Search

It is shown that generically, in Baire’s sense, the bound states of the Hamiltonian of the hydrogen atom have spectral measures with exact 0-lower and 1/3-upper generalized fractal dimensions; the relation to (a weak form of) dynamical delocalization along orthonormal bases is also discussed. Such result is a consequence of certain distributions of the Hamiltonian eigenvalues for which the hydrogen atom is a particular case. [ABSTRACT FROM AUTHOR]

AIM: To apply the multifractal analysis method as a quantitative approach to a comprehensive description of the microvascular network architecture of the normal human retina. METHODS: Fifty volunteers were enrolled in this study in the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and January 2014. A set of 100 segmented and skeletonised human retinal images, corresponding to normal states of the retina were studied. An automatic unsupervised method for retinal vessel segmentation was applied before multifractal analysis. The multifractal analysis of digital retinal images was made with computer algorithms, applying the standard box-counting method. Statistical analyses were performed using the GraphPad InStat software. RESULTS: The architecture of normal human retinal microvascular network was able to be described using the multifractal geometry. The average of generalized dimensions (Dq) for q=0, 1, 2, the width of the multifractal spectrum (Δα=αmax - αmin) and the spectrum arms’ heights difference (│Δf│) of the normal images were expressed as mean±standard deviation (SD): for segmented versions, D0=1.7014±0.0057; D1=1.6507±0.0058; D2=1.5772±0.0059; Δα=0.92441±0.0085; │Δf│= 0.1453±0.0051; for skeletonised versions, D0=1.6303±0.0051; D1=1.6012±0.0059; D2=1.5531±0.0058; Δα=0.65032±0.0162; │Δf│= 0.0238±0.0161. The average of generalized dimensions (Dq) for q=0, 1, 2, the width of the multifractal spectrum (Δα) and the spectrum arms’ heights difference (│Δf│) of the segmented versions was slightly greater than the skeletonised versions. CONCLUSION: The multifractal analysis of fundus photographs may be used as a quantitative parameter for the evaluation of the complex three-dimensional structure of the retinal microvasculature as a potential marker for early detection of topological changes associated with retinal diseases.

Micro-fractures act as a bridge between pores and macro-fractures and directly affect the permeability of coal reservoirs. Understanding the micro-fracture heterogeneity and its relationships with deformation can enrich our knowledge on the gas transportation behavior of tectonically deformed coals. The multifractal theory was applied to characterize the variations and heterogeneity of micro-fractures in high-rank brittle tectonically deformed coals. The results indicated that the micro-fractures exhibit multifractal behavior at ε = 1–1/16 (ε being the size of the boxes considered), and that the obtained generalized dimension spectra Dq-q, multifractal singularity spectrum f(α)-α, and characteristic parameters, including the capacity dimension D0, information dimension D1, correlation dimension D2, D1/D0, the width of multifractal spectrum Δα, and the difference of the fractal dimensions Δf(α), can be applied to accurately describe the differences and variations of the micro-fracture distributions and scales. Moreover, the deformation has a significant influence on the variation and heterogeneity of micro-fracture distributions and scales, as demonstrated by the change rules of the characteristic parameters. The D1, D2, and D1/D0 increased with the increase of deformation degree, while Δα and Δf(α) decreased. Consequently, the deformation leads to higher density, more uniform distributions, and more dispersed and complex scales in the micro-fractures of brittle tectonically deformed coals, and controls the permeability of coal reservoirs. [ABSTRACT FROM AUTHOR]

Entropy-based tools are commonly used to describe the dynamics of complex systems. In the last few decades, non-extensive statistics, based on Tsallis entropy, and multifractal techniques have shown to be useful to characterize long-range interaction and scaling behavior. In this paper, an approach based on generalized Tsallis dimensions is used for the formulation of mutual-information-related dependence coefficients in the multifractal domain. Different versions according to the normalizing factor, as well as to the inclusion of the non-extensivity correction term are considered and discussed. An application to the assessment of dimensional interaction in the structural dynamics of a seismic real series is carried out to illustrate the usefulness and comparative performance of the measures introduced.

The box counting dimension d B of a complex network G , and the generalized dimensions { D q , q ∈ R } of G , have been well studied. However, the Hausdorff dimension d H of a geometric object, which generalizes d B by not assuming equal-diameter boxes, has not previously been extended to G . Similarly, the generalized Hausdorff dimensions { D q H , q ∈ R } of a geometric object (defined by Grassberger in 1985), which extend the generalized dimensions D q by not assuming equal-diameter boxes, have not previously been extended to G . In this paper we first develop a definition of d H for G and compare d H to d B on both constructed and real-world networks. Then we extend Grassberger’s work by defining the generalized Hausdorff dimensions D q H of G , and computing the D q H vs. q multifractal spectrum for several networks. Given a minimal covering B ( s ) of G for a range S of box sizes, computing d H utilizes the diameter of each box in B ( s ) for s ∈ S , and computing D q H utilizes the diameter and mass of each box in B ( s ) . Also, computing d B and D q (for a given q ) typically utilizes linear regression; in contrast, computing d H and D q H (for a given q ) requires minimizing a function of one variable. Computational results show that d H can sometimes be more useful than d B in quantifying changes in the topology of a network. However, d H is harder to compute than d B , and D q H is less well behaved than D q . We conclude that d H and D q H should be added to the set of useful metrics for characterizing a complex network, but they cannot be expected to replace d B and D q . [ABSTRACT FROM AUTHOR]

Computing the generalized dimensions D q of a complex network requires covering the network by a minimal number of “boxes” of size s , for a range of s . We show that, unlike the case for a geometric multifractal, for a complex network the shape of the D q vs. q curve can be monotone increasing, or monotone decreasing, or even have both a local maximum and a local minimum, depending on the range of box sizes used to compute D q . We provide insight into this behavior by deriving a simple closed-form expression for the derivative of D q at q = 0 . The estimate depends on the ratio of the geometric mean of the box masses (where the mass of a box is the number of nodes it contains) to the arithmetic mean of the box masses. [ABSTRACT FROM AUTHOR]

Computing the generalized dimensions D q of a complex network requires covering the network by a minimal number of “boxes” of size s . We show that the current definition of D q is ambiguous, since there are in general multiple minimal coverings of size s . We resolve the ambiguity by first computing, for each s , the minimal covering that is summarized by the lexicographically minimal vector x ( s ) . We show that x ( s ) is unique and easily obtained from any box counting method. The x ( s ) vectors can then be used to unambiguously compute D q . Moreover, x ( s ) is related to the partition function, and the first component of x ( s ) can be used to compute D ∞ without any partition function evaluations. We compare the box counting dimension and D ∞ for three networks. [ABSTRACT FROM AUTHOR]

Entropy-based tools are commonly used to describe the dynamics of complex systems. In the last few decades, non-extensive statistics, based on Tsallis entropy, and multifractal techniques have shown to be useful to characterize long-range interaction and scaling behavior. In this paper, an approach based on generalized Tsallis dimensions is used for the formulation of mutual-information-related dependence coefficients in the multifractal domain. Different versions according to the normalizing factor, as well as to the inclusion of the non-extensivity correction term are considered and discussed. An application to the assessment of dimensional interaction in the structural dynamics of a seismic real series is carried out to illustrate the usefulness and comparative performance of the measures introduced. [ABSTRACT FROM AUTHOR]

Different approaches and tools have been adopted for the analysis and characterization of regional seismicity based on spatio-temporal series of event occurrences. Two main aspects of interest in this context concern scaling properties and dimensional interaction. This paper is focused on the statistical use of information-theoretic concepts and measures in the analysis of structural complexity of seismic distributional patterns. First, contextual significance is motivated, and preliminary elements related to informational entropy, complexity and multifractal analysis are introduced. Next, several technical and methodological extensions are proposed. Specifically, limiting behaviour of some complexity measures in connection with generalized dimensions is established, justifying a concept of multifractal complexity. Under scaling behaviour, a mutual-information-related dependence coefficient for assessing spatio-temporal interaction is defined in terms of generalized dimensions. Also, an alternative form of generalized dimensions based on Tsallis entropy convergence rates is formulated. Further, possible incorporation of effects, such as earthquake magnitude, is achieved in terms of weighted box-counting distributions. Different aspects in relation to the above elements are analyzed and illustrated using two well-known series of seismic event data of an underlying different nature, occurred in the areas of Agrón (Granada, Spain) and El Hierro (Canary Islands, Spain). Finally, various related directions for continuing research are indicated. [ABSTRACT FROM AUTHOR]

Abstract: The spatial and temporal distribution of land price plays a key role in urban development and redevelopment processes. Identifying the features of land price distribution (LPD) is essential for improving urban planning and modeling land use changes. The purposes of this study were to determine if LPD can be characterized by multifractal models and to develop multifractal methods for characterizing the properties of LPD at various scales. An analysis was performed for a study site in Wuhan City, central China. Land prices were sampled in the years 2001, 2004, and 2007. The LPD patterns were represented by multifractal spectra estimated using the method of moments and characterized by five quantitative multifractal parameters. The results showed that the dimension spectra calculated from the LPD data in various regions and at different times indeed depict multifractality, the curves of the multifractal spectra are continuous, displaying the same characteristics of asymmetric and convex curves at the same times in different regions, where the common transitional trend was from shorter toward the left but much longer toward the right in 2001, comparatively symmetric with a slight right deviation in 2004, shorter toward the right but much longer toward the left in 2007, which implying continuous multifractality observed for LPD, and this trend indicates that the singularity of land prices in the different areas keeps step with urban development. In addition, the horizontal characteristics of the curves also differed in different development stages in the city. These results also demonstrated that we may characterize the spatial and temporal differences of different LPD patterns using multifractal methods, which may thus be utilized as a quantitative measure in understanding how land price affects changes in urban land use. [Copyright &y& Elsevier]

Laser Doppler flowmetry (LDF) signals - that reflect the peripheral cardiovascular system - are now widespread in blood microcirculation research. Over the last few years, the central cardiovascular system has been the subject of many fractal and multifractal works. However, only very few multifractal studies of LDF signals have been published. Such multifractal analyses have shown that LDF data can be weakly multifractal but the origin of such characteristics are still unknown. We therefore herein propose a multifractal analysis of LDF signals recorded on the forearm of twelve healthy subjects, before and after skin local heating. The results show that the partition functions for all the signals have power-law characteristics. Moreover, generalized dimensions present very few variations with q for the signals recorded before heating; these variations are larger 20 minutes after local heating. Physiological activities may therefore play a role in the weak multifractal properties of LDF data. [ABSTRACT FROM AUTHOR]

Abstract: The pore-size distribution (PSD) has an important influence on the complex gas transport phenomena (O2 and CO2) that occur in apple tissue during storage under controlled atmosphere conditions. It defines the apple tissue microstructure that is correlated to many other apple properties. In this article multifractal analysis (MFA) has been used to study the multiscale structure of the PSD using generalized dimensions in three varieties of apples (Jonagold, Greenstar, and Kanzi) based on X-ray imaging technology (8.5μm resolution). Tomographic images of apple samples were taken at two positions within the parenchyma tissue: close to the peel and near to the core. The images showed suitable scaling properties. The generalized dimensions were determined with an R 2 greater than 0.98 in the range of moment orders between −1 and +10. The variation of Dq with respect to q and the shape of the multifractal generalized spectrum revealed that the PSD structure of apple tissue has properties close to multifractal self-similarity measures. Comparisons among cultivars showed that, in spite of the complexity and variability of the pore space of these apple samples, the extracted generalized dimensions from PSD were significantly different (p <0.05). The generalized dimensions D 0, D 1, D 2, and the quantity D 0–D 2 could be used to discriminate tissue samples from different positions or cultivars. Also, high correlations were found between these parameters and the porosity (R 2 ⩾0.935). These results demonstrate that MFA is an appropriate tool for characterizing the internal pore-size distribution of apple tissue and thus may be used as a quantitative measure to understand how tissue microstructure affects important physical properties of apple. [Copyright &y& Elsevier]

Multifractal box counting analysis has been widely applied to study the scaling characteristics of grayscale images. Since bit depth is an important property of such images it is desirable know the impact of varying bit depths on the estimation of the generalized dimensions (Dq). We generated random geometrical multifractal grayscale fields, which were then transformed from double precision to 16, 13, and 8 bit depths. Digitized grayscale images of soil thin sections at 13 bit depth were also selected for study and transformed to 8 bit depth. The moment based box counting method was applied to evaluate the bit depth effects on Dq. The partition functions for the multifractal fields became noticeably nonlinear on a log-log scale when q ≪ 0 as the bit depth decreased. This trend can be attributed to loss of grayscale details, changes in the local mass distribution, and the occurrence of zeros due to the bit depth transformation and data normalization processes. These effects were most pronounced for positively skewed multifractal fields, with a high proportion of extremely small mass fractions. As a result, the generalized dimensions estimated by linear regression were not always accurate, and an alternative method based on numerical derivatives was explored. The numerical method significantly improved the accuracy of the multifractal analyses; the maximum absolute difference between the analytical and numerically-derived estimates of Dq was only 9.62 × 10-3. However, when applied to situations in which the box counting scale factor did not match the scale factor used to generate the multifractal field, the numerically-derived estimates of Dq were severely biased. In this case, the linear regression method is preferable even though some error may occur due to limited bit depths. All of the soil grayscale images exhibited multifractal scaling characteristics, although there was little effect of bit depth on the resulting Dq values. Because of random fluctuations in the partition functions, the linear regression method proved to be more robust than the numerical derivative method for estimating the generalized dimensions of natural grayscale images. [ABSTRACT FROM AUTHOR]

Abstract: Fractality is a common property in nature. It can also be observed in time series representing dynamics of complex processes. Therefore fractal analysis could be a useful tool to describe the dynamics of brain electrical activities in physiological and pathological conditions. In this study, we carried out a spatio-temporal analysis of monofractal and multifractal properties of whole-night sleep EEG recordings. We estimated the Hurst exponent (H) and the range of fractal spectra (dD) in 10 healthy subjects. We found higher H values during NREM4 compared to NREM2 and REM in all electrodes. Measure dD showed an opposite trend. Differences of H and dD between NREM2 and REM reached significancy at circumscribed regions only. Our results contribute to a deeper understanding of the fractal nature of brain electrical activities and may have implications for automatic classification of sleep stages. [Copyright &y& Elsevier]

Most soil parameters as the spatial variability of preferential pathways for water and chemical transport in field soils show complex variations at different scales that cannot accurately described with stationary assumptions. This is why multifractal formalism or the wavelet transform reveals as useful tools for classifying and quantifying the spatial variability of these preferential pathways, as visualized through dye infiltration experiments. The high-resolution images resulting from these experiments are analyzed using both box-counting methods and wavelet transform analysis (WTA). The box-counting methods reveals global scaling patterns while the WTA focuses on local distribution of singularities. So, in the context of multiscaling structure analysis the wavelet methods can complement box-counting analysis which could be useful for statistically describing preferential flow path geometry and flow processes. The methodology is illustrated using well-known fractal structures as multifractal Sierpinsky carpets and results are illustrated with images of horizontal planes of the subsoil, acquired after dye infiltration into a 4m2 plot located on a Vertisol soil near College Station, Texas. [ABSTRACT FROM AUTHOR]

The moment-based box counting method of multifractal analysis is widely used for estimating generalized dimensions, Dq, from two-dimensional grayscale images. An evaluation of the accuracy of this method is needed to establish confidence in the resulting estimates of Dq. We estimated Dq from q = -10 to +10 for 23 random geometrical multifractal fields with different grid sizes, and known analytical Dq versus q functions. The fields were transformed to give normalized grayscale values between zero and one. Comparison of the estimated and analytical functions indicated the moment-based box counting method overestimates Dq by as much as 6.9% when q ≪ 0. The root mean square error, RMSE, for the entire range of q values examined ranged from 7.81 × 10-6 to 1.35 × 10-1, with a geometric mean of 6.50 × 10-3. The RMSE decreased with decreasing grid size and increasing heterogeneity. These trends appear to be largely due to the presence of zeros in the normalized grayscale fields. Variations in the slope of the log-transformed partition function, ln[χ(q,δ)], with box size resulted in the overestimation of Dq when q ≪ 0. An alternative procedure for estimating Dq was developed based on the numerical first derivatives of ln[χ(q,δ)]. Using this approach the maximum deviation in Dq values was only 1.2%, while the RMSE varied from 3.11 × 10-6 to 2.72 × 10-2, with a geometric mean of 2.57 × 10-4. When analyzing normalized grayscale fields, moment-based estimates of Dq should be interpreted with care. An order of magnitude increase in the accuracy of Dq can be achieved for such fields if the numerical first derivatives of ln[χ(q,δ)] are used in the analysis instead of standard linear regression. [ABSTRACT FROM AUTHOR]

Abstract: We use multifractal analysis (MFA) to investigate how the Rényi dimensions of the solid mass and the pore space in porous structures are related to each other. To our knowledge, there is no investigation about the relationship of Rényi or generalized dimensions of two phases of the same structure. Images of three different natural porous structures covering three orders of magnitude were investigated: a microscopic soil structure, a soil void system visible without magnification and a mineral dendrite. Image size was always 1024×1024 pixels and box sizes were chosen as powers of 2. MFA was carried out according to the method of moments, i.e., the probability distribution was estimated for moments ranging from −10< q <10 and the Rényi dimensions were calculated from the log/log slope of the probability distribution for the respective moments over box sizes. A meaningful interval of box sizes was determined by estimating the characteristic length of the pore space and taking the next higher power of 2 value as the smallest box size, whereas the greatest box size was determined by optimizing the coefficients of determination of the log/log fits for all q. The optimized box size range spans from 32 to 1024 pixels for all images. Good generalized dimension (Dq) spectra were obtained for this box size range, which are capable of characterizing heterogeneous spatial porous structure. They are alike for all images and phases which the exception of the solid mass of the soil void system, which shows a rather flat D q behavior. A closer examination reveals that similar patterns of structure gain similar spectra of generalized dimensions. The capacity dimension for q =0 is close to the Euclidian dimension 2 for all investigated images and phases. [Copyright &y& Elsevier]

A statistical estimate for generalized dimensions of a set $$A \subset \mathbb{R}^m$$ based on the computation of average distances to the closest points in a sample of elements of A is given. For smooth manifolds with Lebesgue measures and for self-similar fractals with self-similar measures, the estimate is proved to be consistent. [ABSTRACT FROM AUTHOR]

The multifractal dimension D q of β-Si 3 N 4 agglomerates in β-Si 3 N 4 /316 L stainless steel (SUS316L) composites was measured. In addition to capacity dimension D 0 related to the morphology of β-Si 3 N 4 agglomerates, information dimension D 1 for theses samples was related to the entropy of configuration affected by the dispersion of β-Si 3 N 4 particles was confirmed, and the correlation dimension D 2 reflected the connectivity between the particles. The obtained results showed that the multifractal dimensions D q (q = 0, 1, and 2) were effective as indices for evaluating the morphology, arrangement, and connectivity (or dispersibility) of dispersed particles in solid state reaction, and suggested a new method of quantitative stereology. Image 1 • Material texture of the Si 3 N 4 /SUS composite was investigated by multifractal analysis. • The capacity dimension D 0 was related to the morphology of aggregates. • The information dimension D 1 was related to the entropy of configuration of particles. • The correlation dimension D 2 reflected the connectivity between the particles. • The D 0 , D 1 and D 2 were effective as indices for evaluating the self-assembly of texture. [ABSTRACT FROM AUTHOR]

Aim: To apply the multifractal analysis method as a quantitative approach to a comprehensive description of the microvascular network architecture of the normal human retina., Methods: Fifty volunteers were enrolled in this study in the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and January 2014. A set of 100 segmented and skeletonised human retinal images, corresponding to normal states of the retina were studied. An automatic unsupervised method for retinal vessel segmentation was applied before multifractal analysis. The multifractal analysis of digital retinal images was made with computer algorithms, applying the standard box-counting method. Statistical analyses were performed using the GraphPad InStat software., Results: The architecture of normal human retinal microvascular network was able to be described using the multifractal geometry. The average of generalized dimensions ( D q ) for q =0, 1, 2, the width of the multifractal spectrum ( Δα=α max - α min ) and the spectrum arms' heights difference ( |Δf| ) of the normal images were expressed as mean±standard deviation (SD): for segmented versions, D 0 =1.7014±0.0057; D 1 =1.6507±0.0058; D 2 =1.5772±0.0059; Δα =0.92441±0.0085; |Δf| = 0.1453±0.0051; for skeletonised versions, D 0 =1.6303±0.0051; D 1 =1.6012±0.0059; D 2 =1.5531±0.0058; Δα =0.65032±0.0162; |Δf| = 0.0238±0.0161. The average of generalized dimensions ( D q ) for q =0, 1, 2, the width of the multifractal spectrum ( Δα ) and the spectrum arms' heights difference ( |Δf| ) of the segmented versions was slightly greater than the skeletonised versions., Conclusion: The multifractal analysis of fundus photographs may be used as a quantitative parameter for the evaluation of the complex three-dimensional structure of the retinal microvasculature as a potential marker for early detection of topological changes associated with retinal diseases.