Lacunarity analysis: A general technique for the analysis of spatial patterns

Using isolation-by-distance to jointly estimate effective population density and dispersal distance: a practical evaluation using bumble bees

Effective population density and intergenerational dispersal distance are key aspects of population biology, but obtaining empirical estimates of these parameters can be difficult. This is especially true for my study taxa, wild bees. In this paper, I apply and evaluate an existing but underutilized method to estimate effective density and dispersal distance of bumble bees (Bombus, Apidae). Specifically, using 10 datasets of bumble bees in North America, I use the relationship between genetic isolation-by-distance and Wright's neighborhood size to define a density-dispersal isocline-that is, a curve describing pairs of density and dispersal values consistent with observed rates of isolation-by-distance. These parameters are inversely related; as one increases the other decreases. I then use outside estimates of bumble bee dispersal distances to make more specific estimates of effective colony density. Compared to some prior estimates of census density (100s to 1000s colonies/km2), my estimated effective colony densities were very low (1-41 effective colonies/km2). I also hypothesize, however, that these estimates are affected by the spatial extent of sampling, due to scale-dependent patterns in the distribution of individuals. To test this hypothesis, I subsampled each dataset to simulate varying study extent, and repeated my analysis. Within populations, effective densities tended to decrease when measured across larger spatial extents. Altogether, I demonstrate a useful and under-appreciated tool for studying population biology, especially of small, mobile animals like bees, but also show that researchers must interpret their results carefully within the context of their study design.

Classifying breast cancer and fibroadenoma tissue biopsies from paraffined stain-free slides by fractal biomarkers in Fourier Ptychographic Microscopy

Breast cancer is one of the most spread and monitored pathologies in high-income countries. After breast biopsy, histological tissue is stored in paraffin, sectioned and mounted. Conventional inspection of tissue slides under benchtop light microscopes involves paraffin removal and staining, typically with H&E. Then, expert pathologists are called to judge the stained slides. However, paraffin removal and staining are operator-dependent, time and resources consuming processes that can generate ambiguities due to non-uniform staining. Here we propose a novel method that can work directly on paraffined stain-free slides. We use Fourier Ptychography as a quantitative phase-contrast microscopy method, which allows accessing a very wide field of view (i.e., mm2) in one single image while guaranteeing high lateral resolution (i.e., 0.5 µm). This imaging method is multi-scale, since it enables looking at the big picture, i.e. the complex tissue structure and connections, with the possibility to zoom-in up to the single-cell level. To handle this informative image content, we introduce elements of fractal geometry as multi-scale analysis method. We show the effectiveness of fractal features in describing and classifying fibroadenoma and breast cancer tissue slides from ten patients with very high accuracy. We reach 94.0 ± 4.2% test accuracy in classifying single images. Above all, we show that combining the decisions of the single images, each patient's slide can be classified with no error. Besides, fractal geometry returns a guide map to help pathologist to judge the different tissue portions based on the likelihood these can be associated to a breast cancer or fibroadenoma biomarker. The proposed automatic method could significantly simplify the steps of tissue analysis and make it independent from the sample preparation, the skills of the lab operator and the pathologist.

Parrondo's effects with aperiodic protocols

In this work, we study the effectiveness of employing archetypal aperiodic sequencing-namely, Fibonacci, Thue-Morse, and Rudin-Shapiro-on the Parrondian effect. From a capital gain perspective, our results show that these series do yield a Parrondo's paradox with the Thue-Morse based strategy outperforming not only the other two aperiodic strategies but benchmark Parrondian games with random and periodical (AABBAABB…) switching as well. The least performing of the three aperiodic strategies is the Rudin-Shapiro. To elucidate the underlying causes of these results, we analyze the cross correlation between the capital generated by the switching protocols and that of the isolated losing games. This analysis reveals that a strong anticorrelation with both isolated games is typically required to achieve a robust manifestation of Parrondo's effect. We also study the influence of the sequencing on the capital using the lacunarity and persistence measures. In general, we observe that the switching protocols tend to become less performing in terms of the capital as one increases the persistence and, thus, approaches the features of an isolated losing game. For the (log-)lacunarity, a property related to heterogeneity, we notice that for small persistence (less than 0.5), the performance increases with the lacunarity with a maximum around 0.4. In respect of this, our work shows that the optimization of a switching protocol is strongly dependent on a fine-tuning between persistence and heterogeneity.

The perceived beauty of art is not strongly calibrated to the statistical regularities of real-world scenes

Aesthetic judgements are partly predicted by image statistics, although the extent to which they are calibrated to the statistics of real-world scenes and the 'visual diet' of daily life is unclear. Here, we investigated the extent to which the beauty ratings of Western oil paintings from the JenAesthetics dataset can be accounted for by real-world scene statistics. We computed spatial and chromatic image statistics for the paintings and a set of real-world scenes captured by a head-mounted camera as participants went about daily lives. Partial least squares regression (PLSR) indicated that 6-15% of the variance in beauty ratings of the art can be accounted for by the art's image statistics. The luminance contrast of paintings made an important contribution to the PLSR models: paintings were perceived as more beautiful the greater the variation in luminance. PLSR models which expressed the art's image statistics relative to real-world scene statistics explained a similar amount of variance to models using the art's image statistics. The importance of an image statistic to perceived beauty was not related to how closely art reproduces the value from the real world. The findings suggest that beauty judgements of art are not strongly calibrated to the scene statistics of the real world.

Edible oil based optical clearing for optical coherence tomography angiography imaging

In optical imaging, optical clearing agents are commonly used to enhance the structural details of a sample. The current study investigates how to use it to improve the data obtained by an optical coherence tomography angiography system. A natural edible oil with no chemical base has been used for optical clearing. In-vivo testing on mice and humans yielded excellent optical clearing. Using computational techniques, the improvement in angiography signal caused by the optical clearing agent is investigated qualitatively and quantitatively. Compared to the control group, applying the edible oil-based optical clearing agent demonstrated improved vessel percentage and refined vascular signal intensity along depth.

Box-Counting Fractal Analysis: A Primer for the Clinician

This chapter lays out the elementary principles of fractal geometry underpinning much of the rest of this book. It assumes a minimal mathematical background, defines the key principles and terms in context, and outlines the basics of a fractal analysis method known as box counting and how it is used to perform fractal, lacunarity, and multifractal analyses. As a standalone reference, this chapter grounds the reader to be able to understand, evaluate, and apply essential methods to appreciate and heal the exquisitely detailed fractal geometry of the brain.

Geographic uncertainties in external exposome studies: A multi-scale approach to reduce exposure misclassification

BACKGROUND

Many studies on environment-health associations have emphasized that the selected buffer size (i.e., the scale of the geographic context when exposures are assigned at people's address location) may affect estimated effect sizes. However, there is limited methodological progress in addressing these buffer size-related uncertainties.

AIM

We aimed to 1) develop a statistical multi-scale approach to address buffer-related scale effects in cohort studies, and 2) investigate how environment-health associations differ between our multi-scale approach and ad hoc selected buffer sizes.

METHODS

We used lacunarity analyses to determine the largest meaningful buffer size for multiple high-resolution exposure surfaces (i.e., fine particulate matter [PM2.5], noise, and the normalized difference vegetation index [NDVI]). Exposures were linked to 7.7 million Dutch adults at their home addresses. We assigned exposure estimates based on buffers with fine-grained distance increments until the lacunarity-based upper limit was reached. Bayesian Cox model averaging addressed geographic uncertainties in the estimated exposure effect sizes within the exposure-specific upper buffer limits on mortality. Z-tests assessed statistical differences between averaged effect sizes and those obtained through pre-selected 100, 300, 1200, and 1500 m buffers.

RESULTS

The estimated lacunarity curves suggested exposure-specific upper buffer size limits; the largest was for NDVI (960 m), followed by noise (910 m) and PM2.5 (450 m). We recorded 845,229 deaths over eight years of follow-up. Our multi-scale approach indicated that higher values of NDVI were health-protectively associated with mortality risk (hazard ratio [HR]: 0.917, 95 % confidence interval [CI]: 0.886-0.948). Increased noise exposure was associated with an increased risk of mortality (HR: 1.003, 95 % CI: 1.002-1.003), while PM2.5 showed null associations (HR:0.998, 95 % CI: 0.997-1.000). Effect sizes of NDVI and noise differed significantly across the averaged and prespecified buffers (p < 0.05).

CONCLUSIONS

Geographic uncertainties in residential-based exposure assessments may obscure environment-health associations or risk spurious ones. Our multi-scale approach produced more consistent effect estimates and mitigated contextual uncertainties.

Fractal dimension of heights facilitates mesoscopic mechanical properties in ternary hard film surfaces

Hardness of thin films is a noteworthy property in the electronic and mechanical industry and is generally observed to be dependent on the degree of roughening facilitated from surface heights' surface spatial heterogeneity at the mesoscopic observation scale. Nonetheless, owing to enhanced scale fluctuations and higher-order central moments, conventional parameters provide limitations and errors in capturing the spatial heterogeneity of surfaces. Herein, we have utilized scale-independent fractal parameters to analyze the spatial heterogeneity of surface heights in Ti1−xSixN ternary hard films deposited with varying Si doping concentrations using sputtering technique. The fractal dimension, lacunarity coefficient, Moran index, surface entropy, Otsu's separability, and fractal succolarity were computed to provide an overarching understanding of the surface heights' spatial heterogeneity. Principal component analysis was employed on the data sets to identify the parameter(s) accounting for the maximum variance and accordingly, the structure–property relation between spatial heterogeneity of surface and hardness is analyzed and discussed in the context of the fractal dimension of surface heights. The results indicate the possibility of mesoscopic surface engineering and, consequently, tuning of hardness and modulus of elasticity in Ti1−xSixN hard films by mere changing of surface spatial heterogeneity facilitated by the fractal dimension of surface heights.

Lacunarity as a quantitative measure of mixing—a micro-CT analysis-based case study on granular materials

In practically every industry, mixing is a fundamental process, yet its 3D analysis is scarce in the literature. High-resolution computed tomography (micro-CT) is the perfect X-ray imaging tool to investigate the mixing of granular materials. Other than qualitative analysis, 3D micro-CT images provide an opportunity for quantitative analysis, which is of utmost importance, in terms of efficiency (time and budget) and environmental impact of the mixing process. In this work, lacunarity is proposed as a measure of mixing. By the lacunarity calculation on the repeated micro-CT measurements, a temporal description of the mixing can be given in three dimensions. As opposed to traditional mixing indices, the lacunarity curve provides additional information regarding the spatial distribution of the grains. Discrete element method simulations were also performed and showed similar results to the experiments.

Prediction of textile pilling resistance using optical coherence tomography

This paper describes a new method of textile pilling prediction, based on multivariate analysis of the spatial layer above the surface. The original idea of the method is the acquisition of 3D fabric image using optical coherence tomography (OCT) with infrared light, which allows for the fabric fuzz visualization with high sensitivity. The pilling layer, reconstructed with the resolution of [Formula: see text], includes reliable textural information related to the amount of loose fibers and bunches appearing as a result of abrasion. Pilling intensity was assigned by supervised classification of the textural features using both linear (PLS-DA - partial least squares discriminant analysis, LDA - linear discriminant analysis) and non-linear (SVM - support vector machine) classifiers. The results demonstrated that the method is more suitable for fabrics after short-term abrasion, when the fuzz prevails over tangled fibers in the pilling layer. In that case, pilling grades were predicted with [Formula: see text] accuracy, sensitivity and specificity (for SVM model). The validation accuracy of the tested models after machine abrasion achieves lower values (up to [Formula: see text] for LDA model). With our method, we clearly showed that OCT can be used to quantitatively trace appearance changes of fabric samples due to test abrasion.

The Longitudinal Assessment of Vascular Parameters of the Retina and Their Correlations with Systemic Characteristics in Type 2 Diabetes-A Pilot Study

The aim of the study was to assess various retinal vessel parameters of diabetes mellitus (DM) patients and their correlations with systemic factors in type 2 DM. A retrospective exploratory study in which 21 pairs of baseline and follow-up images of patients affected by DM were randomly chosen from the Sankara Nethralaya−Diabetic Retinopathy Study (SN DREAMS) I and II datasets. Patients’ fundus was photographed, and the diagnosis was made based on Klein classification. Vessel thickness parameters were generated using a web-based retinal vascular analysis platform called VASP. The thickness changes between the baseline and follow-up images were computed and normalized with the actual thicknesses of baseline images. The majority of parameters showed 10~20% changes over time. Vessel width in zone C for the second vein was significantly reduced from baseline to follow-up, which showed positive correlations with systolic blood pressure and serum high-density lipoproteins. Fractal dimension for all vessels in zones B and C and fractal dimension for vein in zones A, B and C showed a minimal increase from baseline to follow-up, which had a linear relationship with diastolic pressure, mean arterial pressure, serum triglycerides (p < 0.05). Lacunarity for all vessels and veins in zones A, B and C showed a minimal decrease from baseline to follow-up which had a negative correlation with pulse pressure and positive correlation with serum triglycerides (p < 0.05). The vessel widths for the first and second arteries significantly increased from baseline to follow-up and had an association with high-density lipoproteins, glycated haemoglobin A1C, serum low-density lipoproteins and total serum cholesterol. The central reflex intensity ratio for the second artery was significantly decreased from baseline to follow-up, and positive correlations were noted with serum triglyceride, serum low-density lipoproteins and total serum cholesterol. The coefficients for branches in zones B and C artery and the junctional exponent deviation for the artery in zone A decreased from baseline to follow-up showed positive correlations with serum triglycerides, serum low-density lipoproteins and total serum cholesterol. Identifying early microvascular changes in diabetic patients will allow for earlier intervention, improve visual outcomes and prevent vision loss.

Mitochondrial DNA Profiling by Fractal Lacunarity to Characterize the Senescent Phenotype as Normal Aging or Pathological Aging

Biocomplexity, chaos, and fractality can explain the heterogeneity of aging individuals by regarding longevity as a “secondary product” of the evolution of a dynamic nonlinear system. Genetic-environmental interactions drive the individual senescent phenotype toward normal, pathological, or successful aging. Mitochondrial dysfunctions and mitochondrial DNA (mtDNA) mutations represent a possible mechanism shared by disease(s) and the aging process. This study aims to characterize the senescent phenotype and discriminate between normal (nA) and pathological (pA) aging by mtDNA mutation profiling. MtDNA sequences from hospitalized and non-hospitalized subjects (age-range: 65–89 years) were analyzed and compared to the revised Cambridge Reference Sequence (rCRS). Fractal properties of mtDNA sequences were displayed by chaos game representation (CGR) method, previously modified to deal with heteroplasmy. Fractal lacunarity analysis was applied to characterize the senescent phenotype on the basis of mtDNA sequence mutations. Lacunarity parameter β, from our hyperbola model function, was statistically different (p < 0.01) between the nA and pA groups. Parameter β cut-off value at 1.26 × 10−3 identifies 78% nA and 80% pA subjects. This also agrees with the presence of MT-CO gene variants, peculiar to nA (C9546m, 83%) and pA (T9900w, 80%) mtDNA, respectively. Fractal lacunarity can discriminate the senescent phenotype evolving as normal or pathological aging by individual mtDNA mutation profile.

Analysis of Regional Variations of the Interstitial Cells of Cajal in the Murine Distal Stomach Informed by Confocal Imaging and Machine Learning Methods

Introduction

The network of Interstitial Cells of Cajal (ICC) plays a plethora of key roles in maintaining, coordinating, and regulating the contractions of the gastrointestinal (GI) smooth muscles. Several GI functional motility disorders have been associated with ICC degradation. This study extended a previously reported 2D morphological analysis and applied it to 3D spatial quantification of three different types of ICC networks in the distal stomach guided by confocal imaging and machine learning methods. The characterization of the complex changes in spatial structure of the ICC network architecture contributes to our understanding of the roles that different types of ICC may play in post-prandial physiology, pathogenesis, and/or amelioration of GI dsymotility- bridging structure and function.

Methods

A validated classification method using Trainable Weka Segmentation was applied to segment the ICC from a confocal dataset of the gastric antrum of a transgenic mouse, followed by structural analysis of the segmented images.

Results

The machine learning model performance was compared to manually segmented subfields, achieving an area under the receiver-operating characteristic (AUROC) of 0.973 and 0.995 for myenteric ICC (ICC-MP; n = 6) and intramuscular ICC (ICC-IM; n = 17). The myenteric layer in the distal antrum increased in thickness (from 14.5 to 34 μm) towards the lesser curvature, whereas the thickness decreased towards the lesser curvature in the proximal antrum (17.7 to 9 μm). There was an increase in ICC-MP volume from proximal to distal antrum (406,960 ± 140,040 vs. 559,990 ± 281,000 μm³; p = 0.000145). The % of ICC volume was similar for ICC-LM and for ICC-CM between proximal (3.6 ± 2.3% vs. 3.1 ± 1.2%; p = 0.185) and distal antrum (3.2 ± 3.9% vs. 2.5 ± 2.8%; p = 0.309). The average % volume of ICC-MP was significantly higher than ICC-IM at all points throughout sample (p < 0.0001).

Conclusions

The segmentation and analysis methods provide a high-throughput framework of investigating the structural changes in extended ICC networks and their associated physiological functions in animal models.

Evaluating the Ability to Use Contextual Features Derived from Multi-Scale Satellite Imagery to Map Spatial Patterns of Urban Attributes and Population Distributions

With an increasing global population, accurate and timely population counts are essential for urban planning and disaster management. Previous research using contextual features, using mainly very-high-spatial-resolution imagery (<2 m spatial resolution) at subnational to city scales, has found strong correlations with population and poverty. Contextual features can be defined as the statistical quantification of edge patterns, pixel groups, gaps, textures, and the raw spectral signatures calculated over groups of pixels or neighborhoods. While they correlated with population and poverty, which components of the human-modified landscape were captured by the contextual features have not been investigated. Additionally, previous research has focused on more costly, less frequently acquired very-high-spatial-resolution imagery. Therefore, contextual features from both very-high-spatial-resolution imagery and lower-spatial-resolution Sentinel-2 (10 m pixels) imagery in Sri Lanka, Belize, and Accra, Ghana were calculated, and those outputs were correlated with OpenStreetMap building and road metrics. These relationships were compared to determine what components of the human-modified landscape the features capture, and how spatial resolution and location impact the predictive power of these relationships. The results suggest that contextual features can map urban attributes well, with out-of-sample R2 values up to 93%. Moreover, the degradation of spatial resolution did not significantly reduce the results, and for some urban attributes, the results actually improved. Based on these results, the ability of the lower resolution Sentinel-2 data to predict the population density of the smallest census units available was then assessed. The findings indicate that Sentinel-2 contextual features explained up to 84% of the out-of-sample variation for population density.

Choroidal Neovascularization and Haller Vessel Morphology Associated with Vision and Treatment Number after 1 year in Age-related Macular Degeneration

Purpose

The present study aimed to quantify various factors of vessel morphology, including vessel diameter, length, and complexity (fractal dimension and lacunarity) of both choroidal neovascularization (CNV) and en face Haller vessels using optical coherence tomography angiography (OCTA) and en face structural optical coherence tomography in typical neovascular age-related macular degeneration (nAMD) and polypoidal choroidal vasculopathy (PCV) and to identify factors associated with visual acuity (VA) loss and number of injections within a year after the day of OCTA.

Methods

We retrospectively analyzed 43 eyes of nAMD patients and 33 eyes of PCV patients whose OCTA was performed at least 12 months after an initial anti-vascular endothelial growth factor treatment. Quantitative parameters, including vessel area, vessel diameter, vessel length, fractal dimension, and lacunarity were analyzed from en face images of CNV and Haller vessels. Clinical information, including logarithm of the minimum angle of resolution visual acuity and injection number of anti-vascular endothelial growth factor were acquired after 12 months from OCTA date. Using logistic regression analyses, parameters associated with logarithm of the minimum angle of resolution VA loss of 0.2 or more (VA loss group) and a number of injections of four or more (unstable group) after 12 months were analyzed.

Results

In typical nAMD, the VA loss group was associated with a smaller number of intersections of Haller vessels. The unstable group was associated with an increased lacunarity of CNV in typical nAMD. In PCV, both VA loss and unstable groups were associated with a higher maximal diameter of Haller vessels.

Conclusions

VA loss and injection number of nAMD and PCV 12 months after OCTA imaging were associated with different morphological parameters of CNV and Haller vessels. Therefore, quantitative analyses of both CNV and Haller vessels from OCTA and en face optical coherence tomography might provide prognostic information about visual outcome and injection frequency within 12 months after OCTA imaging.

Environmentally adaptive automated recognition of underwater mines with synthetic aperture sonar imagery

This work demonstrates that automated mine countermeasure (MCM) tasks are greatly facilitated by characterizing the seafloor environment in which the sensors operate as a first step within a comprehensive strategy for how to exploit information from available sensors, multiple detector types, measured features, and target classifiers, depending on the specific seabed characteristics present within the high-frequency synthetic aperture sonar (SAS) imagery used to perform MCM tasks. This approach is able to adapt as environmental characteristics change and includes the ability to recognize novel seabed types. Classifiers are then adaptively retrained through active learning in these unfamiliar seabed types, resulting in improved mitigation of challenging environmental clutter as it is encountered. Further, a segmentation constrained network algorithm is introduced to enable enhanced generalization abilities for recognizing mine-like objects from underrepresented environments within the training data. Additionally, a fusion approach is presented that allows the combination of multiple detectors, feature types spanning both measured expert features and deep learning, and an ensemble of classifiers for the particular seabed mixture proportions measured around each detected target. The environmentally adaptive approach is demonstrated to provide the best overall performance for automated mine-like object recognition.

Cost-Effective Fabrication of Fractal Silicon Nanowire Arrays

Silicon nanowires (Si NWs) emerged in several application fields as a strategic element to surpass the bulk limits with a flat compatible architecture. The approaches used for the Si NW realization have a crucial impact on their final performances and their final cost. This makes the research on a novel and flexible approach for Si NW fabrication a crucial point for Si NW-based devices. In this work, the novelty is the study of the flexibility of thin film metal-assisted chemical etching (MACE) for the fabrication of Si NWs with the possibility of realizing different doped Si NWs, and even a longitudinal heterojunction p-n inside the same single wire. This point has never been reported by using thin metal film MACE. In particular, we will show how this approach permits one to obtain a high density of vertically aligned Si NWs with the same doping of the substrate and without any particular constraint on doping type and level. Fractal arrays of Si NWs can be fabricated without any type of mask thanks to the self-assembly of gold at percolative conditions. This Si NW fractal array can be used as a substrate to realize controllable artificial fractals, integrating other interesting elements with a cost-effective microelectronics compatible approach.

Statistical Analysis of Nanofiber Mat AFM Images by Gray-Scale-Resolved Hurst Exponent Distributions

Two-dimensional structures, either periodic or random, can be classified by diverse mathematical methods. Quantitative descriptions of such surfaces, however, are scarce since bijective definitions must be found to measure unique dependency between described structures and the chosen quantitative parameters. To solve this problem, we use statistical analysis of periodic fibrous structures by Hurst exponent distributions. Although such a Hurst exponent approach was suggested some years ago, the quantitative analysis of atomic force microscopy (AFM) images of nanofiber mats in such a way was described only recently. In this paper, we discuss the influence of typical AFM image post-processing steps on the gray-scale-resolved Hurst exponent distribution. Examples of these steps are polynomial background subtraction, aligning rows, deleting horizontal errors and sharpening. Our results show that while characteristic features of these false-color images may be shifted in terms of gray-channel and Hurst exponent, they can still be used to identify AFM images and, in the next step, to quantitatively describe AFM images of nanofibrous surfaces. Such a gray-channel approach can be regarded as a simple way to include some information about the 3D structure of the image.

Erbium emission in Er:Y2O3 decorated fractal arrays of silicon nanowires

Disordered materials with new optical properties are capturing the interest of the scientific community due to the observation of innovative phenomena. We present the realization of novel optical materials obtained by fractal arrays of silicon nanowires (NWs) synthesized at low cost, without mask or lithography processes and decorated with Er:Y2O3, one of the most promising material for the integration of erbium in photonics. The investigated structural properties of the fractal Er:Y2O3/NWs demonstrate that the fractal morphology can be tuned as a function of the sputtering deposition angle (from 5° to 15°) of the Er:Y2O3 layer. We demonstrate that by this novel approach, it is possible to simply change the Er emission intensity by controlling the fractal morphology. Indeed, we achieved the increment of Er emission at 560 nm, opening new perspectives on the control and enhancement of the optical response of novel disordered materials.

Hymenophore configuration of the oak mazegill (Daedalea quercina)

The complex hymenophore configuration of the oak mazegill (Daedalea quercina, Polyporales) is rarely quantified, although quantifications are important analytical tools to assess form and growth. We quantified the hymenophore configuration of the oak mazegill by manual counting of tubes and tubular branches and ends. Complementary measurements were made with the software AngioTool. We found that the number of tubular branches and ends varied substantially between specimens, with a positive correlation with hymenophore area (5-51 cm²). We then measured complexity as tubular branches and ends per area, and complexity was not correlated with the size of the basidiocarps. Basidiocarps from two locations were compared (Hald ege, N = 11; Hvidding krat, N = 7), and the prevalence of branches and that of ends were greater in the Hvidding krat hymenophores (P < 0.001 and P = 0.029, respectively). Additionally, lacunarity, a measure of complexity ("gappiness"), gave a higher score for the Hald ege hymenophores (P = 0.002). Lacunarity analysis of multiple species of Polyporales showed that the oak mazegill hymenophore is comparatively complex. Concerning factors that affect hymenophore complexity of the oak mazegill, we observed that greater hymenophore complexity was associated with abrupt boundaries between growth zones on the pileus surface. Several years of monitoring documented that basidiocarps can remodel to gravitational changes and heal from damage. In conclusion, intra- and interspecies differences of hymenophore configuration can be quantified. In oak mazegill, hymenophore complexity is not dependent on size per se, although abrupt borders between growth zones are associated with increased complexity. Some of the variation between basidiocarps may reflect aspects of the ecology of the individual fungus.

Reproductive Homeostasis and Senescence in Drosophila melanogaster

The homeostatic properties of reproduction in aging female Drosophila melanogaster are investigated. Classic studies based on cohort analysis suggest that homeostatic capacity declines gradually as daily oviposition rates decline in aging flies. Analysis at the level of individuals gives a very different picture: reproductive homeostasis remains relatively constant for most of adult life until a critical point when oviposition either ceases entirely or continues in dysregulated fashion. The collapse of homeostatic capacity is abrupt. Enhanced homeostasis is associated with increased lifetime fecundity and improved prospects for survival. The fractal concept of lacunarity can be used to parameterize the "roughness" of individual fecundity trajectories and is inversely related to homeostatic capacity.

Detecting moving targets in active sonar echograph of harbor environment using high-order time lacunarity

This paper presents a feature for detecting potential targets from high level littoral clutters in active sonar echographs. Based on lacunarity, which describes the image texture statistics, an extension to the time domain is made in order to measure the dynamic behavior of target echoes and background clutters. Moreover, as high-order moments have been shown to well characterize the non-Rayleigh tails of littoral clutter, high-order computation is incorporated in the proposed high-order time lacunarity (HOT-Lac). The potential of HOT-Lac is demonstrated using a series of active sonar echographs with diverse cooperative targets detected in real-world harbor environments in the South China Sea. Specifically, it is shown how HOT-Lac can effectively distinguish different moving small targets from high-level background clutter, and how this ability can be exploited to highlight an invasion target in harbor security.

Biocomplexity and Fractality in the Search of Biomarkers of Aging and Pathology: Mitochondrial DNA Profiling of Parkinson's Disease

Increasing evidence implicates mitochondrial dysfunction in the etiology of Parkinson's disease (PD). Mitochondrial DNA (mtDNA) mutations are considered a possible cause and this mechanism might be shared with the aging process and with other age-related neurodegenerative disorders such as Alzheimer's disease (AD). We have recently proposed a computerized method for mutated mtDNA characterization able to discriminate between AD and aging. The present study deals with mtDNA mutation-based profiling of PD. Peripheral blood mtDNA sequences from late-onset PD patients and age-matched controls were analyzed and compared to the revised Cambridge Reference Sequence (rCRS). The chaos game representation (CGR) method, modified to visualize heteroplasmic mutations, was used to display fractal properties of mtDNA sequences and fractal lacunarity analysis was applied to quantitatively characterize PD based on mtDNA mutations. Parameter β, from the hyperbola model function of our lacunarity method, was statistically different between PD and control groups when comparing mtDNA sequence frames corresponding to GenBank np 5713-9713. Our original method, based on CGR and lacunarity analysis, represents a useful tool to analyze mtDNA mutations. Lacunarity parameter β is able to characterize individual mutation profile of mitochondrial genome and could represent a promising index to discriminate between PD and aging.

Paradoxical impact of sprawling intra-Urban Heat Islets: Reducing mean surface temperatures while enhancing local extremes

Extreme heat is one of the deadliest health hazards that is projected to increase in intensity and persistence in the near future. Here, we tackle the problem of spatially heterogeneous heat distribution within urban areas. We develop a novel multi-scale metric of identifying emerging heat clusters at various percentile-based thermal thresholds and refer to them collectively as intra-Urban Heat Islets. Using remotely sensed Land Surface Temperatures, we first quantify the spatial organization of heat islets in cities at various degrees of sprawl and densification. We then condense the size, spacing, and intensity information about heterogeneous clusters into probability distributions that can be described using single scaling exponents (denoted by β, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{\Lambda }}}_{{\boldsymbol{s}}{\boldsymbol{c}}{\boldsymbol{o}}{\boldsymbol{r}}{\boldsymbol{e}}}$$\end{document}Λscore, and λ, respectively). This allows for a seamless comparison of the heat islet characteristics across cities at varying spatial scales and improves on the traditional Surface Urban Heat Island (SUHI) Intensity as a bulk metric. Analysis of Heat Islet Size distributions demonstrates the emergence of two classes where the dense cities follow a Pareto distribution, and the sprawling cities show an exponential tempering of Pareto tail. This indicates a significantly reduced probability of encountering large heat islets for sprawling cities. In contrast, analysis of Heat Islet Intensity distributions indicates that while a sprawling configuration is favorable for reducing the mean SUHI Intensity of a city, for the same mean, it also results in higher local thermal extremes. This poses a paradox for urban designers in adopting expansion or densification as a growth trajectory to mitigate the UHI.

Lacunarity of the zero crossings of Gaussian processes

A lacunarity analysis of the zero crossings derived from Gaussian stochastic processes with oscillatory autocorrelation functions is evaluated and reveals distinct multiscaling signatures depending on the smoothness and degree of anticorrelation of the process. These bear qualitative similarities and quantitative distinctions from an oscillatory deterministic signal and a Poisson random process both possessing the same mean interval size between crossings. At very small and large scales compared with the correlation length of the random processes, the lacunarity is similar to the Poisson but exhibits significant departures from Poisson behavior if there is a zero-frequency component to the process's power spectrum. A comparison of exact results with the gliding box technique that is frequently used to determine lacunarity demonstrates its inherent bias.

Adsorptive Spin Coating To Study Thin-Film Stability in Both Wetting and Nonwetting Regimes

A new thin-film fabrication method, adsorptive spin coating, was evaluated in the preparation of poly(vinyl alcohol) (PVOH) thin films on silicon-wafer-supported poly(dimethylsiloxane) (PDMS) substrates. This method takes advantage of the rapid spontaneous adsorption of PVOH at the substrate-solution interface during the brief contact period and the directionality of drying during spinning. Similar to the results obtained using dip coating, the PVOH thin films wet the 2 kDa PDMS substrate and exhibit dewetted fractal morphologies on thicker PDMS substrates. This method generated PVOH films with thicknesses that were comparable to those prepared by dip coating except that thicker PVOH films were obtained at lower spin rates, following the Meyerhofer relationship in the wetting regime. Stepwise dewetting dynamics of confined PVOH drops were captured using high-speed photography. Drying and polymer aggregation initiate at the periphery of the drop and propagate toward the center of the drop. Each dewetted thin film adopts the footprint of the original drop and shows globally ordered patterns, which depend on both initial drop size and spin rate. The PVOH thin films have excellent stability toward water rinse if they are continuous and are given sufficient time to dry. This new adsorptive spin-coating method is not only straightforward but also unique in its ability to generate globally ordered morphologies that are the outcome of fast spontaneous adsorption, spin symmetry, and temporally and spatially adjustable drying rates. It is a valuable tool for fabricating a wide range of thin-film systems where surface adsorption/reaction is rapid, in both wetting and nonwetting regimes.

Primality, Fractality, and Image Analysis

This paper deals with the hidden structure of prime numbers. Previous numerical studies have already indicated a fractal-like behavior of prime-indexed primes. The construction of binary images enables us to generalize this result. In fact, two-integer sequences can easily be converted into a two-color image. In particular, the resulting method shows that both the coprimality condition and Ramanujan primes resemble the Minkowski island and Cantor set, respectively. Furthermore, the comparison between prime-indexed primes and Ramanujan primes is introduced and discussed. Thus the Cantor set covers a relevant role in the fractal-like description of prime numbers. The results confirm the feasibility of the method based on binary images. The link between fractal sets and chaotic dynamical systems may allow the characterization of the Hénon map only in terms of prime numbers.

Handwriting movements for assessment of motor symptoms in schizophrenia spectrum disorders and bipolar disorder

The main aim of the present study was to explore the value of several measures of handwriting in the study of motor abnormalities in patients with bipolar or psychotic disorders. 54 adult participants with a schizophrenia spectrum disorder or bipolar disorder and 44 matched healthy controls, participated in the study. Participants were asked to copy a handwriting pattern consisting of four loops, with an inking pen on a digitizing tablet. We collected a number of classical, non-linear and geometrical measures of handwriting. The handwriting of patients was characterized by a significant decrease in velocity and acceleration and an increase in the length, disfluency and pressure with respect to controls. Concerning non-linear measures, we found significant differences between patients and controls in the Sample Entropy of velocity and pressure, Lempel-Ziv of velocity and pressure, and Higuchi Fractal Dimension of pressure. Finally, Lacunarity, a measure of geometrical heterogeneity, was significantly greater in handwriting patterns from patients than from controls. We did not find differences in any handwriting measure on function of the specific diagnosis or the antipsychotic dose. Results indicate that participants with a schizophrenia spectrum disorder or bipolar disorder exhibit significant motor impairments and that these impairments can be readily quantified using measures of handwriting movements. Besides, they suggest that motor abnormalities are a core feature of several mental disorders and they seem to be unrelated to the pharmacological treatment.

Comparison of a Novel Ultrasonic Scaler Tip vs. Conventional Design on a Titanium Surface

The aim of this in vitro study was to evaluate the alterations of a titanium surface after treatment with two different types of ultrasonic tips: conventional steel versus an innovative copper alloy silver-plated one. Twenty smooth-surface, grade IV unalloyed titanium discs were divided into two groups. The discs were ultrasonically instrumented and the scaler was connected with a loading machine. The surface morphology was examined using scanning electron microscopy (SEM) and fractal analysis of lacunarity was calculated to highlight the alteration of the surface using the two different tips. The SEM analysis showed different degrees of surface roughness between the two types of scaler tips. Moreover, these observations demonstrated that the new tip showed fewer irregularities on the disc’s surface than the conventional steel tip. The statistical and fractal analysis showed a statistically significant difference between the two groups. Surface alterations of titanium induced by the conventional ultrasonic tips were much greater than those made by copper alloy silver plated tips. The presented results suggest that the use of this new ultrasonic tip may reduce the alterations on the implant surface during its use in dental practice.

Morphometrical evaluation of the choriocapillaris imaged by swept-source optical coherence tomography angiography

Purpose

To assess the reproducibility of quantitative morphometrical evaluation of the choriocapillaris imaged with swept-source optical coherence tomography angiography (SS-OCTA).

Subjects and methods

This observational, cross-sectional case series included 35 eyes of healthy individuals and 32 eyes of 32 patients. Two images of the fovea were taken using SS-OCTA with 3×3 mm squares. Images of the choriocapillaris within 800×800 pixel squares centered at the fovea were analyzed morphometrically using open-source software "AngioTool" that applies a Gaussian recursive filter and multiscale Hessian enhancement. This program's vessel thickness and intensity parameters can be changed to aid vessel detection. We measured the pairs of images per eye with different parameter sets and calculated the intraclass correlation (ICC) for the morphometrical results. After determining the parameters that produced high reproducibility, we evaluated regional variations in 800×800 pixel mm squares within the fovea.

Results

The ICCs for vessel area, total vessel length, vessel diameter index, and mean lacunarity were over 0.9 using the parameters of "vessel thickness" 3-4 and intensity 15 in the group including all subjects. When measurements were performed using these same parameter values, the vessel density and mean vessel diameter index were 60.5% and 19.1±0.389, respectively. Vessel density, vessel length, vessel diameter index, and mean lacunarity did not change significantly within an 800×800 pixel square centered at the fovea except for the 200×200 pixel square at the foveal center.

Conclusion

SS-OCTA images of the choriocapillaris can be measured with high reproducibility by morphometrical evaluation using open-source software with multiscale Hessian enhancement. Such automated morphometric analysis can provide an objective evaluation of the choriocapillaris.

Multilacunarity as a spatial multiscale multi-mass morphometric of change in the meso-architecture of plant parenchyma tissue

The lacunarity index (monolacunarity) averages the behavior of variable size structures in a binary image. The generalized lacunarity concept (multilacunarity) on the basis of generalized distribution moments is an appealing model that can account for differences in the mass content at different scales. The model was tested previously on natural images [J. Vernon-Carter et al., Physica A 388, 4305 (2009)]. Here, the computational aspects of multilacunarity are validated using synthetic binary images that consist of random maps, spatial stochastic patterns, patterns with circular or polygonal elements, and a plane fractal. Furthermore, monolacunarity and detrended fluctuation analysis were employed to quantify the mesostructural changes in the intercellular air spaces of frozen-thawed parenchymatous tissue of pome fruit [N. A. Valous et al., J. Appl. Phys. 115, 064901 (2014)]. Here, the aim is to further examine the coherence of the multilacunarity model for quantifying the mesostructural changes in the intercellular air spaces of parenchymatous tissue of pome and stone fruit, acquired with X-ray microcomputed tomography, after storage and ripening, respectively. The multilacunarity morphometric is a multiscale multi-mass fingerprint of spatial pattern composition, assisting the exploration of the effects of metabolic and physiological activity on the pore space of plant parenchyma tissue.

Pattern formation of stains from dried drops to identify spermatozoa motility

We study how cell motility affects the stains left by the evaporation of droplets of a biofluid suspension containing mouse spermatozoa. The suspension, which contains also a high concentration of salts usually needed by motile cells, forms, upon drying, a crystallized pattern. We examine the structural characteristics of such patterns by optical microscopy. The analysis reveals that cell motility affects the formation of elongated crystals with lateral tips, as well as the creation of interlocked aggregates. We prove that a lacunarity algorithm based on polar symmetry, distinguishes among deposits generated by motile and non-motile cells with an accuracy greater than 95%.

Microenvironmental influence on microtumour infiltration patterns: 3D-mathematical modelling supported by in vitro studies

Mathematical modelling approaches have become increasingly abundant in cancer research. Tumour infiltration extent and its spatial organization depend both on the tumour type and stage and on the bio-physicochemical characteristics of the microenvironment. This sets a complex scenario that often requires a multidisciplinary and individually adjusted approach. The ultimate goal of this work is to present an experimental/numerical combined method for the development of a three-dimensional mathematical model with the ability to reproduce the growth and infiltration patterns of a given avascular microtumour in response to different microenvironmental conditions. The model is based on a diffusion-convection reaction equation that considers logistic proliferation, volumetric growth, a rim of proliferative cells at the tumour surface, and invasion with diffusive and convective components. The parameter values of the model were fitted to experimental results while radial velocity and diffusion coefficients were made spatially variable in a case-specific way through the introduction of a shape function and a diffusion-limited-aggregation (DLA)-derived fractal matrix, respectively, according to the infiltration pattern observed. The in vitro model consists of multicellular tumour spheroids (MTSs) of an epithelial mammary tumour cell line (LM3) immersed in a collagen I gel matrix with a standard culture medium ("naive" matrix) or a conditioned medium from adipocytes or preadipocytes ("conditioned" matrix). It was experimentally determined that both adipocyte and preadipocyte conditioned media had the ability to change the MTS infiltration pattern from collective and laminar to an individual and atomized one. Numerical simulations were able to adequately reproduce qualitatively and quantitatively both kinds of infiltration patterns, which were determined by area quantification, analysis of fractal dimensions and lacunarity, and Bland-Altman analysis. These results suggest that the combined approach presented here could be established as a new framework with interesting potential applications at both the basic and clinical levels in the oncology area.

Fractal dimension and lacunarity analysis of retinal microvascular morphology in hypertension and diabetes

Hypertension and diabetes mellitus represent modifiable risk factors for vascular disease. They cause microvascular remodeling, and ultimately result in end-organ damage. Therefore, development of methods for noninvasive quantification of the effects of hypertension and diabetes mellitus on microvasculature is of paramount importance. The two goals of the study were: 1) to characterize the geometric complexity and inhomogeneity of retinal vasculature in hypertensive retinopathy (HR) and in proliferative diabetic retinopathy (PDR) by using box counting fractal dimension and lacunarity analysis, and 2) to determine if the combination of these two parameters can be used to describe differences in the vascular tree geometry between HR and PDR. The extended set of retinal images from the publicly available STARE database was manually segmented by our expert, validated, and made available for other researchers to use. The healthy retinal vascular network has a higher complexity (fractal dimension) compared to that in HR and in PDR. However, there is no difference in microvascular complexity between HR and PDR. The inhomogeneity of the retinal microvascular tree (lacunarity) was higher in PDR compared to HR. Lacunarity and fractal dimension together quantitatively characterize microvascular geometry in the retina with higher specificity than fractal analysis alone.

Rooting strategies in a subtropical savanna: a landscape-scale three-dimensional assessment

In resource-limited savannas, the distribution and abundance of fine roots play an important role in acquiring essential resources and structuring vegetation patterns and dynamics. However, little is known regarding the three-dimensional distribution of fine roots in savanna ecosystems at the landscape scale. We quantified spatial patterns of fine root density to a depth of 1.2 m in a subtropical savanna landscape using spatially specific sampling. Kriged maps revealed that fine root density was highest at the centers of woody patches, decreased towards the canopy edges, and reached lowest values within the grassland matrix throughout the entire soil profile. Lacunarity analyses indicated that spatial heterogeneities of fine root density decreased continuously to a depth of 50 cm and then increased in deeper portions of the soil profile across this landscape. This vertical pattern might be related to inherent differences in root distribution between trees/shrubs and herbaceous species, and the presence/absence of an argillic horizon across this landscape. The greater density of fine roots beneath woody patches in both upper and lower portions of the soil profile suggests an ability to acquire disproportionately more resources than herbaceous species, which may facilitate the development and persistence of woody patches across this landscape.

Changes in Dimensionality and Fractal Scaling Suggest Soft-Assembled Dynamics in Human EEG

Humans are high-dimensional, complex systems consisting of many components that must coordinate in order to perform even the simplest of activities. Many behavioral studies, especially in the movement sciences, have advanced the notion of soft-assembly to describe how systems with many components coordinate to perform specific functions while also exhibiting the potential to re-structure and then perform other functions as task demands change. Consistent with this notion, within cognitive neuroscience it is increasingly accepted that the brain flexibly coordinates the networks needed to cope with changing task demands. However, evaluation of various indices of soft-assembly has so far been absent from neurophysiological research. To begin addressing this gap, we investigated task-related changes in two distinct indices of soft-assembly using the established phenomenon of EEG repetition suppression. In a repetition priming task, we assessed evidence for changes in the correlation dimension and fractal scaling exponents during stimulus-locked event-related potentials, as a function of stimulus onset and familiarity, and relative to spontaneous non-task-related activity. Consistent with predictions derived from soft-assembly, results indicated decreases in dimensionality and increases in fractal scaling exponents from resting to pre-stimulus states and following stimulus onset. However, contrary to predictions, familiarity tended to increase dimensionality estimates. Overall, the findings support the view from soft-assembly that neural dynamics should become increasingly ordered as external task demands increase, and support the broader application of soft-assembly logic in understanding human behavior and electrophysiology.

Spatial intratumoral heterogeneity of proliferation in immunohistochemical images of solid tumors

PURPOSE

The interactions of neoplastic cells with each other and the microenvironment are complex. To understand intratumoral heterogeneity, subtle differences should be quantified. Main factors contributing to heterogeneity include the gradient ischemic level within neoplasms, action of microenvironment, mechanisms of intercellular transfer of genetic information, and differential mechanisms of modifications of genetic material/proteins. This may reflect on the expression of biomarkers in the context of prognosis/stratification. Hence, a rigorous approach for assessing the spatial intratumoral heterogeneity of histological biomarker expression with accuracy and reproducibility is required, since patterns in immunohistochemical images can be challenging to identify and describe.

METHODS

A quantitative method that is useful for characterizing complex irregular structures is lacunarity; it is a multiscale technique that exhaustively samples the image, while the decay of its index as a function of window size follows characteristic patterns for different spatial arrangements. In histological images, lacunarity provides a useful measure for the spatial organization of a biomarker when a sampling scheme is employed and relevant features are computed. The proposed approach quantifies the segmented proliferative cells and not the textural content of the histological slide, thus providing a more realistic measure of heterogeneity within the sample space of the tumor region. The aim is to investigate in whole sections of primary pancreatic neuroendocrine neoplasms (pNENs), using whole-slide imaging and image analysis, the spatial intratumoral heterogeneity of Ki-67 immunostains. Unsupervised learning is employed to verify that the approach can partition the tissue sections according to distributional heterogeneity.

RESULTS

The architectural complexity of histological images has shown that single measurements are often insufficient. Inhomogeneity of distribution depends not only on percentage content of proliferation phase but also on how the phase fills the space. Lacunarity curves demonstrate variations in the sampled image sections. Since the spatial distribution of proliferation in each case is different, the width of the curves changes too. Image sections that have smaller numerical variations in the computed features correspond to neoplasms with spatially homogeneous proliferation, while larger variations correspond to cases where proliferation shows various degrees of clumping. Grade 1 (uniform/nonuniform: 74%/26%) and grade 3 (uniform: 100%) pNENs demonstrate a more homogeneous proliferation with grade 1 neoplasms being more variant, while grade 2 tumor regions render a more diverse landscape (50%/50%). Hence, some cases show an increased degree of spatial heterogeneity comparing to others with similar grade. Whether this is a sign of different tumor biology and an association with a more benign/malignant clinical course needs to be investigated further. The extent and range of spatial heterogeneity has the potential to be evaluated as a prognostic marker.

CONCLUSIONS

The association with tumor grade as well as the rationale that the methodology reflects true tumor architecture supports the technical soundness of the method. This reflects a general approach which is relevant to other solid tumors and biomarkers. Drawing upon the merits of computational biomedicine, the approach uncovers salient features for use in future studies of clinical relevance.

Bidimensional unconstrained optimization approach to EMD: An algorithm revealing skin perfusion alterations in pseudoxanthoma elasticum patients

BACKGROUND AND OBJECTIVE

Pseudoxanthoma elasticum (PXE) is an inherited and systemic metabolic disorder that affects the skin, leading among other things to a peau d'orange appearance. Unfortunately, PXE is still poorly understood and there is no existing therapy to treat the disease. Because the skin is the first organ to be affected in PXE, we propose herein a study of skin microvascular perfusion. By means of this analysis, our goal is to increase knowledge of PXE.

METHODS

For this purpose, microvascular data from patients suffering from PXE and from healthy control subjects were recorded using the laser speckle contrast imaging (LSCI) modality. These data were processed using the recent 2D version of the unconstrained optimization approach to empirical mode decomposition (UOA-EMD). Our work therefore corresponds to the first time this algorithm has been applied to biomedical data.

RESULTS

Our study shows that the 2D-UOA-EMD is able to reveal spatial patterns on local textures of LSCI data. Moreover, these spatial patterns differ between PXE patients and control subjects. Quantification measure of these spatial patterns reveals statistical significant differences between PXE and control subjects, in the neck (p=0.0004) and in the back (p=0.0052).

CONCLUSIONS

For the first time, alterations in microvascular perfusion in PXE patients have been revealed. Our findings open new avenues for our understanding of pathophysiologic skin changes in PXE.

Biocomplexity and Fractality in the Search of Biomarkers of Aging and Pathology: Focus on Mitochondrial DNA and Alzheimer's Disease

Alzheimer’s disease (AD) represents one major health concern for our growing elderly population. It accounts for increasing impairment of cognitive capacity followed by loss of executive function in late stage. AD pathogenesis is multifaceted and difficult to pinpoint, and understanding AD etiology will be critical to effectively diagnose and treat the disease. An interesting hypothesis concerning AD development postulates a cause-effect relationship between accumulation of mitochondrial DNA (mtDNA) mutations and neurodegenerative changes associated with this pathology. Here we propose a computerized method for an easy and fast mtDNA mutations-based characterization of AD. The method has been built taking into account the complexity of living being and fractal properties of many anatomic and physiologic structures, including mtDNA. Dealing with mtDNA mutations as gaps in the nucleotide sequence, fractal lacunarity appears a suitable tool to differentiate between aging and AD. Therefore, Chaos Game Representation method has been used to display DNA fractal properties after adapting the algorithm to visualize also heteroplasmic mutations. Parameter β from our fractal lacunarity method, based on hyperbola model function, has been measured to quantitatively characterize AD on the basis of mtDNA mutations. Results from this pilot study to develop the method show that fractal lacunarity parameter β of mtDNA is statistically different in AD patients when compared to age-matched controls. Fractal lacunarity analysis represents a useful tool to analyze mtDNA mutations. Lacunarity parameter β is able to characterize individual mutation profile of mitochondrial genome and appears a promising index to discriminate between AD and aging.

An Initial Exploration of a Lacunarity-Based Segregation Measure

It has been recognized that urban residential segregation is a scale-dependent phenomenon, and yet all existing structural or spatial indices measure only certain dimensions of segregation at one single scale. Inspired by new metrics developed by landscape ecologists to measure landscape heterogeneity, in this paper we explore the feasiblity of a multiscale, lacunarity-based segregation measure. We also develop a straightforward GIS-based procedure to calculate this new measure. Our initial simulation results show that lacunarity is an effective measure that can capture multiple dimensions of segregation patterns at multiple scales.

Changing Patterns of Residential Segregation in a Prismatic Metropolis: A Lacunarity-Based Study in Houston, 1980–2000

The role of race versus class in shaping residential segregation patterns has been a contentious issue in segregation studies for decades. Despite the voluminous, interdisciplinary literature, scholars have reported conflicting evidence on the role of race versus class in residential segregations. We attribute the current inconclusive literature partially to the failure to consider scale explicitly in residential segregation measures, and partially to the growing complexity of a multiethnic melting pot in most cities in the United States. Inspired by new metrics employed by landscape ecologists to measure landscape heterogeneity, residential segregation is reconceived as a scale-dependent social phenomenon in this paper. We also present an alternative to existing structural or spatial segregation measures, considered as less efficient because most of the existing indices measure only a few dimensions of segregation at a single scale. We have developed a multiscale, lacunarity-based segregation measure, and have used it to examine the role of race versus class in residential segregation patterns in Houston, Texas. Using census-tract-level data from 1980 to 2000, we found that race is still the most important factor in explaining residential segregation despite the overall decline of segregation by both income and race. It was also found that the changing segregation patterns over time are contingent upon the scale as well as the race or income group considered.

Villus packing density and lacunarity: Markers of placental efficiency?

We evaluate, in routine H&E histology slides, villus quantity in a given area (villous packing density, VPD) and the pattern or "gappiness" of villous distribution (lacunarity), and test for correlations with a proxy for fetoplacental metabolic rate, β calculated as (ln (placental weight)/ln (birthweight)) from Kleiber's law [1]. Three ∼4.3 mm² images each were obtained from 88 term placentas. Ranges of VPD and lacunarity were each correlated with β (r = 0.31, p = 0.003, r = 0.23, p = 0.03 and respectively). The relationship between β and within-placenta variation in VPD and lacunarity highlights the need to study not merely the mean but the variance of villous geometries and spatial distributions.

Tug-of-war lacunarity-A novel approach for estimating lacunarity

Modern instrumentation provides us with massive repositories of digital images that will likely only increase in the future. Therefore, it has become increasingly important to automatize the analysis of digital images, e.g., with methods from pattern recognition. These methods aim to quantify the visual appearance of captured textures with quantitative measures. As such, lacunarity is a useful multi-scale measure of texture's heterogeneity but demands high computational efforts. Here we investigate a novel approach based on the tug-of-war algorithm, which estimates lacunarity in a single pass over the image. We computed lacunarity for theoretical and real world sample images, and found that the investigated approach is able to estimate lacunarity with low uncertainties. We conclude that the proposed method combines low computational efforts with high accuracy, and that its application may have utility in the analysis of high-resolution images.