Box-counting Analyses Research Articles

Optical coherence tomography angiography reveals progressive worsening of retinal vascular geometry in diabetic retinopathy and improved geometry after panretinal photocoagulation.

PurposeTo quantify vessel tortuosity and fractal dimension of the superficial capillary plexus (SCP) of the macula in different stages of diabetic retinopathy (DR), and following panretinal photocoagulation (PRP) using optical coherence tomography angiography (OCTA).Methods75 eyes of 75 subjects were divided into five groups; healthy controls, diabetes with no clinical DR, non-proliferative diabetic retinopathy (NPDR), proliferative diabetic retinopathy (PDR) and patients who received PRP for PDR (PDR+PRP).For vessel tortuosity, SCP slabs from 3x3 mm macular OCTA scans were processed using imageJ (NIH, USA), where large perifoveal vessels were traced and their length was measured with tortuosity calculated as the ratio between the actual length and the straight Euclidean length. For fractal dimension, SCP slabs were processed and imported to Fractalyse (ThéMA, France), where box-counting analyses produced fractal dimension values.ResultsWe found a significant difference in vessel tortuosity and fractal dimension between the five groups (one-way ANOVA, p < 0.001both). NPDR and PDR had significantly more tortuous vessels and lower fractal dimension compared to healthy controls (Tukey HSD: p = 0.02, 0.015,0.015 and <0.001, respectively). Fractal dimension was also significantly lower in NPDR and PDR compared to eyes with no clinical DR (p <0.001 both), and in PDR compared to NPDR (p = 0.014). Following PRP, vessel tortuosity was significantly lower and fractal dimension was higher in PDR+PRP compared to PDR (p = 0.001 and 0.031, respectively).ConclusionsWe used macular OCTA scans to demonstrate significantly higher perifoveal large vessel tortuosity, and lower fractal dimension in NPDR and PDR compared to healthy controls. Vessel tortuosity shows more dramatic normalization than fractal dimension and could be explored as a sensitive marker for successful PRP.

VALUE OF FRACTAL ANALYSIS OF OPTICAL COHERENCE TOMOGRAPHY ANGIOGRAPHY IN VARIOUS STAGES OF DIABETIC RETINOPATHY.

To use fractal dimensional analysis to investigate retinal vascular disease patterns in patients with diabetic retinopathy using spectral domain optical coherence tomography angiography. A retrospective study was conducted which included 49 eyes from 26 control subjects and 58 eyes from 35 patients known to have diabetic retinopathy. Of the 58 eyes with known retinopathy, 31 were categorized as nonproliferative diabetic retinopathy (13 mild, 9 moderate, and 9 severe) and 27 were categorized as proliferative diabetic retinopathy. Optical coherence tomography angiography images were acquired using the RTVue XR Avanti (Optovue, Inc). Automated segmentation was obtained through both the superficial and deep capillary plexuses for each eye. Grayscale optical coherence tomography angiography images were standardized and binarized using ImageJ (National Institutes of Health). Fractal box-counting analyses were conducted using Fractalyse (ThéMA). Fractal dimensions (FDs) and correlation coefficient of the superficial and deep capillary plexuses were compared between control eyes and those in various stages of diabetic retinopathy. The superficial and deep capillary plexuses from diabetic and control eyes were analyzed. The average FD for diabetic eyes was significantly lower than in control eyes in the superficial plexus (P = 2.4 × 10) and in the deep capillary plexus (P = 1.87 × 10 ) with a more statistically significant difference noted in the deep capillary plexus. When analyzing diabetic patients without edema noted on optical coherence tomography, the FD was significantly reduced in the superficial (P = 0.001) and deep (P = 1.49 × 10) plexuses. When analyzing diabetic patients with edema noted on optical coherence tomography, the FD was significantly reduced in the superficial (P = 2.0 × 10) and deep (P = 1.85 × 10) plexuses. The optical coherence tomography angiography FD is significantly lower in both superficial and deep capillary plexuses in eyes with all stages studied of diabetic retinopathy. The results were more often significant for the deep capillary plexus. The use of fractal analysis provides an objective criterion to assess microvascular disease burden in diabetic retinopathy.

Simulating clear-sky index increment correlations under mixed sky conditions using a fractal cloud model

Modelling clear-sky index increments (i.e. changes in normalized surface irradiance over specified intervals of time) and their spatial autocorrelation structures is important for the reliable grid integration of photovoltaic power systems. In order to capture increment correlation structures under mixed sky conditions, we apply a fractal cloud model. First, we use thousands of fish-eye sky camera and satellite images to estimate cloud edge fractal dimension D on scales between tens of meters and hundreds of kilometers. Box-counting analyses of cloud edges extracted from these images result in best-fit estimates of fractal dimension between 1.4 and 1.6, with satellite approximations being consistently higher than sky camera estimates. Contrary to previous studies, we find no evidence of any scale break and attribute the systematic discrepancy between the two estimates of D to intrinsic differences in the instrument-specific techniques of cloud detection. Next, we synthesize small-scale fractal cloud fields, with D=1.5 and satellite-derived cloud images as input. Using cloud motion vectors from the same satellite images, we then translate the small-scale cloud fields across a set of model pyranometer locations to obtain corresponding clear-sky index time series. From the simulated time series we calculate increment correlation structures for distances between tens of meters and about ten kilometers, and compare them to observation-based results from 1Hz measurements of up to 99 pyranometers. The observed isotropic, along-wind, and across-wind spatial autocorrelation structures of clear-sky index increments are captured well by the model, both in terms of overall value and shape. While there are some systematic differences between modelled and observed structures (e.g. underevaluation for very small time scales of a few seconds), the differences are essentially small. In general, the simulated correlations are not strongly sensitive to variations in the fractal model parameters.

Fractal Dimensional Analysis of Optical Coherence Tomography Angiography in Eyes With Diabetic Retinopathy.

We used fractal dimensional analysis to analyze retinal vascular disease burden in eyes with diabetic retinopathy using spectral-domain optical coherence tomography angiography (OCTA). A retrospective study was performed of 13 eyes with diabetic retinopathy without diabetic macular edema and 56 control eyes. Optical coherence tomography angiography images were acquired using the RTVue XR Avanti. Automated segmentation was obtained through the superficial and deep capillary plexuses for each eye. Grayscale OCTA images were standardized and binarized using ImageJ. Fractal box-counting analyses were performed using Fractalyse. Fractal dimensions (FD) as well as software-generated vascular density analyses of the superficial and deep capillary plexuses were compared between diabetic and control eyes using 2-tailed t-tests and 1-way multivariate ANOVA (MANOVA) analyses. The superficial and deep plexuses from diabetic and control eyes were analyzed. The average FD for diabetic eyes was significantly lower than control eyes for the superficial (P = 4.513 × 10-3) and deep (P = 2.653 × 10-3) capillary plexuses. In diabetic eyes, the vascular density also was significantly reduced in the superficial (P = 8.068 × 10-5) and deep (P = 3.120 × 10-6) capillary plexuses. One-way MANOVA showed a significant difference between diabetic and control eyes. The OCTA FD is significantly reduced in the superficial and deep capillary plexuses in eyes with diabetic retinopathy. Applying fractal analysis to OCTA imaging holds the potential to establish quantitative parameters for microvascular pathology.

Dislocation arrangement in small crystal volumes determines power-law size dependence of yield strength

It is by now well-known that micron-sized metallic crystals exhibit a smaller-being-stronger size effect: the yield strength σ varies with specimen size D approximately as a power-law σ∼D−m, and the exponent m has been found to vary within a range of ∼0.3–1.0 for different metals. However, little is known about why such a power-law comes into play, and what determines the actual value of the exponent m involved. This work shows that if the yield strength is determined by the Taylor interaction mechanism within the initial dislocation network, then for the size dependence of strength to be of the power-law relation observed, it is necessary for the mesh lengths L of the dislocation network to be power-law distributed, i.e. p(L)∼L−q. In such a case, the exponent m of the size effect is predicted to be inversely proportional to the sum of q the exponent of the mesh-length distribution and n the exponent of the dislocation velocity vs. stress law. To verify these predictions, compression experiments on aluminum micro-pillars with different pre-strains from 0% to 15% were carried out. The different pre-strains led to different initial dislocation networks, as well as different exponent m in the size dependence of strength. Box-counting analyses of transmission electron micrographs of the initial dislocation networks showed that the 2-D projected dislocation patterns were approximate fractals. On increasing pre-strain, the exponent m for the size dependence of strength was found to decrease while the fractal dimension of the initial dislocation patterns increased, thus verifying the inverse relationship between the two quantities. These findings show that the commonly observed power-law scaling of strength with size is due to an approximate power-law distribution of the initial dislocation mesh lengths, which also appears to be a robust feature in deformed metals. Furthermore, for a given metal, it is the exponent q of the initial mesh-length distribution which determines the value of the exponent m in the size dependence of strength.

Scale-invariance and self-organized criticality in migmatites of the southern Hualapai Mountains, Arizona

Whether or not melt-flow networks are scale-invariant may have broad implications for the understanding of the development of magmatic systems. Leucosome distributions in natural migmatite samples from the Hualapai Mountains, Arizona, are used to investigate the scaling characteristics of an inferred syndeformational melt-flow network. One-dimensional line and two-dimensional box-counting analyses yield distributions that are consistent with a scale-invariant relationship between the leucosome size and leucosome frequency. We infer from these results that the interactions of anatexis and deformation gave rise to a self-organized critical system in which melt distribution and melt movement were linked over a range of scales (at least millimeter- to meter-scale). We conclude that, even in metatexite migmatites volumetrically dominated by solid phases, melts can play an active role in determining the architecture of the melt-flow network. One implication of the envisioned self-organized critical system is the potential for rapid, large-scale melt coalescence and escape without long-term maintenance of a fully interconnected melt-flow network.

Fractal analysis of the evolution of a fracture network in a granite outcrop, SE Korea

Kinematical and fractal analyses were conducted on a fracture network in a well-exposed granite outcrop in SE Korea. The objective of this study is to examine the temporal and spatial evolution of the fracture network. From the orientation and abutting relationships of fracture sets, six fracturing events and their relative ages were established, several of which included strike-slip reactivations of earlier formed fractures. These events may be correlated with the Cenozoic to Recent evolution of the Yangsan Fault and surrounding areas. 2-D box counting analyses were performed on maps of the fracture network at the six stages of its evolution. For the earliest event D = 1.11, and represents a clustered development of fractures. Event 2 leads to a large increase of D = 1.51, with subsequent events producing a gradual increase up to 1.55 after the final event. This stabilization of the fractal dimension involved a change from the widespread development of new fractures (event 2) to the greater importance of reactivation of existing fractures, with only localized development of new fractures, usually as tip and linkage damage around pre-existing fractures. Fracture density was greatest for the event 2 fractures and increased only gradually after that. This pattern is similar to that of the fractal dimension and a good linear correlation was found between these two parameters. Although fractal dimensions and density show little change during the late stages of deformation, the fracture connectivity continued to increase due to the formation of local secondary fractures developed during reactivation of earlier formed fractures.

Three-dimensional wavelet-based multifractal method: the need for revisiting the multifractal description of turbulence dissipation data.

We generalize the wavelet transform modulus maxima (WTMM) method to multifractal analysis of 3D random fields. This method is calibrated on synthetic 3D monofractal fractional Brownian fields and on 3D multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration. Then we apply the 3D WTMM method to the dissipation field issued from 3D isotropic turbulence simulations. We comment on the need to revisit previous box-counting analyses which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master nonconservative singular cascade measures.

Fractal Representation of Turbulent Dispersing Plumes

Abstract Fractal analysis techniques have been applied to the concentration fields from large-eddy simulations of plume dispersion in a turbulent boundary layer. Fractal dimensions between 1.3 and 1.35 are obtained from area-perimeter and box-counting analyses for neutral and convective conditions. These values are close to previous estimates from atmospheric data. Methods for generating fractal fields with given statistical moments are examined and the simplest of these, the recursive refinement technique, is shown to be inadequate. The problem is shown to be the interpolation step of the procedure, which intrinsically reduces the variance with each refinement. Accurate statistical representation is obtained by replacing the interpolation step of the refinement technique with a sum of random pulses of appropriate width and random location. The pulse technique can easily he adapted to generate either clipped-normal or lognormal one-point probability distributions. Results from the fractal generation techn...

Fractal Analyses of High-Resolution Cloud Droplet Measurements

Abstract 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 inte...