Maximum entropy, fractal dimension and lacunarity in quantification of cellular rejection in myocardial biopsy of patients submitted to heart transplantation

This paper presents a method for the quantification of cellular rejection in endomyocardial biopsies of patients submitted to heart transplant. The model is based on automatic multilevel thresholding, which employs histogram quantification techniques, histogram slope percentage analysis and the calculation of maximum entropy. The structures were quantified with the aid of the multi-scale fractal dimension and lacunarity for the identification of behavior patterns in myocardial cellular rejection in order to determine the most adequate treatment for each case.

Multi-scale lacunarity as an alternative to quantify and diagnose the behavior of prostate cancer

Prostate cancer is a serious public health problem accounting for up to 30% of clinical tumors in men. The diagnosis of this disease is made with clinical, laboratorial and radiological exams, which may indicate the need for transrectal biopsy. Prostate biopsies are discerningly evaluated by pathologists in an attempt to determine the most appropriate conduct. This paper presents a set of techniques for identifying and quantifying regions of interest in prostatic images. Analyses were performed using multi-scale lacunarity and distinct classification methods: decision tree, support vector machine and polynomial classifier. The performance evaluation measures were based on area under the receiver operating characteristic curve (AUC). The most appropriate region for distinguishing the different tissues (normal, hyperplastic and neoplasic) was defined: the corresponding lacunarity values and a rule's model were obtained considering combinations commonly explored by specialists in clinical practice. The best discriminative values (AUC) were 0.906, 0.891 and 0.859 between neoplasic versus normal, neoplasic versus hyperplastic and hyperplastic versus normal groups, respectively. The proposed protocol offers the advantage of making the findings comprehensible to pathologists. (C) 2014 Elsevier Ltd. All rights reserved.

Lacunarity of the Spatial Distributions of Soil Types in Europe.

Lacunarity as a means of quantifying textural properties of spatial distributions suggests a classification into three main classes of the most abundant soils that cover 92% of Europe. Soils with a well-defined self-similar structure of the linear class are related to widespread spatial patterns that are nondominant but ubiquitous at continental scale. Fractal techniques have been increasingly and successfully applied to identify and describe spatial patterns in natural sciences. However, objects with the same fractal dimension can show very different optical properties because of their spatial arrangement. This work focuses primary attention on the geometrical structure of the geographical patterns of soils in Europe. We made use of the European Soil Database to estimate lacunarity indexes of the most abundant soils that cover 92% of the surface of Europe and investigated textural properties of their spatial distribution. We observed three main classes corresponding to three different patterns that displayed the graphs of lacunarity functions, that is, linear, convex, and mixed. They correspond respectively to homogeneous or self-similar, heterogeneous or clustered and those in which behavior can change at different ranges of scales. Finally, we discuss the pedological implications of that classification.

鄂尔多斯高原沙地草地的生态异质性

空间异质性的重要性在于其在自然生态系统中的普遍性和它所引起的生态学效应。空间异质性不仅在生态学理论中占有重要位置，而且也是生态系统管理和景观规划等实践中需要解决的基本问题。本文对鄂尔多斯高原(37038，～40052'N，106027'～111028'E)沙地草地的主要景观单元一沙地和风蚀沙化梁地植被与土壤的空间异质性格局与动态进行了定量的描述和实验研究。 在鄂尔多斯高原沙地景观中，通过对两条相互垂直的Skm样线上植被盖度、植物物种数的半方差分析和分形分析，揭示了植被的依赖于空间尺度的自相关特征和等级结构;沙地景观植被结构以小尺度空间变异占优势，植被斑块多集中在5-10 m的尺度上。对植被盖度低于20%的流沙斑点或斑块沿样线分布的空隙度分析表明流沙斑点或斑块呈适度聚集的格局。植被盖度和植物物种数与土壤有机碳和全氮含量的空间异质性之间的关系随空间尺度和生境类型的不同而有所不同。对植物群落内风蚀沙化斑块中的植物再生动态和克隆植物沙鞭的克隆生长观测实验表明，克隆植物的存在有利于风蚀沙化斑块的固定，跨越不同斑块边界的克隆植物的生长导致了斑块间的物质能量流动，从而影响了景观空间异质性的动态。 在风蚀沙化梁地中，梁坡处的土壤最为瘠薄，土壤颗粒组成以中细沙为主，这反映了风沙活动在梁坡处形成的沙粒堆积。相应地，梁坡处是以油蒿为优势种的沙生植物群落，与梁顶的典型草原植物（本氏针茅（Stipa bungeana)+百里香（Thymus serpyllum）+冷蒿（Artemisiafrigida））群落截然不同。从梁顶到梁底，植物群落受控于较大尺度上的与地形有关的环境梯度，因而其在较大尺度上的空间异质性占优势。

Fractal dimension, wavelet shrinkage and anomaly detection for mine hunting

An anomaly detection approach is considered for the mine hunting in sonar imagery problem. The authors exploit previous work that used dual-tree wavelets and fractal dimension to adaptively suppress sand ripples and a matched filter as an initial detector. Here, lacunarity inspired features are extracted from the remaining false positives, again using dual-tree wavelets. A one-class support vector machine is then used to learn a decision boundary, based only on these false positives. The approach exploits the large quantities of 'normal' natural background data available but avoids the difficult requirement of collecting examples of targets in order to train a classifier. © 2012 The Institution of Engineering and Technology.

Scale-wise evolution of rainfall probability density functions fingerprints the rainfall generation mechanism

© 2010 by the American Geophysical Union.The cross-scale probabilistic structure of rainfall intensity records collected over time scales ranging from hours to decades at sites dominated by both convective and frontal systems is investigated. Across these sites, intermittency build-up from slow to fast time-scales is analyzed in terms of heavy tailed and asymmetric signatures in the scale-wise evolution of rainfall probability density functions (pdfs). The analysis demonstrates that rainfall records dominated by convective storms develop heavier-Tailed power law pdfs toward finer scales when compared with their frontal systems counterpart. Also, a concomitant marked asymmetry build-up emerges at such finer time scales. A scale-dependent probabilistic description of such fat tails and asymmetry appearance is proposed based on a modified q-Gaussian model, able to describe the cross-scale rainfall pdfs in terms of the nonextensivity parameter q, a lacunarity (intermittency) correction and a tail asymmetry coefficient, linked to the rainfall generation mechanism.

Fractal species distributions do not produce power-law species-area relationships

We derive the species-area relationship (SAR) expected from an assemblage of fractally distributed species. If species have truly fractal spatial distributions with different fractal dimensions, we show that the expected SAR is not the classical power-law function, as suggested recently in the literature. This analytically derived SAR has a distinctive shape that is not commonly observed in nature: upward-accelerating richness with increasing area (when plotted on log-log axes). This suggests that, in reality, most species depart from true fractal spatial structure. We demonstrate the fitting of a fractal SAR using two plant assemblages (Alaskan trees and British grasses). We show that in both cases, when modelled as fractal patterns, the modelled SAR departs from the observed SAR in the same way, in accord with the theory developed here. The challenge is to identify how species depart from fractality, either individually or within assemblages, and more importantly to suggest reasons why species distributions are not self-similar and what, if anything, this can tell us about the spatial processes involved in their generation.

Gorgonians of the South of Portugal: biology, ecology and conservation

O papel ecológico das gorgónias (Octocorallia: Alcyonacea) nos fundos marinhos rochosos é mundialmente reconhecido. Contudo, a informação acerca da ecologia e biologia das espécies de gorgónias nas zonas temperadas do NE Atlântico é manifestamente escassa, especialmente tendo em consideração as actuais perturbações globais, regionais e locais. Nos fundos rochosos da costa algarvia até aos 30 m, verificouse que várias espécies de gorgónias são abundantes e frequentes, nomeadamente Eunicella labiata, Eunicella gazella, Eunicella verrucosa, Leptogorgia lusitanica e Leptogorgia sarmentosa. As populações de gorgónias são co-dominadas por diferentes espécies que apresentaram elevados índices de associação, indicando reduzidos níveis de competição entre elas. Em todo o caso, a estrutura dos povoamentos diferiu com as condições locais. Todas as espécies evidenciaram padrões de distribuição semelhantes ao longo do gradiente de profundidade, i.e. a abundância aumenta significamente com a profundidade após os 15 m. A profundidades mais baixas (até aos 15 m), a distribuição das gorgónias parece ser condicionada por factores abióticos e pela competição com algas. Com efeito, os padrões de distribuição espacial das espécies de gorgónias na costa algarvia são determinados pela interacção de pressões naturais e antropogénicas (ex. pesca). Ainda que as colónias de maior tamanho não tenham sido restritas a áreas menos pescadas, em áreas mais perturbadas pela pesca, a distribuição dos tamanhos das colónias estava maioritariamente desviada para tamanhos mais pequenos. Os efeitos das perturbações naturais nas populações de gorgónias foram evidenciados pela ocorrência de padrões demográficos distintos em áreas vizinhas sujeitas a níveis semelhantes de pressões antropogénicas. Estes estudos demonstraram, ainda, que os efeitos na distribuição de frequências de tamanho das colónias são dependentes das espécies de gorgónias em causa: Eunicella labiata não parece ser afectada; Leptogorgia sarmentosa é tendencialmente afectada por pressões antropogénicas; Eunicella gazella e Leptogorgia lusitanica aparentam ser afectadas, quer por pressões naturais, quer por pressões antropogénicas. Os efeitos verificados nos padrões da distribuição de frequências de tamanho, particularmente a tendência para o desvio destas frequências para tamanhos mais pequenos em áreas sujeitas a perturbações, poderão ter consequências para a biodiversidade dos fundos sublitorais rochosos na costa algarvia. Com efeito, o presente estudo apoia o paradigma geral de que os corais são habitats que suportam comunidades de elevada biodiversidade e abundância. Num dos poucos estudos que examinam a relação entre as gorgónias e as suas comunidades de invertebrados epibentónicos, foi verificado que as gorgónias (Eunicella gazella e Leptogorgia lusitanica) sustentam comunidades ricas (11 phyla, 181 taxa) e abundantes (7284 indivíduos). Estas comunidades são dominadas por anfípodes, mas os poliquetas tiveram um grande contributo para os níveis elevados de biodiversidade. Verificou-se, igualmente, que o tamanho da colónia desempenha um papel fundamental na biodiversidade, na medida em que as colónias de menor tamanho apresentaram um contributo mais baixo, comparativamente às médias e grandes. Ainda que ambas as gorgónias partilhem a maioria das espécies amostradas, 11 e 18 taxa foram exclusivos de Eunicella gazella e Leptogorgia lusitanica, respectivamente (excluindo indivíduos com presenças únicas). No entanto, a maioria destes taxa eram ou pouco abundantes ou pouco frequentes. A excepção foi a presença de planárias (Turbellaria) de coloração branca nas colónias de Eunicella gazella, provavelmente beneficiando do efeito de camuflagem proporcionado pelos ramos com a mesma coloração. Com efeito, a complementaridade entre as comunidades epibentónicas associadas a ambas as gorgónias diminuiu quando usados os dados de presença/ausência, sugerindo que os padrões de biodiversidade são mais afectados pelas alterações na abundância relativa das espécies dominantes do que pela composição faunística. As comunidades de epifauna bentónica associadas a estas gorgónias não só apresentaram valores elevados de ®-diversidade, como de ¯- diversidade, resultantes de padrões intrincados de variabilidade na sua composição e estrutura. Ainda que o conjunto de espécies disponíveis para colonização seja, na generalidade, o mesmo para ambos os locais, cada colónia apresenta uma parte deste conjunto. Na sua totalidade, as colónias de gorgónias poderão funcionar como uma metacomunidade, mas a estrutura das comunidades associadas a cada colónia (ex. número total de espécies e abundância) parecem depender dos atributos da colónia, nomeadamente superfície disponível para colonização (altura, largura e área), complexidade e heterogeneidade (dimensão fractal e lacunaridade, respectivamente) e cobertura epibentónica “colonial” (ex. fauna colonial e algas macroscópicas; CEC). Numa primeira tentativa para quantificar a relação entre as gorgónias e os invertebrados epibentónicos a elas associados (em termos de abundância e riqueza específica), verificou-se que a natureza e a intensidade destas relações dependem da espécie hospedeira e variam para os grupos taxonómicos principais. No entanto, independentemente do grupo taxonómico, a riqueza específica e a abundância estão significativamente correlacionadas com a CEC. Com efeito, a CEC provavelmente devido a um efeito trófico (aumento da disponibilidade alimentar directo ou indirecto), combinado com a superfície disponível para colonização (efeito espécies-área) foram as variáveis mais relacionadas com os padrões de abundância e riqueza específica. Por outro lado, ainda que a complexidade estrutural seja frequentemente indicada como um dos factores responsáveis pela elevada diversidade e abundância das comunidades bentónicas associadas a corais, a dimensão fractal e a lacunaridade apenas foram relevantes nas comunidades associadas a Leptogorgia lusitanica. A validade do paradigma que defende que a complexidade estrutural promove a biodiversidade poderá ser, então, dependente da escala a que se realizam os estudos. No caso das gorgónias, o efeito da complexidade ao nível dos agregados de gorgónias poderá ser muito mais relevante do que ao nível da colónia individual, reforçando a importância da sua conservação como um todo, por forma a preservar a diversidade de espécies hospedeiras, o seu tamanho e estrutura. Actividades antropogénicas como a pesca, podem, ainda, ter efeitos negativos ao nível da reprodução de espécies marinhas. Analogamente ao verificado para os padrões de distribuição espacial das populações de gorgónias na costa algarvia, a informação relativa à sua reprodução é igualmente escassa. Os estudos realizados em populações de Eunicella gazella a 16m de profundidade, demonstraram que o desenvolvimento anual das estruturas reprodutivas é altamente sincronizado entre os sexos. A razão entre sexos na população foi de 1.09 (F:M), encontrando-se perto da paridade. A espermatogénese estende-se por 6 a 8 meses, enquanto que a oogénese é mais demorada, levando mais de um ano para que os oócitos se desenvolvam até estarem maduros. Antes da libertação dos gâmetas, foi observada uma elevada fecundidade nas fêmeas (27.30§13.24 oócitos pólipo−1) e nos machos (49.30§31.14 sacos espermáticos pólipo−1). Estes valores encontram-se entre os mais elevados reportados à data para zonas temperadas. A libertação dos gâmetas (não há evidência de desenvolvimento larvar, nem à superfície da colónia, nem no seu interior) occorre em Setembro/ Outubro, após um período de elevada temperatura da água do mar. As fêmeas emitem oócitos maduros de elevadas dimensões, retendo, todavia, os oócitos imaturos que se desenvolvem apenas na época seguinte. Ainda que o efeito da pesca nas populações de gorgónias da costa do Algarve seja perceptível, às taxas actuais, o mergulho recreativo não aparenta afectar seriamente estas populações. Contudo, sendo uma indústria em expansão e conhecendo-se a preferência de mergulhadores por áreas rochosas naturais ricas em espécies bentónicas, futuramente poderá vir a afectar estes habitats. A monitorização de mergulhadores na costa algarvia mostrou que a sua maioria (88.6 %) apresenta comportamentos que podem impactar o habitat, com uma taxa média de contactos de 0.340§0.028 contactos min−1. Esta taxa foi mais elevada em mergulhadores com moderada experiência e na fase inicial do mergulho (0–10 min). Os contactos com as barbatanas e mãos foram comuns, resultando, maioritariamente, na resuspensão do sedimento, mas geralmente apresentando um impacto reduzido. Todavia, a fauna também foi afectada, quer por danos físicos, quer pela interacção com os mergulhadores, e num cenário de expansão significativa desta actividade, os impactos na fauna local poderão aumentar, com consequências para os ecossistemas de fundos rochosos da costa sul de Portugal. Na sua globalidade, a informação recolhida nos estudos que contemplam esta tese, por ser em grande parte totalmente nova para a região, espera-se que contribua para a gestão da zona costeira do Algarve.

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

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

Multiscale Fractal Descriptors and Polynomial Classifier for Partial Pixels Identification in Regions of Interest of Mammographic Images

Computer systems are used to support breast cancer diagnosis, with decisions taken from measurements carried out in regions of interest (ROIs). We show that support decisions obtained from square or rectangular ROIs can to include background regions with different behavior of healthy or diseased tissues. In this study, the background regions were identified as Partial Pixels (PP), obtained with a multilevel method of segmentation based on maximum entropy. The behaviors of healthy, diseased and partial tissues were quantified by fractal dimension and multiscale lacunarity, calculated through signatures of textures. The separability of groups was achieved using a polynomial classifier. The polynomials have powerful approximation properties as classifiers to treat characteristics linearly separable or not. This proposed method allowed quantifying the ROIs investigated and demonstrated that different behaviors are obtained, with distinctions of 90% for images obtained in the Cranio-caudal (CC) and Mediolateral Oblique (MLO) views.

Aprendizado de máquina simbólico e técnicas fractais para caracterizar rejeição em biópsia miocárdica

This work combines symbolic machine learning and multiscale fractal techniques to generate models that characterize cellular rejection in myocardial biopsies and that can base a diagnosis support system. The models express the knowledge by the features threshold, fractal dimension, lacunarity, number of clusters, spatial percolation and percolation probability, all obtained with myocardial biopsies processing. Models were evaluated and the most significant was the one generated by the C4.5 algorithm for the features spatial percolation and number of clusters. The result is relevant and contributes to the specialized literature since it determines a standard diagnosis protocol. © 2013 Springer.

Automatic method to classify images based on multiscale fractal descriptors and paraconsistent logic

In this study is presented an automatic method to classify images from fractal descriptors as decision rules, such as multiscale fractal dimension and lacunarity. The proposed methodology was divided in three steps: quantification of the regions of interest with fractal dimension and lacunarity, techniques under a multiscale approach; definition of reference patterns, which are the limits of each studied group; and, classification of each group, considering the combination of the reference patterns with signals maximization (an approach commonly considered in paraconsistent logic). The proposed method was used to classify histological prostatic images, aiming the diagnostic of prostate cancer. The accuracy levels were important, overcoming those obtained with Support Vector Machine (SVM) and Bestfirst Decicion Tree (BFTree) classifiers. The proposed approach allows recognize and classify patterns, offering the advantage of giving comprehensive results to the specialists.

Toward Multi-Scale Object-Based Data Fusion

This paper proposes a new methodology for object based 2-D data fu- sion, with a multiscale character. This methodology is intended to be use in agriculture, specifically in the characterization of the water status of different crops, so as to have an appropriate water management at a farm-holding scale. As a first approach to its evaluation, vegetation cover vigor data has been integrated with texture data. For this purpose, NDVI maps have been calculated using a multispectral image and Lacunarity maps from the panchromatic image. Preliminary results show this methodology is viable in the integration and management of large volumes of data, which characterize the behavior of agricultural covers at farm-holding scale.

Fractal scaling of apparent soil moisture estimated from vertical planes of Vertisol pit images

Image analysis could be a useful tool for investigating the spatial patterns of apparent soil moisture at multiple resolutions. The objectives of the present work were (i) to define apparent soil moisture patterns from vertical planes of Vertisol pit images and (ii) to describe the scaling of apparent soil moisture distribution using fractal parameters. Twelve soil pits (0.70 m long × 0.60 m width × 0.30 m depth) were excavated on a bare Mazic Pellic Vertisol. Six of them were excavated in April/2011 and six pits were established in May/2011 after 3 days of a moderate rainfall event. Digital photographs were taken from each Vertisol pit using a Kodak™ digital camera. The mean image size was 1600 × 945 pixels with one physical pixel ≈373 μm of the photographed soil pit. Each soil image was analyzed using two fractal scaling exponents, box counting (capacity) dimension (DBC) and interface fractal dimension (Di), and three prefractal scaling coefficients, the total number of boxes intercepting the foreground pattern at a unit scale (A), fractal lacunarity at the unit scale (Λ1) and Shannon entropy at the unit scale (S1). All the scaling parameters identified significant differences between both sets of spatial patterns. Fractal lacunarity was the best discriminator between apparent soil moisture patterns. Soil image interpretation with fractal exponents and prefractal coefficients can be incorporated within a site-specific agriculture toolbox. While fractal exponents convey information on space filling characteristics of the pattern, prefractal coefficients represent the investigated soil property as seen through a higher resolution microscope. In spite of some computational and practical limitations, image analysis of apparent soil moisture patterns could be used in connection with traditional soil moisture sampling, which always renders punctual estimates