Spectral Signatures for Non-rigid 3D Shape Retrieval

This thesis addresses problems associated with computing spectral shape signatures for non-rigid 3D object retrieval. More specifically, we use spectral shape analysis tools to describe the characteristics of different 3D object representations. This thesis tries to answer whether spectral shape analysis tools can enhance classical shape signatures to improve the performance of the non-rigid shape retrieval problem. Furthermore, it describes the stages of the framework for composing non-rigid shape signatures, built from the shape Laplacian. This thesis presents four methods to improve each part of the framework for computing spectral shape signatures. The first stage comprises computing the right shape spectrum to describe 3D objects. We introduce the Kinetic Laplace-Beltrami operator which computes enhanced spectral components from 3D meshes specific to non-rigid shape retrieval and we also introduce the Mesh-Free Laplace Operator which computes more precise and robust spectral components from 3D point clouds. After computing the shape spectrum, we propose the Improved Wave Kernel Signature, a more discriminative local descriptor built from the Laplacian eigenfunctions. This descriptor is used throughout this thesis and it achieves, in most cases, state-of-the-art performances. Then, we define a new framework for encoding sparse local descriptors into shape signatures that can be compared to each other. Here, we show how to use the Fisher Vector and Super Vector to encode spectral descriptors and also how to compute dissimilarities between shape signatures using the Efficient Manifold Ranking. Furthermore, we describe the construction of the Point-Cloud Shape Retrieval of Non-Rigid Toys dataset, aimed in testing non-rigid shape signatures on point clouds, after we evidenced a lack of point-cloud benchmarks in the literature. With these ingredients, we are able to construct shape signatures which are specially built for non-rigid shape retrieval

A machine learning approach to the unsupervised segmentation of mitochondria in subcellular electron microscopy data

Recent advances in cellular and subcellular microscopy demonstrated its potential towards unravelling the mechanisms of various diseases at the molecular level. The biggest challenge in both human- and computer-based visual analysis of micrographs is the variety of nanostructures and mitochondrial morphologies. The state-of-the-art is, however, dominated by supervised manual data annotation and early attempts to automate the segmentation process were based on supervised machine learning techniques which require large datasets for training. Given a minimal number of training sequences or none at all, unsupervised machine learning formulations, such as spectral dimensionality reduction, are known to be superior in detecting salient image structures. This thesis presents three major contributions developed around the spectral clustering framework which is proven to capture perceptual organization features. Firstly, we approach the problem of mitochondria localization. We propose a novel grouping method for the extracted line segments which describes the normal mitochondrial morphology. Experimental findings show that the clusters obtained successfully model the inner mitochondrial membrane folding and therefore can be used as markers for the subsequent segmentation approaches. Secondly, we developed an unsupervised mitochondria segmentation framework. This method follows the evolutional ability of human vision to extrapolate salient membrane structures in a micrograph. Furthermore, we designed robust non-parametric similarity models according to Gestaltic laws of visual segregation. Experiments demonstrate that such models automatically adapt to the statistical structure of the biological domain and return optimal performance in pixel classification tasks under the wide variety of distributional assumptions. The last major contribution addresses the computational complexity of spectral clustering. Here, we introduced a new anticorrelation-based spectral clustering formulation with the objective to improve both: speed and quality of segmentation. The experimental findings showed the applicability of our dimensionality reduction algorithm to very large scale problems as well as asymmetric, dense and non-Euclidean datasets

[Activity of Institute for Computer Applications in Science and Engineering]

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science

Theoretical analysis for convex and non-convex clustering algorithms

Clustering is one of the most important unsupervised learning problem in the machine learning and statistics community. Given a set of observations, the goal is to find the latent cluster assignment of the data points. The observations can be either some covariates corresponding to each data point, or the relational networks representing the affinity between pair of nodes. We study the problem of community detection in stochastic block models and clustering mixture models. The two kinds of problems bear a lot of resemblance, and similar techniques can be applied to solve them. It is common practice to assume some underlying model for the data generating process in order to analyze it properly. With some pre-defined partitions of all data points, generative models can be defined to represent those two types of data observations. For the covariates, the mixture model is one of the most flexible and widely-used models, where each cluster i comes from some distribution D [subscript i], and the entire distribution is a convex sum over all distributions [mathematical equation]. We assume that the data is Gaussian or sub-gaussian, and analyze two algorithms: 1) Expectation-Maximization algorithm, which is notoriously non-convex and sensitive to local optima, and 2) Convex relaxation of the k-means algorithm. We show both methods are consistent under certain conditions when the signal to noise ratio is relatively high. And we obtain the upper bounds for error rate if the signal to noise ration is low. When there are outliers in the data set, we show that the semi-definite relaxation exhibits more robust result compared to spectral methods. For the networks, we consider the Stochastic Block Model (SBM), in which the probability of edge presence is fully determined by the cluster assignments of the pair of nodes. We use a semi-definite programming (SDP) relaxation to learn the clustering matrix, and discuss the role of model parameters. In most SDP relaxations of SBM, the number of communities is required for the algorithm, which is a strong requirement for many real-world applications. In this thesis, we propose to introduce a regularization to the nuclear norm, which is shown to be able to exactly recover both the number of communities and cluster memberships even when the number of communities is unknown. In many real-world networks, it is more common to see both network structure and node covariates simultaneously. In this case, we present a regularization based method to effectively combine the two sources of information. The proposed method works especially well when the covariates and network contain complementary information.Statistic

Recent developments in mathematical aspects of relativistic fluids

We review some recent developments in mathematical aspects of relativistic fluids. The goal is to provide a quick entry point to some research topics of current interest that is accessible to graduate students and researchers from adjacent fields, as well as to researches working on broader aspects of relativistic fluid dynamics interested in its mathematical formalism. Instead of complete proofs, which can be found in the published literature, here we focus on the proofs' main ideas and key concepts. After an introduction to the relativistic Euler equations, we cover the following topics: a new wave-transport formulation of the relativistic Euler equations tailored to applications; the problem of shock formation for relativistic Euler; rough (i.e., low-regularity) solutions to the relativistic Euler equations; the relativistic Euler equations with a physical vacuum boundary; relativistic fluids with viscosity. We finish with a discussion of open problems and future directions of research.Comment: Minor typos correcte

Efficient Algorithms for Solving Structured Eigenvalue Problems Arising in the Description of Electronic Excitations

Matrices arising in linear-response time-dependent density functional theory and many-body perturbation theory, in particular in the Bethe-Salpeter approach, show a 2 × 2 block structure. The motivation to devise new algorithms, instead of using general purpose eigenvalue solvers, comes from the need to solve large problems on high performance computers. This requires parallelizable and communication-avoiding algorithms and implementations. We point out various novel directions for diagonalizing structured matrices. These include the solution of skew-symmetric eigenvalue problems in ELPA, as well as structure preserving spectral divide-and-conquer schemes employing generalized polar decompostions

Analysis and Visualization of Higher-Order Tensors: Using the Multipole Representation

Materialien wie Kristalle, biologisches Gewebe oder elektroaktive Polymere kommen häufig in verschiedenen Anwendung, wie dem Prothesenbau oder der Simulation von künstlicher Muskulatur vor. Diese und viele weitere Materialien haben gemeinsam, dass sie unter gewissen Umständen ihre Form und andere Materialeigenschaften ändern. Um diese Veränderung beschreiben zu können, werden, abhängig von der Anwendung, verschiedene Tensoren unterschiedlicher Ordnung benutzt. Durch die Komplexität und die starke Abhängigkeit der Tensorbedeutung von der Anwendung, gibt es bisher kein Verfahren Tensoren höherer Ordnung darzustellen, welches standardmäßig benutzt wird. Auch bezogen auf einzelne Anwendungen gibt es nur sehr wenig Arbeiten, die sich mit der visuellen Darstellung dieser Tensoren auseinandersetzt. Diese Arbeit beschäftigt sich mit diesem Problem. Es werden drei verschiedene Methoden präsentiert, Tensoren höherer Ordnung zu analysieren und zu visualisieren. Alle drei Methoden basieren auf der sogenannte deviatorischen Zerlegung und der Multipoldarstellung. Mit Hilfe der Multipole können die Symmetrien des Tensors und damit des beschriebenen Materials bestimmt werden. Diese Eigenschaft wird in für die Visualisierung des Steifigkeitstensors benutzt. Die zweite Methode basiert direkt auf den Multipolen und kann damit beliebige Tensoren in drei Dimensionen darstellen. Dieses Verfahren wird anhand des Kopplungs Tensors, ein Tensor dritter Ordnung, vorgestellt. Die ersten zwei Verfahren sind lokale Glyph-basierte Verfahren. Das dritte Verfahren ist ein erstes globales Tensorvisualisierungsverfahren, welches Tensoren beliebiger Ordnung und Symmetry in drei Dimensionen mit Hilfe eines linienbasierten Verfahrens darstellt.Materials like crystals, biological tissue or electroactive polymers are frequently used in applications like prosthesis construction or the simulation of artificial musculature. These and many other materials have in common that they change their shape and other material properties under certain circumstances. To describe these changes, different tensors of different order, dependent of the application, are used. Due to the complexity and the strong dependency of the tensor meaning of the application, there is, by now, no visualization method that is used by default. Also for specific applications there are only a few methods that address the visual analysis of higher-order tensors. This work adresses this problem. Three different methods to analyse and visualize tensors of higher order will be provided. All three methods are based on the so called deviatoric decomposition and the multipole representation. Using the multipoles the symmetries of a tensor and, therefore, of the described material, can be calculated. This property is used to visualize the stiffness tensor. The second method uses the multipoles directly and can be used for each tensor of any order in three dimensions. This method is presented by analysing the third-order coupling tensor. These two techniques are glyph-based visualization methods. The third one, a line-based method, is, according to our knowledge, a first global visualization method that can be used for an arbitrary tensor in three dimensions

Large Scale Inverse Problems

This book is thesecond volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation &amp Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications. The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary. This collection of survey articles focusses on the large inverse problems commonly arising in simulation and forecasting in the earth sciences

Foundations of Quantum Theory: From Classical Concepts to Operator Algebras

Quantum physics; Mathematical physics; Matrix theory; Algebr