Showing papers on "Fractal dimension published in 2009"

Fractal organization of trabecular bone images on calcaneus radiographs

Abstract: Bone density is not the unique factor conditioning bone strength. Trabecular bone microarchitecture also plays an important role. We have developed a fractal evaluation of trabecular bone microarchitecture on calcaneus radiographs. Fractal models may provide a single numeric evaluation (the fractal dimension) of such complex structures. Our evaluation results from an analysis of images with a varying range of gray levels, without binarization of the image. It is based on the fractional brownian motion model, or more precisely on the analysis of its increment, the fractional gaussian noise (FGN). The use of this model may be considered validated if two conditions are fulfilled: the gaussian repartition and the self-similarity of our data. The gaussian repartition of intermediate lines of these images was tested on a sample of 32,800 lines from 82 images. Following a chi-square goodness-of-fit test, it was checked in 86% of these lines for alpha = 0.01. The self-similarity was tested on 20 images by two estimators, the variance method of Pentland and the spectrum method of Fourier. Self-similarity is defined by lined-up points in a log-log plot of the FGN spectrum or of the variance as a function of the lag. We found two self-similarity areas between scales of analysis ranging from 105 to 420 microns, then above 900 microns, where linear regression produced high mean correlation coefficients (r > or = 0.97). Following this validation, we studied the reproducibility of this new technique. Intra- and interobserver reproducibility, influence of transferring the region of interest, and long-term reproducibility were assessed and given CV of 0.61 +/- 0.15, 0.68 +/- 0.47, 0.53 +/- 0.16, and 2.07 +/- 0.84%, respectively. These data have allowed us to validate the use of this fractal model by checking the fractal organization of our radiographic images analyzed by the model. The good reproducibility of successive x-rays in the same subject allows us to undertake population studies and to envisage longitudinal series.

Plant leaf identification based on volumetric fractal dimension

Abstract: Texture is an important visual attribute used to describe the pixel organization in an image. As well as it being easily identified by humans, its analysis process demands a high level of sophistic...

Fractals and Multifractals in Ecology and Aquatic Science

Abstract: Introduction About Geometries and Dimensions From Euclidean to Fractal Geometry Dimensions Self-Similar Fractals Self-Similarity, Power Laws, and the Fractal Dimension Methods for Self-Similar Fractals Self-Affine Fractals Several Steps toward Self-Affinity Methods for Self-Affine Fractals Frequency Distribution Dimensions Cumulative Distribution Functions and Probability Density Functions The Patch-Intensity Dimension, Dpi The Korcak Dimension, DK Fragmentation and Mass-Size Dimensions, Dfr and Dms Rank-Frequency Dimension, Drf Fractal-Related Concepts Some Clarifications Fractals and Deterministic Chaos Fractals and Self-Organization Fractals and Self-Organized Criticality Estimating Dimensions with Confidence Scaling or Not Scaling? That Is the Question Errors Affecting Fractal Dimension Estimates Defining the Confidence Limits of Fractal Dimension Estimates Performing a Correct Analysis From Fractals to Multifractals A Random Walk toward Multifractality Methods for Multifractals Cascade Models for Intermittency Multifractals: Misconceptions and Ambiguities Joint Multifractals Intermittency and Multifractals: Biological and Ecological Implications

Electrical treeing characteristics in XLPE power cable insulation in frequency range between 20 and 500 Hz

Abstract: Electrical treeing is one of the main reasons for long term degradation of polymeric materials used in high voltage AC applications. In this paper we report on an investigation of electrical tree growth characteristics in XLPE samples from a commercial XLPE power cable. Electrical trees have been grown over a frequency range from 20 Hz to 500 Hz and images of trees were taken using CCD camera without interrupting the application of voltage. The fractal dimension of electric tree is obtained using a simple box-counting technique. Contrary to our expectation it has been found that the fractal dimension prior to the breakdown shows no significant change when frequency of the applied voltage increases. Instead, the frequency accelerates tree growth rate and reduces the time to breakdown. A new approach for investigating the frequency effect on trees has been devised. In addition to looking into the fractal analysis of tree as a whole, regions of growth are being sectioned to reveal differences in terms of growth rate, accumulated damage and fractal dimension.

Fractal solids, product measures and fractional wave equations

Abstract: This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d and a resolution length scale R. The focus is on pre-fractal media (i.e. those with lower and upper cut-offs) through a theory based on a dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D, d and R. This procedure allows a specification of a geometry configuration of continua by ‘fractal metric’ coefficients, on which the continuum mechanics is subsequently constructed. While all the derived relations depend explicitly on D, d and R, upon setting D = 3 and d = 2, they reduce to conventional forms of governing equations for continuous media with Euclidean geometries. Whereas the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure, making it capable of grasping local fractal anisotropy. Finally, the one-, two- and three-dimensional wave equations are developed, showing that the continuum mechanics approach is consistent with that obtained via variational energy principles.

Characterization of atrophic changes in the cerebral cortex using fractal dimensional analysis

Abstract: 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 (D f) of the cortical ribbons were then computed using a box-counting algorithm. The mean D f 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.

Breakup of dense colloidal aggregates under hydrodynamic stresses.

Abstract: Flow-induced aggregation of colloidal particles leads to aggregates with fairly high fractal dimension (df approximately 2.4-3.0) which are directly responsible for the observed rheological properties of sheared dispersions. We address the problem of the decrease in aggregate size with increasing hydrodynamic stress, as a consequence of breakup, by means of a fracture-mechanics model complemented by experiments in a multipass extensional (laminar) flow device. Evidence is shown that as long as the inner density decay with linear size within the aggregate (due to fractality) is not negligible (as for df approximately 2.4-2.8), this imposes a substantial limitation to the hydrodynamic fragmentation process as compared with nonfractal aggregates (where the critical stress is practically size independent). This is due to the fact that breaking up a fractal object leads to denser fractals which better withstand stress. In turbulent flows, accounting for intermittency introduces just a small deviation with respect to the laminar case, while the model predictions are equally in good agreement with experiments from the literature. Our findings are summarized in a diagram for the breakup exponent (governing the size versus stress scaling) as a function of fractal dimension.

The fractal geometry of life.

Abstract: 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".

NMR Measure of Translational Diffusion and Fractal Dimension. Application to Molecular Mass Measurement

Abstract: Experimental NMR diffusion measure on polymers and on globular proteins are presented. These results, complemented with results found in the literature, enable a general description of effective fractal dimension for objects such as small organic molecules, sugars, polymers, DNA, and proteins. Results are compared to computational simulations as well as to theoretical values. A global picture of the diffusion phenomenon emerges from this description. A power law relating molecular mass with diffusion coefficients is described and found to be valid over 4 orders of magnitude. From this law, the fractal dimension of the molecular family can be measured, with experimental values ranging from 1.41 to 2.56 in full agreement with theoretical approaches. Finally, a method for evaluating the molecular mass of unknown solutes is described and implemented as a Web page.

Role of surface roughness characterized by fractal geometry on laminar flow in microchannels.

Abstract: A three-dimensional model of laminar flow in microchannels is numerically analyzed incorporating surface roughness effects as characterized by fractal geometry. The Weierstrass-Mandelbrot function is proposed to characterize the multiscale self-affine roughness. The effects of Reynolds number, relative roughness, and fractal dimension on laminar flow are all investigated and discussed. The results indicate that unlike flow in smooth microchannels, the Poiseuille number in rough microchannels increases linearly with the Reynolds number, Re, and is larger than what is typically observed in smooth channels. For these situations, the flow over surfaces with high relative roughness induces recirculation and flow separation, which play an important role in single-phase pressure drop. More specifically, surfaces with the larger fractal dimensions yield more frequent variations in the surface profile, which result in a significantly larger incremental pressure loss, even though at the same relative roughness. The accuracy of the predicted Poiseuille number as calculated by the present model is verified using experimental data available in the literature.

On the Physical Properties of Apparent Two-Phase Fractal Porous Media

Abstract: AU›o®ƒa®EAÝ : DLA, diff usion-limited aggregation; SSA, specifi c surface area. SO›‘®ƒ½ S›‘a®EA : FUƒ‘aƒ½Ý In this study, we summarized some basic characters of fractal porous media, including the fractal pore or parƟ cle size distribuƟ on, pore or parƟ cle density funcƟ on, the fractal dimensions for the pore and solid phases, and their relaƟ ons. The geometric porosiƟ es vs. the fractal dimensions and microstructures of porous media were reviewed and discussed in two and three dimensions. The specifi c surface areas of fractal porous media in two and three dimensions were derived and were expressed as a funcƟ on of the fractal dimensions and microstructural parameters. The fl uid velociƟ es in fractal porous media were also derived and found to be a funcƟ on of the fractal dimensions and microstructural parameters of the medium. The parameters presented are the fundamental ones and may have potenƟ al in analysis of transport properƟ es in fractal porous media.

Scaling law for the radius of gyration of proteins and its dependence on hydrophobicity

Abstract: The scaling law between the radius of gyration and the length of a polymer chain has long been an interesting topic since the Flory theory. In this article, we seek to derive a unified formula for the scaling exponent of proteins under different solvent conditions. The formula is obtained by considering the balance between the excluded volume effect and elastic interactions among monomers. Our results show that the scal- ing exponent is closely related to the fractal dimension of a protein's structure at the equilibrium state.Applying this formula to natural proteins yields a 2/5 law with fractal dimension 2 at the native state, which is in good agreement with other studies based on Protein Data Bank analysis. We also study the dependence of the scaling exponent on the hydrophobicity of a protein chain through a simple two-letters HP model. The results provides a way to estimate the globular structure of a protein, and could be help- ful for the investigation of the mechanisms of protein folding. © 2008 Wiley Periodicals, Inc.

Generation and Geometrical Analysis of Dense Clusters with Variable Fractal Dimension

Abstract: The generation and geometrical analysis of clusters composed of rigid monodisperse primary particles with variable fractal dimension, df, in the range from 2.2 to 3 are presented. For all generated aggregate populations, it was found that the dimensionless aggregate mass, i, and the aggregate size, characterized by the radius of gyration, Rg, normalized by the primary particle radius, Rp, follow a fractal scaling, i = kf(Rg/Rp)df. Furthermore, the obtained prefactor of the fractal scaling, kf, is related to df according to kf = 4.46df−2.08, which is in agreement with literature data. For cases when df cannot be directly determined from light scattering or confocal laser scanning microscopy, it can be estimated from its relation with a perimeter fractal dimension, dpf, or a chord fractal dimension, dcf, both obtained from 2D projection of aggregates. A relation between df and dpf of the form df = −1.5dpf + 4.4 was developed by fitting data obtained in this work for 2.2 < df < 3 together with data of Lee an...

Fractal analysis of effective thermal conductivity for three-phase (unsaturated) porous media

Abstract: A fractal analysis of effective thermal conductivity for unsaturated fractal porous media is presented based on the thermal-electrical analogy and statistical self-similarity of porous media. Here, we derive a dimensionless expression of effective thermal conductivity without any empirical constant. The effects of the parameters of fractal porous media on the dimensionless effective thermal conductivity are discussed. From this study, it is shown that, when the thermal conductivity of solid phase and wet phase are greater than that of the gas phase (viz., ks∕kg>1, kw∕kg>1), the dimensionless effective thermal conductivity of unsaturated fractal porous media decreases with decreasing degree of saturation (Sw) and increasing fractal dimension for pore area (Df), fractal dimension for tortuosity (Dt), and porosity (ϕ); when the thermal conductivities of solid phase and wet phase are lower than that of the gas phase (viz., ks∕kg<1, kw∕kg<1), the trends were just opposite. Our model was validated by comparing ...

Fractal analysis of damage detected in concrete structural elements under loading

Abstract: In Civil Engineering materials subjected to stress or strain states a quantitative evaluation of damage is of great importance due to the critical character of this phenomenon, which at a certain point suddenly turns into a catastrophic failure. An effective damage assessment criterion is represented by the statistical analysis of the amplitude distribution of acoustic emission (AE) signals emerging from the growing microcracks. The amplitudes of such signals are distributed according to the Gutenberg–Richter (GR) law and characterised through the b-value which decreases systematically with damage growth. On the other hand, the damage process is also characterised by the progressive coalescence of microcracks to form fracture surfaces. Geometrically the fractal dimension D of the damaged domain is expected to decrease from an initial value comprised between 2 and 3 towards a final value nearly equal to 2. The b-value and the fractal analysis, are here applied to two case studies of concrete specimens loaded up to failure, and the obtained results are compared and discussed. In particular, we emphasize that a single fractal dimension does not adequately describe a crack network, since two damaged domains with the same fractal dimension could have significantly different properties.

Characterization and classification of tumor lesions using computerized fractal-based texture analysis and support vector machines in digital mammograms

Abstract: This paper presents a detailed study of fractal-based methods for texture characterization of mammographic mass lesions and architectural distortion. The purpose of this study is to explore the use of fractal and lacunarity analysis for the characterization and classification of both tumor lesions and normal breast parenchyma in mammography. We conducted comparative evaluations of five popular fractal dimension estimation methods for the characterization of the texture of mass lesions and architectural distortion. We applied the concept of lacunarity to the description of the spatial distribution of the pixel intensities in mammographic images. These methods were tested with a set of 57 breast masses and 60 normal breast parenchyma (dataset1), and with another set of 19 architectural distortions and 41 normal breast parenchyma (dataset2). Support vector machines (SVM) were used as a pattern classification method for tumor classification. Experimental results showed that the fractal dimension of region of interest (ROIs) depicting mass lesions and architectural distortion was statistically significantly lower than that of normal breast parenchyma for all five methods. Receiver operating characteristic (ROC) analysis showed that fractional Brownian motion (FBM) method generated the highest area under ROC curve (A z = 0.839 for dataset1, 0.828 for dataset2, respectively) among five methods for both datasets. Lacunarity analysis showed that the ROIs depicting mass lesions and architectural distortion had higher lacunarities than those of ROIs depicting normal breast parenchyma. The combination of FBM fractal dimension and lacunarity yielded the highest A z value (0.903 and 0.875, respectively) than those based on single feature alone for both given datasets. The application of the SVM improved the performance of the fractal-based features in differentiating tumor lesions from normal breast parenchyma by generating higher A z value. FBM texture model is the most appropriate model for characterizing mammographic images due to self-affinity assumption of the method being a better approximation. Lacunarity is an effective counterpart measure of the fractal dimension in texture feature extraction in mammographic images. The classification results obtained in this work suggest that the SVM is an effective method with great potential for classification in mammographic image analysis.

Complexity analysis of EEG in patients with schizophrenia using fractal dimension.

Abstract: We computed Higuchi's fractal dimension (FD) of resting, eyes closed EEG recorded from 30 scalp locations in 18 male neuroleptic-naive, recent-onset schizophrenia (NRS) subjects and 15 male healthy control (HC) subjects, who were group-matched for age. Schizophrenia patients showed a diffuse reduction of FD except in the bilateral temporal and occipital regions, with the reduction being most prominent bifrontally. The positive symptom (PS) schizophrenia subjects showed FD values similar to or even higher than HC in the bilateral temporo-occipital regions, along with a co-existent bifrontal FD reduction as noted in the overall sample of NRS. In contrast, this increase in FD values in the bilateral temporo-occipital region was absent in the negative symptom (NS) subgroup. The regional differences in complexity suggested by these findings may reflect the aberrant brain dynamics underlying the pathophysiology of schizophrenia and its symptom dimensions. Higuchi's method of measuring FD directly in the time domain provides an alternative for the more computationally intensive nonlinear methods of estimating EEG complexity.

Assessment of the spatial pattern of colorectal tumour perfusion estimated at perfusion CT using two-dimensional fractal analysis.

Abstract: The aim was to evaluate the feasibility of fractal analysis for assessing the spatial pattern of colorectal tumour perfusion at dynamic contrast-enhanced CT (perfusion CT). Twenty patients with colorectal adenocarcinoma underwent a 65-s perfusion CT study from which a perfusion parametric map was generated using validated commercial software. The tumour was identified by an experienced radiologist, segmented via thresholding and fractal analysis applied using in-house software: fractal dimension, abundance and lacunarity were assessed for the entire outlined tumour and for selected representative areas within the tumour of low and high perfusion. Comparison was made with ten patients with normal colons, processed in a similar manner, using two-way mixed analysis of variance with statistical significance at the 5% level. Fractal values were higher in cancer than normal colon (p ≤ 0.001): mean (SD) 1.71 (0.07) versus 1.61 (0.07) for fractal dimension and 7.82 (0.62) and 6.89 (0.47) for fractal abundance. Fractal values were lower in ‘high’ than ‘low’ perfusion areas. Lacunarity curves were shifted to the right for cancer compared with normal colon. In conclusion, colorectal cancer mapped by perfusion CT demonstrates fractal properties. Fractal analysis is feasible, potentially providing a quantitative measure of the spatial pattern of tumour perfusion.