images were analyzed using a computer program to determine the 3-dimensional histogram fractal dimension. A composite cytology/histology/clinical reference standard was used to assess sensitivity, specificity, positive predictive value, negative predictive value, and area under the receiver operating curve. Overall, 102 SPLs from 100 patients were studied. At final diagnosis, 69 (68%) were malignant and 33 benign. At elastography, both pSR and wSR appeared to be significantly higher in malignant as compared with benign SPLs (pSR, 24.5 vs 6.4 [P < .001]; wSR, 56.6 vs 15.3 [P < .001]). When the best cut-off levels of pSR and wSR at 9.10 and 16.2, respectively, were used, sensitivity, specificity, positive predictive value, negative predictive value, and area under the receiver operating curve were 88.4%, 78.8%, 89.7%, 76.9%, and 86.7% and 91.3%, 69.7%, 86.5%, 80%, and 85.7%, respectively. Fractal analysis showed a significant statistical difference (P = .0087) between the mean surface fractal dimension of malignant lesions (D = 2.66 ± .01) versus neuroendocrine tumor (D = 2.73 ± .03) and a statistical difference for all 3 channels red, green, and blue (P < .0001). EUS elastography with pSR and fractal-based analysis are useful in characterizing SPLs. (Clinical trial registration number: NCT02855151.). Copyright © 2018 American Society for Gastrointestinal Endoscopy. Published by Elsevier Inc. All rights

Noninvasive fractal biomarker of clock neurotransmitter disturbance in humans with dementia

PubMed Central

Hu, Kun; Harper, David G.; Shea, Steven A.; Stopa, Edward G.; Scheer, Frank A. J. L.

2013-01-01

Human motor activity has a robust, intrinsic fractal structure with similar patterns from minutes to hours. The fractal activity patterns appear to be physiologically important because the patterns persist under different environmental conditions but are significantly altered/reduced with aging and Alzheimer's disease (AD). Here, we report that dementia patients, known to have disrupted circadian rhythmicity, also have disrupted fractal activity patterns and that the disruption is more pronounced in patients with more amyloid plaques (a marker of AD severity). Moreover, the degree of fractal activity disruption is strongly associated with vasopressinergic and neurotensinergic neurons (two major circadian neurotransmitters) in postmortem suprachiasmatic nucleus (SCN), and can better predict changes of the two neurotransmitters than traditional circadian measures. These findings suggest that the SCN impacts human activity regulation at multiple time scales and that disrupted fractal activity may serve as a non-invasive biomarker of SCN neurodegeneration in dementia. PMID:23863985

On fractal properties of arterial trees.

PubMed

Zamir, M

1999-04-21

The question of fractal properties of arterial trees is considered in light of data from the extensive tree structure of the right coronary artery of a human heart. Because of the highly non-uniform structure of this tree, the study focuses on the purely geometrical rather than statistical aspects of fractal properties. The large number of arterial bifurcations comprising the tree were found to have a mixed degree of asymmetry at all levels of the tree, including the depth of the tree where it has been generally supposed that they would be symmetrical. Cross-sectional area ratios of daughter to parent vessels were also found to be highly mixed at all levels, having values both above and below 1.0, rather than consistently above as has been generally supposed in the past. Calculated values of the power law index which describes the theoretical relation between the diameters of the three vessel segments at an arterial bifurcation were found to range far beyond the two values associated with the cube and square laws, and not clearly favoring one or the other. On the whole the tree structure was found to have what we have termed "pseudo-fractal" properties, in the sense that vessels of different calibers displayed the same branching pattern but with a range of values of the branching parameters. The results suggest that a higher degree of fractal character, one in which the branching parameters are constant throughout the tree structure, is unlikely to be attained in non-uniform vascular structures. Copyright 1999 Academic Press.

Two Dimensional Drug Diffusion Between Nanoparticles and Fractal Tumors

NASA Astrophysics Data System (ADS)

Samioti, S. E.; Karamanos, K.; Tsiantis, A.; Papathanasiou, A.; Sarris, I.

2017-11-01

Drug delivery methods based on nanoparticles are some of the most promising medical applications in nanotechnology to treat cancer. It is observed that drug released by nanoparticles to the cancer tumors may be driven by diffusion. A fractal tumor boundary of triangular Von Koch shape is considered here and the diffusion mechanism is studied for different drug concentrations and increased fractality. A high order Finite Elements method based on the Fenics library is incorporated in fine meshes to fully resolve these irregular boundaries. Drug concentration, its transfer rates and entropy production are calculated in an up to forth order fractal iteration boundaries. We observed that diffusion rate diminishes for successive prefractal generations. Also, the entropy production around the system changes greatly as the order of the fractal curve increases. Results indicate with precision where the active sites are, in which most of the diffusion takes place and thus drug arrives to the tumor.

Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake

NASA Astrophysics Data System (ADS)

Cherny, Alexander Yu.; Anitas, Eugen M.; Osipov, Vladimir A.; Kuklin, Alexander I.

The small-angle scattering (SAS) from the Cantor surface fractal on the plane and Koch snowflake is considered. We develop the construction algorithm for the Koch snowflake, which makes possible the recurrence relation for the scattering amplitude. The surface fractals can be decomposed into a sum of surface mass fractals for arbitrary fractal iteration, which enables various approximations for the scattering intensity. It is shown that for the Cantor fractal, one can neglect with a good accuracy the correlations between the mass fractal amplitudes, while for the Koch snowflake, these correlations are important. It is shown that nevertheless, the correlations can be build in the mass fractal amplitudes, which explains the decay of the scattering intensity $I(q)\\sim q^{D_{\\mathrm{s}}-4}$ with $1 < D_{\\mathrm{s}} < 2$ being the fractal dimension of the perimeter. The curve $I(q)q^{4-D_{\\mathrm{s}}}$ is found to be log-periodic in the fractal region with the period equal to the scaling factor of the fractal. The log-periodicity arises from the self-similarity of sizes of basic structural units rather than from correlations between their distances. A recurrence relation is obtained for the radius of gyration of Koch snowflake, which is solved in the limit of infinite iterations. The present analysis allows us to obtain additional information from SAS data, such as the edges of the fractal regions, the fractal iteration number and the scaling factor.

Fractal and Multifractal Models Applied to Porous Media - Editorial

USDA-ARS?s Scientific Manuscript database

Given the current high level of interest in the use of fractal geometry to characterize natural porous media, a special issue of the Vadose Zone Journal was organized in order to expose established fractal analysis techniques and cutting-edge new developments to a wider Earth science audience. The ...

Fractality and the law of the wall

NASA Astrophysics Data System (ADS)

Xu, Haosen H. A.; Yang, X. I. A.

2018-05-01

Fluid motions in the inertial range of isotropic turbulence are fractal, with their space-filling capacity slightly below regular three-dimensional objects, which is a consequence of the energy cascade. Besides the energy cascade, the other often encountered cascading process is the momentum cascade in wall-bounded flows. Despite the long-existing analogy between the two processes, many of the thoroughly investigated aspects of the energy cascade have so far received little attention in studies of the momentum counterpart, e.g., the possibility of the momentum-transferring scales in the logarithmic region being fractal has not been considered. In this work, this possibility is pursued, and we discuss one of its implications. Following the same dimensional arguments that lead to the D =2.33 fractal dimension of wrinkled surfaces in isotropic turbulence, we show that the large-scale momentum-carrying eddies may also be fractal and non-space-filling, which then leads to the power-law scaling of the mean velocity profile. The logarithmic law of the wall, on the other hand, corresponds to space-filling eddies, as suggested by Townsend [The Structure of Turbulent Shear Flow (Cambridge University Press, Cambridge, 1980)]. Because the space-filling capacity is an integral geometric quantity, the analysis presented in this work provides us with a low-order quantity, with which, one would be able to distinguish between the logarithmic law and the power law.

a Predictive Model of Permeability for Fractal-Based Rough Rock Fractures during Shear

NASA Astrophysics Data System (ADS)

Huang, Na; Jiang, Yujing; Liu, Richeng; Li, Bo; Zhang, Zhenyu

This study investigates the roles of fracture roughness, normal stress and shear displacement on the fluid flow characteristics through three-dimensional (3D) self-affine fractal rock fractures, whose surfaces are generated using the modified successive random additions (SRA) algorithm. A series of numerical shear-flow tests under different normal stresses were conducted on rough rock fractures to calculate the evolutions of fracture aperture and permeability. The results show that the rough surfaces of fractal-based fractures can be described using the scaling parameter Hurst exponent (H), in which H = 3 - Df, where Df is the fractal dimension of 3D single fractures. The joint roughness coefficient (JRC) distribution of fracture profiles follows a Gauss function with a negative linear relationship between H and average JRC. The frequency curves of aperture distributions change from sharp to flat with increasing shear displacement, indicating a more anisotropic and heterogeneous flow pattern. Both the mean aperture and permeability of fracture increase with the increment of surface roughness and decrement of normal stress. At the beginning of shear, the permeability increases remarkably and then gradually becomes steady. A predictive model of permeability using the mean mechanical aperture is proposed and the validity is verified by comparisons with the experimental results reported in literature. The proposed model provides a simple method to approximate permeability of fractal-based rough rock fractures during shear using fracture aperture distribution that can be easily obtained from digitized fracture surface information.

Navigation performance in virtual environments varies with fractal dimension of landscape.

PubMed

Juliani, Arthur W; Bies, Alexander J; Boydston, Cooper R; Taylor, Richard P; Sereno, Margaret E

2016-09-01

Fractal geometry has been used to describe natural and built environments, but has yet to be studied in navigational research. In order to establish a relationship between the fractal dimension (D) of a natural environment and humans' ability to navigate such spaces, we conducted two experiments using virtual environments that simulate the fractal properties of nature. In Experiment 1, participants completed a goal-driven search task either with or without a map in landscapes that varied in D. In Experiment 2, participants completed a map-reading and location-judgment task in separate sets of fractal landscapes. In both experiments, task performance was highest at the low-to-mid range of D, which was previously reported as most preferred and discriminable in studies of fractal aesthetics and discrimination, respectively, supporting a theory of visual fluency. The applicability of these findings to architecture, urban planning, and the general design of constructed spaces is discussed.

Complex Patterns in Financial Time Series Through HIGUCHI’S Fractal Dimension

NASA Astrophysics Data System (ADS)

Grace Elizabeth Rani, T. G.; Jayalalitha, G.

2016-11-01

This paper analyzes the complexity of stock exchanges through fractal theory. Closing price indices of four stock exchanges with different industry sectors are selected. Degree of complexity is assessed through Higuchi’s fractal dimension. Various window sizes are considered in evaluating the fractal dimension. It is inferred that the data considered as a whole represents random walk for all the four indices. Analysis of financial data through windowing procedure exhibits multi-fractality. Attempts to apply moving averages to reduce noise in the data revealed lower estimates of fractal dimension, which was verified using fractional Brownian motion. A change in the normalization factor in Higuchi’s algorithm did improve the results. It is quintessential to focus on rural development to realize a standard and steady growth of economy. Tools must be devised to settle the issues in this regard. Micro level institutions are necessary for the economic growth of a country like India, which would induce a sporadic development in the present global economical scenario.

Fractal scaling in bottlenose dolphin (Tursiops truncatus) echolocation: A case study

NASA Astrophysics Data System (ADS)

Perisho, Shaun T.; Kelty-Stephen, Damian G.; Hajnal, Alen; Houser, Dorian; Kuczaj, Stan A., II

2016-02-01

Fractal scaling patterns, which entail a power-law relationship between magnitude of fluctuations in a variable and the scale at which the variable is measured, have been found in many aspects of human behavior. These findings have led to advances in behavioral models (e.g. providing empirical support for cascade-driven theories of cognition) and have had practical medical applications (e.g. providing new methods for early diagnosis of medical conditions). In the present paper, fractal analysis is used to investigate whether similar fractal scaling patterns exist in inter-click interval and peak-peak amplitude measurements of bottlenose dolphin click trains. Several echolocation recordings taken from two male bottlenose dolphins were analyzed using Detrended Fluctuation Analysis and Higuchi's (1988) method for determination of fractal dimension. Both animals were found to exhibit fractal scaling patterns near what is consistent with persistent long range correlations. These findings suggest that recent advances in human cognition and medicine may have important parallel applications to echolocation as well.

Fuzzy fractals, chaos, and noise

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zardecki, A.

1997-05-01

To distinguish between chaotic and noisy processes, the authors analyze one- and two-dimensional chaotic mappings, supplemented by the additive noise terms. The predictive power of a fuzzy rule-based system allows one to distinguish ergodic and chaotic time series: in an ergodic series the likelihood of finding large numbers is small compared to the likelihood of finding them in a chaotic series. In the case of two dimensions, they consider the fractal fuzzy sets whose {alpha}-cuts are fractals, arising in the context of a quadratic mapping in the extended complex plane. In an example provided by the Julia set, the conceptmore » of Hausdorff dimension enables one to decide in favor of chaotic or noisy evolution.« less

Fractal Measure and Microscopic Modeling of Osseointegration.

PubMed

Santos, Leonardo Cavalcanti Bezerra; Carvalho, Alessandra Albuquerque; Leão, Jair Carneiro; Neto, Paulo Jose; Stosic, Tatijana; Stosic, Borko

2015-01-01

In this study, the process of osseointegration on titanium implant surfaces with different physicochemical treatments subjected to a simulated corporal fluid submersion was evaluated using the concept of fractal dimension. It was found that different treatments led to rather different calcium phosphate crystal growth patterns, with fractal dimension ranging from 1.68 to 1.93. The observed crystal patterns may be explained by a general deposition, diffusion, and aggregation growth mechanism, where diffusing particle sticking probability plays a fundamental role.

Fractal pharmacokinetics of the drug mibefradil in the liver

NASA Astrophysics Data System (ADS)

Fuite, J.; Marsh, R.; Tuszyński, J.

2002-08-01

We explore the ramifications of the fractal geometry of the key organ for drug elimination, the liver, on pharmacokinetic data analysis. A formalism is developed for the use of a combination of well-stirred Euclidean and fractal compartments in the body. Perturbation analysis is carried out to obtain analytical solutions for the drug concentration time evolution. These results are then fitted to experimental data collected from clinically instrumented dogs [see, A. Skerjanec et al., J. Pharm. Sci. 85, 189 (1995)] using the drug mibefradil. The thus obtained spectral fractal dimension has a range of values that is consistent with the value found in independently performed ultrasound experiments on the liver.

Relationship between the anomalous diffusion and the fractal dimension of the environment

NASA Astrophysics Data System (ADS)

Zhokh, Alexey; Trypolskyi, Andrey; Strizhak, Peter

2018-03-01

In this letter, we provide an experimental study highlighting a relation between the anomalous diffusion and the fractal dimension of the environment using the methanol anomalous transport through the porous solid pellets with various pores geometries and different chemical compositions. The anomalous diffusion exponent was derived from the non-integer order of the time-fractional diffusion equation that describes the methanol anomalous transport through the solid media. The surface fractal dimension was estimated from the nitrogen adsorption isotherms using the Frenkel-Halsey-Hill method. Our study shows that decreasing the fractal dimension leads to increasing the anomalous diffusion exponent, whereas the anomalous diffusion constant is independent on the fractal dimension. We show that the obtained results are in a good agreement with the anomalous diffusion model on a fractal mesh.

New 5-adic Cantor sets and fractal string.

PubMed

Kumar, Ashish; Rani, Mamta; Chugh, Renu

2013-01-01

In the year (1879-1884), George Cantor coined few problems and consequences in the field of set theory. One of them was the Cantor ternary set as a classical example of fractals. In this paper, 5-adic Cantor one-fifth set as an example of fractal string have been introduced. Moreover, the applications of 5-adic Cantor one-fifth set in string theory have also been studied.

A simple method for estimating the size of nuclei on fractal surfaces

NASA Astrophysics Data System (ADS)

Zeng, Qiang

2017-10-01

Determining the size of nuclei on complex surfaces remains a big challenge in aspects of biological, material and chemical engineering. Here the author reported a simple method to estimate the size of the nuclei in contact with complex (fractal) surfaces. The established approach was based on the assumptions of contact area proportionality for determining nucleation density and the scaling congruence between nuclei and surfaces for identifying contact regimes. It showed three different regimes governing the equations for estimating the nucleation site density. Nuclei in the size large enough could eliminate the effect of fractal structure. Nuclei in the size small enough could lead to the independence of nucleation site density on fractal parameters. Only when nuclei match the fractal scales, the nucleation site density is associated with the fractal parameters and the size of the nuclei in a coupling pattern. The method was validated by the experimental data reported in the literature. The method may provide an effective way to estimate the size of nuclei on fractal surfaces, through which a number of promising applications in relative fields can be envisioned.

Aqueous synthesis of LiFePO4 with Fractal Granularity.

PubMed

Cabán-Huertas, Zahilia; Ayyad, Omar; Dubal, Deepak P; Gómez-Romero, Pedro

2016-06-03

Lithium iron phosphate (LiFePO4) electrodes with fractal granularity are reported. They were made from a starting material prepared in water by a low cost, easy and environmentally friendly hydrothermal method, thus avoiding the use of organic solvents. Our method leads to pure olivine phase, free of the impurities commonly found after other water-based syntheses. The fractal structures consisted of nanoparticles grown into larger micro-sized formations which in turn agglomerate leading to high tap density electrodes, which is beneficial for energy density. These intricate structures could be easily and effectively coated with a thin and uniform carbon layer for increased conductivity, as it is well established for simpler microstructures. Materials and electrodes were studied by means of XRD, SEM, TEM, SAED, XPS, Raman and TGA. Last but not least, lithium transport through fractal LiFePO4 electrodes was investigated based upon fractal theory. These water-made fractal electrodes lead to high-performance lithium cells (even at high rates) tested by CV and galvanostatic charge-discharge, their performance is comparable to state of the art (but less environmentally friendly) electrodes.

Aqueous synthesis of LiFePO4 with Fractal Granularity

PubMed Central

Cabán-Huertas, Zahilia; Ayyad, Omar; Dubal, Deepak P.; Gómez-Romero, Pedro

2016-01-01

Lithium iron phosphate (LiFePO4) electrodes with fractal granularity are reported. They were made from a starting material prepared in water by a low cost, easy and environmentally friendly hydrothermal method, thus avoiding the use of organic solvents. Our method leads to pure olivine phase, free of the impurities commonly found after other water-based syntheses. The fractal structures consisted of nanoparticles grown into larger micro-sized formations which in turn agglomerate leading to high tap density electrodes, which is beneficial for energy density. These intricate structures could be easily and effectively coated with a thin and uniform carbon layer for increased conductivity, as it is well established for simpler microstructures. Materials and electrodes were studied by means of XRD, SEM, TEM, SAED, XPS, Raman and TGA. Last but not least, lithium transport through fractal LiFePO4 electrodes was investigated based upon fractal theory. These water-made fractal electrodes lead to high-performance lithium cells (even at high rates) tested by CV and galvanostatic charge-discharge, their performance is comparable to state of the art (but less environmentally friendly) electrodes. PMID:27256504

Aqueous synthesis of LiFePO4 with Fractal Granularity

NASA Astrophysics Data System (ADS)

Cabán-Huertas, Zahilia; Ayyad, Omar; Dubal, Deepak P.; Gómez-Romero, Pedro

2016-06-01

Lithium iron phosphate (LiFePO4) electrodes with fractal granularity are reported. They were made from a starting material prepared in water by a low cost, easy and environmentally friendly hydrothermal method, thus avoiding the use of organic solvents. Our method leads to pure olivine phase, free of the impurities commonly found after other water-based syntheses. The fractal structures consisted of nanoparticles grown into larger micro-sized formations which in turn agglomerate leading to high tap density electrodes, which is beneficial for energy density. These intricate structures could be easily and effectively coated with a thin and uniform carbon layer for increased conductivity, as it is well established for simpler microstructures. Materials and electrodes were studied by means of XRD, SEM, TEM, SAED, XPS, Raman and TGA. Last but not least, lithium transport through fractal LiFePO4 electrodes was investigated based upon fractal theory. These water-made fractal electrodes lead to high-performance lithium cells (even at high rates) tested by CV and galvanostatic charge-discharge, their performance is comparable to state of the art (but less environmentally friendly) electrodes.

Fractal geometry in an expanding, one-dimensional, Newtonian universe.

PubMed

Miller, Bruce N; Rouet, Jean-Louis; Le Guirriec, Emmanuel

2007-09-01

Observations of galaxies over large distances reveal the possibility of a fractal distribution of their positions. The source of fractal behavior is the lack of a length scale in the two body gravitational interaction. However, even with new, larger, sample sizes from recent surveys, it is difficult to extract information concerning fractal properties with confidence. Similarly, three-dimensional N-body simulations with a billion particles only provide a thousand particles per dimension, far too small for accurate conclusions. With one-dimensional models these limitations can be overcome by carrying out simulations with on the order of a quarter of a million particles without compromising the computation of the gravitational force. Here the multifractal properties of two of these models that incorporate different features of the dynamical equations governing the evolution of a matter dominated universe are compared. For each model at least two scaling regions are identified. By employing criteria from dynamical systems theory it is shown that only one of them can be geometrically significant. The results share important similarities with galaxy observations, such as hierarchical clustering and apparent bifractal geometry. They also provide insights concerning possible constraints on length and time scales for fractal structure. They clearly demonstrate that fractal geometry evolves in the mu (position, velocity) space. The observed patterns are simply a shadow (projection) of higher-dimensional structure.

Fractal structure of the interplanetary magnetic field

NASA Technical Reports Server (NTRS)

Burlaga, L. F.; Klein, L. W.

1985-01-01

Under some conditions, time series of the interplanetary magnetic field strength and components have the properties of fractal curves. Magnetic field measurements made near 8.5 AU by Voyager 2 from June 5 to August 24, 1981 were self-similar over time scales from approximately 20 sec to approximately 3 x 100,000 sec, and the fractal dimension of the time series of the strength and components of the magnetic field was D = 5/3, corresponding to a power spectrum P(f) approximately f sup -5/3. Since the Kolmogorov spectrum for homogeneous, isotropic, stationary turbulence is also f sup -5/3, the Voyager 2 measurements are consistent with the observation of an inertial range of turbulence extending over approximately four decades in frequency. Interaction regions probably contributed most of the power in this interval. As an example, one interaction region is discussed in which the magnetic field had a fractal dimension D = 5/3.

Wavelet-based 3D reconstruction of microcalcification clusters from two mammographic views: new evidence that fractal tumors are malignant and Euclidean tumors are benign.

PubMed

Batchelder, Kendra A; Tanenbaum, Aaron B; Albert, Seth; Guimond, Lyne; Kestener, Pierre; Arneodo, Alain; Khalil, Andre

2014-01-01

The 2D Wavelet-Transform Modulus Maxima (WTMM) method was used to detect microcalcifications (MC) in human breast tissue seen in mammograms and to characterize the fractal geometry of benign and malignant MC clusters. This was done in the context of a preliminary analysis of a small dataset, via a novel way to partition the wavelet-transform space-scale skeleton. For the first time, the estimated 3D fractal structure of a breast lesion was inferred by pairing the information from two separate 2D projected mammographic views of the same breast, i.e. the cranial-caudal (CC) and mediolateral-oblique (MLO) views. As a novelty, we define the "CC-MLO fractal dimension plot", where a "fractal zone" and "Euclidean zones" (non-fractal) are defined. 118 images (59 cases, 25 malignant and 34 benign) obtained from a digital databank of mammograms with known radiologist diagnostics were analyzed to determine which cases would be plotted in the fractal zone and which cases would fall in the Euclidean zones. 92% of malignant breast lesions studied (23 out of 25 cases) were in the fractal zone while 88% of the benign lesions were in the Euclidean zones (30 out of 34 cases). Furthermore, a Bayesian statistical analysis shows that, with 95% credibility, the probability that fractal breast lesions are malignant is between 74% and 98%. Alternatively, with 95% credibility, the probability that Euclidean breast lesions are benign is between 76% and 96%. These results support the notion that the fractal structure of malignant tumors is more likely to be associated with an invasive behavior into the surrounding tissue compared to the less invasive, Euclidean structure of benign tumors. Finally, based on indirect 3D reconstructions from the 2D views, we conjecture that all breast tumors considered in this study, benign and malignant, fractal or Euclidean, restrict their growth to 2-dimensional manifolds within the breast tissue.

Study of the fractal dimension of the wind and its relationships with turbulent and stability parameters

NASA Astrophysics Data System (ADS)

Tijera, Manuel; Maqueda, Gregorio; Cano, José L.; López, Pilar; Yagüe, Carlos

2010-05-01

The wind velocity series of the atmospheric turbulent flow in the planetary boundary layer (PBL), in spite of being highly erratic, present a self-similarity structure (Frisch, 1995; Peitgen et., 2004; Falkovich et., 2006). So, the wind velocity can be seen as a fractal magnitude. We calculate the fractal dimension (Komolgorov capacity or box-counting dimension) of the wind perturbation series (u' = u- ) in the physical spaces (namely velocity-time). It has been studied the time evolution of the fractal dimension along different days and at three levels above the ground (5.8 m, 13.5 m, 32 m). The data analysed was recorded in the experimental campaign SABLES-98 (Cuxart et al., 2000) at the Research Centre for the Lower Atmosphere (CIBA) located in Valladolid (Spain). In this work the u, v and w components of wind velocity series have been measured by sonic anemometers (20 Hz sampling rate). The fractal dimension versus the integral length scales of the mean wind series have been studied, as well as the influence of different turbulent parameters. A method for estimating these integral scales is developed using the normalized autocorrelation function and a Gaussian fit. Finally, it will be analysed the variation of the fractal dimension versus stability parameters (as Richardson number) in order to explain some of the dominant features which are likely immersed in the fractal nature of these turbulent flows. References - Cuxart J, Yagüe C, Morales G, Terradellas E, Orbe J, Calvo J, Fernández A, Soler MR, Infante C, Buenestado P, Espinalt A, Joergensen HE, Rees JM, Vilá J, Redondo JM, Cantalapiedra IR and Conangla L (2000) Stable atmospheric boundary-layer experiment in Spain (SABLES98): a report. Boundary- Layer Meteorol 96:337-370 - Falkovich G and Kattepalli R. Sreenivasan (2006) Lessons from Hidrodynamic Turbulence. Physics Today 59: 43-49 - Frisch U (1995) Turbulence the legacy of A.N. Kolmogorov Cambridge University Press 269pp - Peitgen H, Jürgens H and