Characterizing the lacunarity of random and deterministic fractal sets

Fractals in the Neurosciences: A Translational Geographical Approach

The chapter presents three new fractal indices (fractal fragmentation index, fractal tentacularity index, and fractal anisotropy index) and normalized Kolmogorov complexity with proven applicability in geographic research, developed by the authors, and the possibility of their future use in neuroscience. The research demonstrates the relevance of fractal analysis in different fields and the basic concepts and principles of fractal geometry being sufficient for the development of models relevant to the studied reality. Also, the research highlighted the need to continue interdisciplinary research based on known fractal indicators, as well as the development of new analysis methods with the translational potential between fields.

Light propagation in two-dimensional and three-dimensional slabs of reflective colloidal particles in solution: The effect of interfaces and interparticle correlations

The propagation of light across 2D and 3D slabs of reflective colloidal particles in a fluidlike state has been investigated by simulation. The colloids are represented as hard spheres with and without an attractive square-well tail. Representative configurations of particles have been generated by Monte Carlo. The path of rays entering the slab normal to its planar surface has been determined by exact geometric scattering conditions, assuming that particles are macroscopic spheres fully reflective at the surface of their hard-core potential. The analysis of light paths provides the transmission and reflection coefficients, the mean-free path, the average length of transmitted and reflected paths, the distribution of scattering events across the slab, the angular spread of the outcoming rays as a function of dimensionality, and thermodynamic state. The results highlight the presence of a sizable population of very long paths, which play an important role in random lasing from solutions of metal particles in an optically active fluid. The output power spectrum resulting from the stimulated emission amplification decays asymptotically as an inverse power law. The present study goes beyond the standard approach based on a random walk confined between two planar interfaces and parametrized in terms of the mean-free path and scattering matrix. Here, instead, the mean-free path, the correlation among scattering events, and memory effects are not assumed a priori, but emerge from the underlying statistical mechanics model of interacting particles. In this way the dependence of properties on the thermodynamic state, the effect of particle-particle and particle-interface correlations and of spatial inhomogeneity, and memory effects are accounted for in a transparent way. Moreover, the approach joins smoothly the ballistic regime of light propagation at low density with the diffusive regime at high density of scattering centers. These properties are exploited to investigate the effect of weak polydispersivity and of large density fluctuations at the critical point of the model with the attractive potential tail.

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.

Classification of Multiple H&E Images via an Ensemble Computational Scheme

In this work, a computational scheme is proposed to identify the main combinations of handcrafted descriptors and deep-learned features capable of classifying histological images stained with hematoxylin and eosin. The handcrafted descriptors were those representatives of multiscale and multidimensional fractal techniques (fractal dimension, lacunarity and percolation) applied to quantify the histological images with the corresponding representations via explainable artificial intelligence (xAI) approaches. The deep-learned features were obtained from different convolutional neural networks (DenseNet-121, EfficientNet-b2, Inception-V3, ResNet-50 and VGG-19). The descriptors were investigated through different associations. The most relevant combinations, defined through a ranking algorithm, were analyzed via a heterogeneous ensemble of classifiers with the support vector machine, naive Bayes, random forest and K-nearest neighbors algorithms. The proposed scheme was applied to histological samples representative of breast cancer, colorectal cancer, oral dysplasia and liver tissue. The best results were accuracy rates of 94.83% to 100%, with the identification of pattern ensembles for classifying multiple histological images. The computational scheme indicated solutions exploring a reduced number of features (a maximum of 25 descriptors) and with better performance values than those observed in the literature. The presented information in this study is useful to complement and improve the development of computer-aided diagnosis focused on histological images.

Estimating fractal dimensions: A comparative review and open source implementations

The fractal dimension is a central quantity in nonlinear dynamics and can be estimated via several different numerical techniques. In this review paper, we present a self-contained and comprehensive introduction to the fractal dimension. We collect and present various numerical estimators and focus on the three most promising ones: generalized entropy, correlation sum, and extreme value theory. We then perform an extensive quantitative evaluation of these estimators, comparing their performance and precision using different datasets and comparing the impact of features like length, noise, embedding dimension, and falsify-ability, among many others. Our analysis shows that for synthetic noiseless data, the correlation sum is the best estimator with extreme value theory following closely. For real experimental data, we found the correlation sum to be more strongly affected by noise vs the entropy and extreme value theory. The recent extreme value theory estimator seems powerful as it has some of the advantages of both alternative methods. However, using four different ways for checking for significance, we found that the method yielded "significant" low-dimensional results for inappropriate data like stock market timeseries. This fact, combined with some ambiguities we found in the literature of the method applications, has implications for both previous and future real-world applications using the extreme value theory approach, as, for example, the argument for small effective dimensionality in the data cannot come from the method itself. All algorithms discussed are implemented as performant and easy to use open source code via the DynamicalSystems.jl library.

Measuring Drug Therapy Effect on Osteoporotic Fracture Risk by Trabecular Bone Lacunarity: The LOTO Study

An MRI method providing one parameter (TBLβ: trabecular-bone-lacunarity-parameter-β) that is sensitive to trabecular bone architecture (TBA) changes with aging and osteoporosis is under study as a new tool in the early diagnosis of bone fragility fracture. A cross-sectional and prospective observational study (LOTO: Lacunarity Of Trabecular bone in Osteoporosis) on over-50s women, at risk for bone fragility fracture, was designed to validate the method. From the baseline data, we observed that in women with prevalent vertebral fractures (VF+), TBA was differently characterized by TBLβ when osteoporosis treatment is considered. Here we verify the potential of TBLβ as an index of osteoporosis treatment efficacy. Untreated (N = 156) and treated (N = 123) women were considered to assess differences in TBLβ related to osteoporosis treatment. Prevalent VFs were found in 31% of subjects, 63% of which were under osteoporosis medications. The results show that TBLβ discriminates between VF+ and VF− patients (p = 0.004). This result is mostly stressed in untreated subjects. Treatment, drug therapy in particular (89% Bisphosphonates), significantly counteracts the difference between VF+ and VF− within and between groups: TBLβ values in treated patients are comparable to untreated VF− and statistically higher than untreated VF+ (p = 0.014) ones. These results highlight the potential role of TBLβ as an index of treatment efficacy.

Application of Lacunarity for Quantification of Single Molecule Localization Microscopy Images

The quantitative analysis of datasets achieved by single molecule localization microscopy is vital for studying the structure of subcellular organizations. Cluster analysis has emerged as a multi-faceted tool in the structural analysis of localization datasets. However, the results it produces greatly depend on the set parameters, and the process can be computationally intensive. Here we present a new approach for structural analysis using lacunarity. Unlike cluster analysis, lacunarity can be calculated quickly while providing definitive information about the structure of the localizations. Using simulated data, we demonstrate how lacunarity results can be interpreted. We use these interpretations to compare our lacunarity analysis with our previous cluster analysis-based results in the field of DNA repair, showing the new algorithm’s efficiency.

Automated assessment of breast margins in deep ultraviolet fluorescence images using texture analysis

Microscopy with ultraviolet surface excitation (MUSE) is increasingly studied for intraoperative assessment of tumor margins during breast-conserving surgery to reduce the re-excision rate. Here we report a two-step classification approach using texture analysis of MUSE images to automate the margin detection. A study dataset consisting of MUSE images from 66 human breast tissues was constructed for model training and validation. Features extracted using six texture analysis methods were investigated for tissue characterization, and a support vector machine was trained for binary classification of image patches within a full image based on selected feature subsets. A weighted majority voting strategy classified a sample as tumor or normal. Using the eight most predictive features ranked by the maximum relevance minimum redundancy and Laplacian scores methods has achieved a sample classification accuracy of 92.4% and 93.0%, respectively. Local binary pattern alone has achieved an accuracy of 90.3%.

Novel Recurrence Relations for Volumes and Surfaces of n-Balls, Regular n-Simplices, and n-Orthoplices in Real Dimensions

This study examines n-balls, n-simplices, and n-orthoplices in real dimensions using novel recurrence relations that remove the indefiniteness present in known formulas. They show that in the negative, integer dimensions, the volumes of n-balls are zero if n is even, positive if n = −4k − 1, and negative if n = −4k − 3, for natural k. The volumes and surfaces of n-cubes inscribed in n-balls in negative dimensions are complex, wherein for negative, integer dimensions they are associated with integral powers of the imaginary unit. The relations are continuous for n ∈ ℝ and show that the constant of π is absent for 0 ≤ n < 2. For n < −1, self-dual n-simplices are undefined in the negative, integer dimensions, and their volumes and surfaces are imaginary in the negative, fractional ones and divergent with decreasing n. In the negative, integer dimensions, n-orthoplices reduce to the empty set, and their real volumes and imaginary surfaces are divergent in negative, fractional ones with decreasing n. Out of three regular, convex polytopes present in all natural dimensions, only n-orthoplices and n-cubes (and n-balls) are defined in the negative, integer dimensions.

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.

Characterization of Two-Phase Flow from Pore-Scale Imaging Using Fractal Geometry under Water-Wet and Mixed-Wet Conditions

High resolution micro-computed tomography images for multiphase flow provide us an effective tool to understand the mechanism of fluid flow in porous media, which is not only fundamental to the understanding of macroscopic measurements but also for providing benchmark datasets to validate pore-scale modeling. In this study, we start from two datasets of pore scale imaging of two-phase flow obtained experimentally under in situ imaging conditions at different water fractional flows under water-wet and mixed-wet conditions. Then, fractal dimension, lacunarity and succolarity are used to quantify the complexity, clustering and flow capacity of water and oil phases. The results show that with the wettability of rock surface altered from water-wet to mixed-wet, the fractal dimension for the water phase increases while for the oil phase, it decreases obviously at low water saturation. Lacunarity largely depends on the degree of wettability alteration. The more uniform wetting surfaces are distributed, the more homogeneous the fluid configuration is, which indicates smaller values for lacunarity. Moreover, succolarity is shown to well characterize the wettability effect on flow capacity. The succolarity of the oil phase in the water-wet case is larger than that in the mixed-wet case while for the water phase, the succolarity value in the water-wet is small compared with that in the mixed-wet, which show a similar trend with relative permeability curves for water-wet and mixed-wet. Our study provides a perspective into the influence that phase geometry has on relative permeability under controlled wettability and the resulting phase fractal changes under different saturations that occur during multiphase flow, which allows a means to understand phase geometric changes that occur during fluid flow.

Phase-ordering kinetics in the Allen-Cahn (Model A) class: Universal aspects elucidated by electrically induced transition in liquid crystals

The two-dimensional (2D) Ising model is the statistical physics textbook example for phase transitions and their kinetics. Quenched through the Curie point with Glauber rates, the late-time description of the ferromagnetic domain coarsening finds its place at the scalar sector of the Allen-Cahn (or Model A) class, which encompasses phase-ordering kinetics endowed with a nonconserved order parameter. Resisting exact results sought for theoreticians since Lifshitz's first account in 1962, the central quantities of 2D Model A-most scaling exponents and correlation functions-remain known up to approximate theories whose disparate outcomes urge experimental assessment. Here we perform such assessment based on a comprehensive study of the coarsening of 2D twisted nematic liquid crystals whose kinetics is induced by a superfast electrical switching from a spatiotemporally chaotic (disordered) state to a two-phase concurrent, equilibrium one. Tracking the dynamics via optical microscopy, we first show the sharp evidence of well-established Model A aspects, such as the dynamic exponent z=2 and the dynamic scaling hypothesis, to then move forward. We confirm the Bray-Humayun theory for Porod's regime describing intradomain length scales of the two-point spatial correlators and show that their nontrivial decay beyond the Porod's scale can be captured in a free-from-parameter fashion by Gaussian theories, namely the Ohta-Jasnow-Kawasaki (OJK) and Mazenko theories. Regarding time-related statistics, we corroborate the aging hypothesis in Model A systems, which includes the collapse of two-time correlators into a master curve whose format is, actually, best accounted for by a solution of the local scaling invariance theory: the same solution that fits the 2D nonconserved Ising model correlator along with the Fisher-Huse conjecture. We also suggest the true value for the local persistence exponent in Model A class, in disfavor of the exact outcome for the diffusion and OJK equations. Finally, we observe a fractal morphology for persistence clusters and extract their universal dimension. Given its accuracy and possibilities, this experimental setup may work as a prototype to address further universality issues in the realm of nonequilibrium systems.

The Study of Usefulness of a Set of Fractal Parameters to Build Classes of Disease Units Based on Images of Pigmented Skin Lesions

Dermatoscopic images are also increasingly used to train artificial neural networks for the future to provide fully automatic diagnostic systems capable of determining the type of pigmented skin lesion. Therefore, fractal analysis was used in this study to measure the irregularity of pigmented skin lesion surfaces. This paper presents selected results from individual stages of preliminary processing of the dermatoscopic image on pigmented skin lesion, in which fractal analysis was used and referred to the effectiveness of classification by fuzzy or statistical methods. Classification of the first unsupervised stage was performed using the method of analysis of scatter graphs and the fuzzy method using the Kohonen network. The results of the Kohonen network learning process with an input vector consisting of eight elements prove that neuronal activation requires a larger learning set with greater differentiation. For the same training conditions, the final results are at a higher level and can be classified as weaker. Statistics of factor analysis were proposed, allowing for the reduction in variables, and the directions of further studies were indicated.

Fractal lacunarity of trabecular bone in vertebral MRI to predict osteoporotic fracture risk in over-fifties women. The LOTO study

BACKGROUND

Osteoporotic fractures are a major cause of morbidity in the elderly. Menopausal women represent the population with the highest risk of early osteoporosis onset, often accompanied by vertebral fractures (VF). Bone mineral density (BMD) is commonly assessed by dual-energy X-ray absorptiometry (DXA) for osteoporosis diagnosis; however, BMD alone does not represent a significant predictor of fracture risk. Bone microarchitecture, instead, arises as a determinant of bone fragility independent of BMD. High-resolution magnetic resonance imaging (MRI) is an effective noninvasive/nonionizing tool for in vivo characterisation of trabecular bone microarchitecture (TBA). We have previously set up an MRI method able to characterise TBA changes in aging and osteoporosis by one parameter, trabecular bone lacunarity parameter β (TBLβ). Fractal lacunarity was used for TBA texture analysis as it describes discontinuity of bone network and size of bone marrow spaces, changes of which increase the risk of bone fracture. This study aims to assess the potential of TBLβ method as a tool for osteoporotic fracture risk.

METHODS

An observational, cross-sectional, and prospective study on over-50s women at risk for VF was designed. TBLβ, our index of osteoporotic fracture risk, is the main outcome measure. It was calculated on lumbar vertebra axial images, acquired by 1.5 T MRI spin-echo technique, from 279 osteopenic/osteoporotic women with/without prior VF. Diagnostic power of TBLβ method, by Receiver Operating Characteristics (ROC) curve and other diagnostic accuracy measurements were compared with lumbar spine DXA-BMD.

RESULTS

Baseline results show that TBLβ is able to discriminate patients with/without prevalent VF (p = 0.003). AUC (area under the curve from ROC) is 0.63 for TBLβ, statistically higher (p = 0.012) than BMD one (0.53). Contribution of TBLβ to prevalent VF is statistically higher (p < 0.001) than BMD (sensitivity: 66% vs. 52% respectively; OR: 3.20, p < 0.0001 for TBLβ vs. 1.31, p = 0.297 for BMD). Preliminary 1-year prospective results suggest that TBA contribution to incident VF is even higher (sensitivity: 73% for TBLβ vs. 55% for BMD; RR: 3.00, p = 0.002 for TBLβ vs. 1.31, p = 0.380 for BMD).

CONCLUSION

Results from this study further highlight the usefulness of TBLβ as a biomarker of TBA degeneration and an index of osteoporotic fracture risk.

Understanding the surface condition of gamma irradiated epoxy alumina nanocomposites adopting wavelets and fractal technique

The influence of alumina nanofiller and gamma irradiation on the surface potential variation of epoxy-alumina nanocomposites was investigated. The surface potential decay rate of nanocomposites has increased and the trap depth decreased with alumina nanoparticles addition to the matrix as well as upon exposure to gamma irradiation, Surface roughness was estimated using the wavelets and fractal technique. Daubechies wavelet of order 4 (db4) wavelet was chosen as the most suitable mother wavelet for surface roughness measurement. Multi resolution signal decomposition (MRSD) analysis of surface profile has revealed that with increasing wt% of alumina nanofiller in the nanocomposites, reduction in surface roughness of nanocomposites was observed. Upon gamma irradiation, the surface roughness factor at each level of MRSD has increased marginally. Fractal dimension and lacunarity were calculated for unaged and gamma ray irradiated samples and it exhibits inverse correlation.

Color Image Complexity versus Over-Segmentation: A Preliminary Study on the Correlation between Complexity Measures and Number of Segments

It is said that image segmentation is a very difficult or complex task. First of all, we emphasize the subtle difference between the notions of difficulty and complexity. Then, in this article, we focus on the question of how two widely used color image complexity measures correlate with the number of segments resulting in over-segmentation. We study the evolution of both the image complexity measures and number of segments as the image complexity is gradually decreased by means of low-pass filtering. In this way, we tackle the possibility of predicting the difficulty of color image segmentation based on image complexity measures. We analyze the complexity of images from the point of view of color entropy and color fractal dimension and for color fractal images and the Berkeley data set we correlate these two metrics with the segmentation results, more specifically the number of quasi-flat zones and the number of JSEG regions in the resulting segmentation map. We report on our experimental results and draw conclusions.

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.

Multifractal-based lacunarity analysis of trabecular bone in radiography

This study presents textural characterization techniques for effective osteoporosis diagnosis using bone radiograph images. The automatic classification of osteoporosis and healthy (control) cases using bone radiograph images in this work presents a major challenge as the images show no visual differences for both cases. The proposed work utilizes multifractals to characterize the trabecular bone texture in the radiographs. Initially, Holder exponents are computed, then Hausdorff dimensions are determined, which quantify the global regularity of the pixels. Finally, lacunarity is computed from the Hausdorff dimensions. Performance metrics show that estimating lacunarity from the Hausdorff dimensions, rather than the input image, directly helps in achieving better textural characterization of bone radiographs, leading to better performance in osteoporosis classification. The proposed lacunarity-based trabecular bone textural characterization method is compared with other multifractal-based methods for trabecular bone textural characterization, such as box-counting and regularization dimensions. The proposed method is also evaluated with the textural characterization of a bone radiograph challenge dataset to demonstrate its effectiveness compared to the other methods used in the challenge.

Pore Structure of Grain-Size Fractal Granular Material

Numerous studies have proven that natural particle-packed granular materials, such as soil and rock, are consistent with the grain-size fractal rule. The majority of existing studies have regarded these materials as ideal fractal structures, while few have viewed them as particle-packed materials to study the pore structure. In this study, theoretical analysis, the discrete element method, and digital image processing were used to explore the general rules of the pore structures of grain-size fractal granular materials. The relationship between the porosity and grain-size fractal dimension was determined based on bi-dispersed packing and the geometric packing theory. The pore structure of the grain-size fractal granular material was proven to differ from the ideal fractal structure, such as the Menger sponge. The empirical relationships among the box-counting dimension, lacunarity, succolarity, grain-size fractal dimension, and porosity were provided. A new segmentation method for the pore structure was proposed. Moreover, a general function of the pore size distribution was developed based on the segmentation results, which was verified by the soil-water characteristic curves from the experimental database.

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.

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.

Breast Cancer Detection in Thermal Infrared Images Using Representation Learning and Texture Analysis Methods

Nowadays, breast cancer is one of the most common cancers diagnosed in women. Mammography is the standard screening imaging technique for the early detection of breast cancer. However, thermal infrared images (thermographies) can be used to reveal lesions in dense breasts. In these images, the temperature of the regions that contain tumors is warmer than the normal tissue. To detect that difference in temperature between normal and cancerous regions, a dynamic thermography procedure uses thermal infrared cameras to generate infrared images at fixed time steps, obtaining a sequence of infrared images. In this paper, we propose a novel method to model the changes on temperatures in normal and abnormal breasts using a representation learning technique called learning-to-rank and texture analysis methods. The proposed method generates a compact representation for the infrared images of each sequence, which is then exploited to differentiate between normal and cancerous cases. Our method produced competitive (AUC = 0.989) results when compared to other studies in the literature.

Autonomous illumination control for localization microscopy

Super-resolution fluorescence microscopy improves spatial resolution, but this comes at a loss of image throughput and presents unique challenges in identifying optimal acquisition parameters. Microscope automation routines can offset these drawbacks, but thus far have required user inputs that presume a priori knowledge about the sample. Here, we develop a flexible illumination control system for localization microscopy comprised of two interacting components that require no sample-specific inputs: a self-tuning controller and a deep learning-based molecule density estimator that is accurate over an extended range of densities. This system obviates the need to fine-tune parameters and enables robust, autonomous illumination control for localization microscopy.

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.

Fractal properties of biophysical models of pericellular brushes can be used to differentiate between cancerous and normal cervical epithelial cells

Fractal behavior is found on the topographies of pericellular brushes on the surfaces of model healthy and cancerous cells, using dissipative particle dynamics models and simulations. The influence of brush composition, chain stiffness and solvent quality on the fractal dimension is studied in detail. Since fractal dimension alone cannot guarantee that the brushes possess fractal properties, their lacunarity was obtained also, which is a measure of the space filling capability of fractal objects. Soft polydisperse brushes are found to have larger fractal dimension than soft monodisperse ones, under poor solvent conditions, in agreement with recent experiments on dried cancerous and healthy human cervical epithelial cells. Additionally, we find that image resolution is critical for the accurate assessment of differences between images from different cells. The images of the brushes on healthy model cells are found to be more textured than those of brushes on model cancerous cells, as indicated by the larger lacunarity of the former. These findings are helpful to distinguish monofractal behavior from multifractality, which has been found to be useful to discriminate between immortal, cancerous and normal cells in recent experiments.

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.

Effect of Repetition Rate on Femtosecond Laser-Induced Homogenous Microstructures

We report on the effect of repetition rate on the formation and surface texture of the laser induced homogenous microstructures. Different microstructures were micromachined on copper (Cu) and titanium (Ti) using femtosecond pulses at 1 and 10 kHz. We studied the effect of the repetition rate on structure formation by comparing the threshold accumulated pulse (FΣpulse) values and the effect on the surface texture through lacunarity analysis. Machining both metals at low FΣpulse resulted in microstructures with higher lacunarity at 10 kHz compared to 1 kHz. On increasing FΣpulse, the microstructures showed higher lacunarity at 1 kHz. The effect of the repetition rate on the threshold FΣpulse values were, however, considerably different on the two metals. With an increase in repetition rate, we observed a decrease in the threshold FΣpulse on Cu, while on Ti we observed an increase. These differences were successfully allied to the respective material characteristics and the resulting melt dynamics. While machining Ti at 10 kHz, the melt layer induced by one laser pulse persists until the next pulse arrives, acting as a dielectric for the subsequent pulse, thereby increasing FΣpulse. However, on Cu, the melt layer quickly resolidifies and no such dielectric like phase is observed. Our study contributes to the current knowledge on the effect of the repetition rate as an irradiation parameter.

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.

Quantification of antiangiogenic treatment effects on tissue heterogeneity in glioma tumour xenograft model using a combination of DCE-MRI and 3D-ultramicroscopy

OBJECTIVES

This study aimed at assessing the effects of an anti-angiogenic treatment, which neutralises vascular endothelial growth factor (VEGF), on tumour heterogeneity.

METHODS

Murine glioma cells have been inoculated into the right brain frontal lobe of 16 mice. Anti-VEGF antibody was administered to a first group (n = 8), while a second group (n = 8) received a placebo. Magnetic resonance acquisitions, performed at days 10, 12, 15 and 23 following the implantation, allowed the derivation of a three-dimensional features dataset characterising tumour heterogeneity. Three-dimensional ultramicroscopy and standard histochemistry analysis have been performed to verify in vivo results.

RESULTS

Placebo-treated mice displayed a highly-vascularised area at the tumour periphery, a monolithic necrotic core and a chaotic dense vasculature across the entire tumour. In contrast, the B20-treated group did not show any highly vascularised regions and presents a fragmented necrotic core. A significant reduction of the number of vessel segments smaller than 17 μm has been observed. There was no difference in overall tumour volume and growth rate between the two groups.

CONCLUSIONS

Region-specific analysis revealed that VEGF inhibition affects only: (1) highly angiogenic compartments expressing high levels of VEGF and characterised by small capillaries, and also (2) the formation and structure of necrotic regions. These effects appear to be transient and limited in time.

KEY POINTS

• VEGF inhibition affects only the highly angiogenic region and small capillaries network • VEGF inhibition is transient in time • Tumour volume is not affected by anti-angiogenic treatment • VEGF inhibition also influences the architecture of necrotic regions.

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.

Detecting Spatiotemporal Dynamic Landscape Patterns Using Remote Sensing and the Lacunarity Index: A Case Study of Haikou City, China

Quantifying the landscape pattern and its dynamics is essential for the monitoring and assessment of the ecological consequences of urbanization. As one of the Special Economic Zones, Haikou is one of the fastest growing regions among all Chinese cities, owing to rapid real estate development. Using a GIS-based land-use dataset from 1986, 1996, and 2000, in combination with a lacunarity index, we attempt to quantify the spatial pattern in the Haikou metropolitan area. After the landscape structure changes over the periods 1986–96 and 1996–2000 are analyzed, a Markov conversion matrix is applied in order to study the sources and destinations of landscape dynamic changes. The lacunarity index is calculated in order to measure the landscape dynamics, with respect to several major land-use types, at a range of spatial scales. The findings indicate that the leapfrog development of real estate and the rapid economic growth of Haikou City have had a great impact on the dynamic landscape patterns. From 1986 to 1996 urban land expanded dramatically and clustered, while cropland was encroached upon and fragmented. From 1996 to 2000, after the government had implemented the strict cropland protection measures, urban expansion and cropland misuse were controlled to a large degree, and a lot of cropland was reclaimed in certain areas. We investigate the dynamic landscape pattern and process, and their implications in policy and economic development.