Enhancing Volumetric Bouligand-Minkowski Fractal Descriptors by using Functional Data Analysis

This work proposes and study the concept of Functional Data Analysis transform, applying it to the performance improving of volumetric Bouligand-Minkowski fractal descriptors. The proposed transform consists essentially in changing the descriptors originally defined in the space of the calculus of fractal dimension into the space of coefficients used in the functional data representation of these descriptors. The transformed decriptors are used here in texture classification problems. The enhancement provided by the FDA transform is measured by comparing the transformed to the original descriptors in terms of the correctness rate in the classification of well known datasets

Multiscale Fractal Descriptors Applied to Nanoscale Images

This work proposes the application of fractal descriptors to the analysis of nanoscale materials under different experimental conditions. We obtain descriptors for images from the sample applying a multiscale transform to the calculation of fractal dimension of a surface map of such image. Particularly, we have used the}Bouligand-Minkowski fractal dimension. We applied these descriptors to discriminate between two titanium oxide films prepared under different experimental conditions. Results demonstrate the discrimination power of proposed descriptors in such kind of application

Characterization of nanostructured material images using fractal descriptors

This work presents a methodology to the morphology analysis and characterization of nanostructured material images acquired from FEG-SEM (Field Emission Gun-Scanning Electron Microscopy) technique. The metrics were extracted from the image texture (mathematical surface) by the volumetric fractal descriptors, a methodology based on the Bouligand-Minkowski fractal dimension, which considers the properties of the Minkowski dilation of the surface points. An experiment with galvanostatic anodic titanium oxide samples prepared in oxalyc acid solution using different conditions of applied current, oxalyc acid concentration and solution temperature was performed. The results demonstrate that the approach is capable of characterizing complex morphology characteristics such as those present in the anodic titanium oxide.Comment: 8 pages, 5 figures, accepted for publication Physica

The role of microhabitats within mangroves: an invertebrate and fish larval perspective

Microhabitats provided through structural complexity are central for the diversity, productivity, connectivity and niche differentiation within and among ecosystems. Mangrove forests afford juvenile fish and invertebrates with nursery and recruitment habitats, facilitated by the fine scale configuration of their specialised root systems. Although the importance of mangroves for resident and transient juveniles is well recognised, the roles that mangrove microhabitats play for larvae is not yet comprehensively understood. This study aimed to determine how microhabitats with varying degrees of complexity influence the composition, abundance and distribution of larval communities that inhabit mangrove forests and the physiological responses of larvae to acute temperature variations in relation to ontogenetic stage and microenvironment exposure. Two relatively pristine study sites were selected to represent a warm temperate and subtropical mangrove system in the Eastern Cape and KwaZulu-Natal on the east coast of South Africa, respectively. The differences in complexity among the root systems of Rhizophora mucronata, Avicennia marina and Bruguiera gymnorhiza were assessed using 3D scanning and the computed 3D models were then analysed using four complexity metrics. Results indicated that A. marina is the most complex in terms of surface-volume ratio, R. mucronata has the most interstitial space among its roots and B. gymnorhiza and R. mucronata differ in their fractal dimensions. Larvae collected in each microhabitat at each site using light traps showed that, despite temperature and salinity homogeneity across microenvironments, spatio-temporal differences occurred in both fish and invertebrate assemblages. This trend suggests that microhabitat structural complexity exerts an influence on larval community composition by acting as a microscape of available habitat, which ensures ecological linkages within and among the mangrove forest and adjacent ecosystems. In addition, the oxygen consumption rates of mangrove-associated brachyuran larvae varied according to mangrove microhabitat, whereby larvae collected at less complex environments had the highest metabolic rates at increased temperatures. Moreover, ontogenetic shifts in physiology were prevalent as older brachyuran larvae were more eurythermal than earlier stages, suggesting that thermally stressful events will have a greater impact on recently spawned larvae. Overall, the interstitial spaces within individual root systems are the most important complexity measure, as utilisation of these mangrove microhabitats is scale-dependent, and larvae will most likely occupy spaces inaccessible to large predators. Likewise, microscale variation in the environmental conditions and ontogenetic stage of brachyuran larvae within the mangrove microscape, can amplify the physiological responses to rapid temperature variations. Results suggest that early stage larvae are the most vulnerable to mass-mortality, and if thermally stressful events increase in frequency, duration and magnitude, the larval supply for the successful recruitment into adult populations could be under threat. Through linking how mangrove microhabitat complexity influences larvae in terms of community metrics and physiology, this study paves the way for further advancement of our understanding of how microscale processes emerge into meso- and macroscale patterns and influence the stability and functioning of highly productive ecosystems

Fast Convergence on Perfect Classification for Functional Data

In this study, we investigate the availability of approaching to perfect classification on functional data with finite samples. The seminal work (Delaigle and Hall (2012)) showed that classification on functional data is easier to define on a perfect classifier than on finite-dimensional data. This result is based on their finding that a sufficient condition for the existence of a perfect classifier, named a Delaigle--Hall (DH) condition, is only available for functional data. However, there is a danger that a large sample size is required to achieve the perfect classification even though the DH condition holds because a convergence of misclassification errors of functional data is significantly slow. Specifically, a minimax rate of the convergence of errors with functional data has a logarithm order in the sample size. This study solves this complication by proving that the DH condition also achieves fast convergence of the misclassification error in sample size. Therefore, we study a classifier with empirical risk minimization using reproducing kernel Hilbert space (RKHS) and analyse its convergence rate under the DH condition. The result shows that the convergence speed of the misclassification error by the RKHS classifier has an exponential order in sample size. Technically, the proof is based on the following points: (i) connecting the DH condition and a margin of classifiers, and (ii) handling metric entropy of functional data. Experimentally, we validate that the DH condition and the associated margin condition have a certain impact on the convergence rate of the RKHS classifier. We also find that some of the other classifiers for functional data have a similar property.Comment: 26 page

Quantitative MR Image Analysis - a Useful Tool in Differentiating Glioblastoma from Solitary Brain Metastasis

Cilj: Prikaz glioblastoma i metastaza na konvencionalnom MRI je često jako sličan, ali se terapijski pristup i prognoza bitno razlikuju. Čak i primenom naprednih MR tehnika, u nekim slučajevima dijagnoza ostaje nejasna. Glavni cilj disertacije bio je da utvrdi da li fraktalna ili teksturna, ili obe kvantitativne analize MR slike mogu doprineti diferencijaciji glioblastoma od solitarne metastaze mozga. Metod: Studija je sprovedena na ukupno 96 pacijenata sa dokazanim dijagnozama glioblastoma (50 pacijenata), odnosno solitarne metastaze (46 pacijenata). Izdvojene su slike sa najinformativnijim prikazom lezije (jedan isti presek u tri različite sekvence: CET1, T2 i SWI), a zatim je učinjena njihova kompjuterska analiza, primenom fraktalne metode brojanja kvadrata i teksturne metode bazirane na matrici zajedničke pojave istog nivoa sive boje (GLCM). Rezultati: Analizom sive skale celog tumora i binarne slike unutrašnjosti tumora sa T2 sekvence dobijen je najveći broj parametara koji značajno razlikuju dve vrste tumora (drugi ugaoni moment SASM, inverzni moment razlike SIDM, kontrast SCON, korelacija SCOR, diferencijalna fraktalna dimenzija DDIFF, odnosno binarna fraktalna dimenzija unutrašnjosti DBIN2, normirana fraktalna dimenzija DNORM, lakunarnost Ʌ2), dok su se druge dve sekvence (CET1 i SWI) pokazale manje pogodnim za kvantifikaciju. Kombinacijom parametara povećala se tačnost testiranja (AUC 0,838±0,041, senzitivnost 78% i specifičnost 76% za kombinaciju SASM i SIDM sa CET1 i T2 + SASM sa SWI + DBIN2 i DNORM sa T2). Zaključak: Kvantifikacija MR slike može doprineti diferencijalno dijagnostičkoj odluci između glioblastoma i solitarne metastaze mozga i potencijalno može postati deo svakodnevne radiološke prakse.Purpose: Presentation of glioblastomas and metastases on conventional MRI is quite similar, however treatment strategy and prognosis are substantially different. Even with advanced MR techniques, in some cases diagnostic uncertainty remains. The main objective of dissertation was to determine whether fractal, texture, or both quantitative MR image analysis could aid in differentiating glioblastoma from solitary brain metastasis. Method: Study embraced 96 patients with proven diagnosis of glioblastoma (50 patients), respectively solitary metastasis (46 patients). Images with the most representative lesion (one same slice on three different sequences: CET1, T2 and SWI) were selected, and computer analysis was done by fractal box-counting and texture gray level co-occurrence matrix (GLCM) methods. Results: Gray scale analysis of whole tumor and binary image analysis of tumor´s inner structures, both derived on T2 sequence, obtained the most significantly different parameters between two types of tumors (angular second moment SASM, inverse difference moment SIDM, contrast SCON, correlation SCOR, differential box dimension DDIFF, respectively binary box dimension DBIN2, normalized box dimension DNORM, lacunarity Ʌ2), while the other two sequences (CET1 and SWI) showed less suitable for quantification. The combinations of parameters yielded better results (AUC-0.838±0.041, sensitivity 78% and specificity 76% for next combination SASM and SIDM from CET1 and T2 + SASM from SWI + DBIN2 and DNORM from T2). Conclusions: MR image quantification may aid in differentiation between glioblastoma and solitary brain metastasis, and potentially could become a part of daily radiology practice

Early screening and diagnosis of diabetic retinopathy

Diabetic retinopathy (DR) is a chronic, progressive and possibly vision-threatening eye disease. Early detection and diagnosis of DR, prior to the development of any lesions, is paramount for more efficiently dealing with it and managing its consequences. This thesis investigates and proposes a number of candidate geometric and haemodynamic biomarkers, derived from fundus images of the retinal vasculature, which can be reliably utilised for identifying the progression from diabetes to DR. Numerous studies exist in literature that investigate only some of these biomarkers in independent normal, diabetic and DR cohorts. However, none exist, to the best of my knowledge, that investigates more than 100 biomarkers altogether, both geometric and haemodynamic ones, for identifying the progression to DR, by also using a novel experimental design, where the same exact matched junctions and subjects are evaluated in a four year period that includes the last three years pre-DR (still diabetic eye) and the onset of DR (progressors’ group). Multiple additional conventional experimental designs, such as non-matched junctions, non-progressors’ group, and a combination of them are also adopted in order to present the superiority of this type of analysis for retinal features. Therefore, this thesis aims to present a complete framework and some novel knowledge, based on statistical analysis, feature selection processes and classification models, so as to provide robust, rigorous and meaningful statistical inferences, alongside efficient feature subsets that can identify the stages of the progression. In addition, a new and improved method for more accurately summarising the calibres of the retinal vessel trunks is also presented. The first original contribution of this thesis is that a series of haemodynamic features (blood flow rate, blood flow velocity, etc.), which are estimated from the retinal vascular geometry based on some boundary conditions, are applied to studying the progression from diabetes to DR. These features are found to undoubtedly contribute to the inferences and the understanding of the progression, yielding significant results, mainly for the venular network. The second major contribution is the proposed framework and the experimental design for more accurately and efficiently studying and quantifying the vascular alterations that occur during the progression to DR and that can be safely attributed only to this progression. The combination of the framework and the experimental design lead to more sound and concrete inferences, providing a set of features, such as the central retinal artery and vein equivalent, fractal dimension, blood flow rate, etc., that are indeed biomarkers of progression to DR. The third major contribution of this work is the new and improved method for more accurately summarising the calibre of an arterial or venular trunk, with a direct application to estimating the central retinal artery equivalent (CRAE), the central retinal vein equivalent (CRVE) and their quotient, the arteriovenous ratio (AVR). Finally, the improved method is shown to truly make a notable difference in the estimations, when compared to the established alternative method in literature, with an improvement between 0.24% and 0.49% in terms of the mean absolute percentage error and 0.013 in the area under the curve. I have demonstrated that some thoroughly planned experimental studies based on a comprehensive framework, which combines image processing algorithms, statistical and classification models, feature selection processes, and robust haemodynamic and geometric features, extracted from the retinal vasculature (as a whole and from specific areas of interest), provide altogether succinct evidence that the early detection of the progression from diabetes to DR can be indeed achieved. The performance that the eight different classification combinations achieved in terms of the area under the curve varied from 0.745 to 0.968

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described