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This thesis makes novel contributions to a problem of practical and theoretical importance, namely how to determine explicitly computable upper bounds for the Hausdorff distance of the spectra of two compact operators on a Hilbert space in terms of the distance of the two operators in operator norm. It turns out that the answer depends crucially on the speed of decay of the sequence of singular values of the two operators. To this end, 'compactness classes', that is, collections of operators the singular values of which decay at a certain speed, are introduced and their functional analytic properties studied in some detail. The main result of the thesis is an explicit formula for the Hausdorff distance of the spectra of two operators belonging to the same compactness class. Along the way, upper bounds for the resolvents of operators belonging to a particular compactness class are established, as well as novel bounds for determinants of trace class operators.

Dans cette thèse nous avons proposé une méthode d’étude des variations paramétriques pour les systèmes mécatroniques continus et hybrides puis une approche du tolérancement mécatronique. Nous avons d’abord étudié les différentes approches existantes pour la prise en compte de la variation de paramètres. Pour les systèmes continus à paramètres variables nous avons choisi la méthode des inclusions différentielles. Nous avons repris l’algorithme de Raczynski et nous avons développé un algorithme d’optimisation qui se base sur la méthode du steepest descent, avec une extension permettant d’obtenir l’optimum global. Pour les systèmes hybrides, contenant des évolutions continues et des sauts discrets, et qui présentent des variations paramétriques, nous avons choisi le formalisme de l’inclusion différentielle impulsionnelle comme outil de modélisation. Nous avons repris ce formalisme et identifié ses éléments sur un système mécatronique. Nous avons développé des algorithmes de résolution des inclusions différentielles impulsionnelles pour un puis pour plusieurs paramètres variables. Pour visualiser les résultats, les algorithmes développés ont été implémentés sous Mathématica. Nous avons fini cette partie par une comparaison entre notre approche et d’autres comme celles autour des automates hybrides à invariant polyèdre, les inclusions différentielles polygonales et l’algorithme pratique de résolution des inclusions différentielles. Nous avons montré alors certains avantages de notre approche. En dernière partie, nous avons repris les différents outils utilisés et résultats obtenus pour définir et affiner notre approche du tolérancement. Nous avons défini la zone du fonctionnement désiré, les différents cas de figures qu’elle peut présenter et son intersection avec le domaine atteignable. Nous avons présenté un outil métrique basé sur la distance topologique de Hausdorff pour le calcul des distances entre ces différents ensembles. Munis de ces éléments, nous avons proposé une démarche itérative pour le tolérancement dans l’espace d’état. / In this thesis we proposed a method for the study of parametric variation for continuous and hybrid systems and an approach for mechatronics tolerancing. We first studied the different existing approaches to take into account the variation of parameters. For continuous systems with variable parameters we chose the method of differential inclusions. We took the Raczynski algorithm and we have developed an optimization algorithm which is based on the steepest descent method with an extension to obtain global optimum. For hybrid systems, containing continuous evolutions and discrete jumps, and have parametric variations, we have chosen the formalism of impulse differential inclusion as a modeling tool. We took this formalism and identified its components on a mechatronic system. We have developed algorithms for solving impulse differential inclusions for several variable parameters. To view the results, the developed algorithms were implemented in Mathematica. We ended this part by a comparison between our approach and others like those around hybrid automata invariant polyhedron, polygonal differential inclusions and practical algorithm for solving differential inclusion. We showed then some advantages of our approach. In the last part, we organized the different tools used and results obtained to define and refine our approach to tolerancing. We defined the area of the desired operation, the various scenarios that may present, and its intersection with reachable area. We presented a metric tool based on topological Hausdorff distance for the calculation of distances between the different sets. With these elements, we proposed an iterative approach to tolerancing in the state space.

Measuring visual closeness of 3-D models is an important issue for different problems and there is still no standardized metric or algorithm to do it. The normal of a surface plays a vital role in the shading of a 3-D object. Motivated by this, we developed two applications to measure visualcloseness, introducing normal difference as a parameter in a weighted metric in Metro’s sampling approach to obtain the maximum and mean distance between 3-D models using 3-D and 6-D correspondence search structures. A visual closeness metric should provide accurate information on what the human observers would perceive as visually close objects. We performed a validation study with a group of people to evaluate the correlation of our metrics with subjective perception. The results were positive since the metrics predicted the subjective rankings more accurately than the Hausdorff distance.

This thesis is broadly divided in two parts. In the first part we give a survey of various distances between metric spaces, namely the uniform distance, Lipschitz distance, Hausdor distance and the Gramoz Hausdor distance. Here we talk about only the most basic of their properties and give a few illustrative examples. As we wish to study collections of metric measure spaces, which are triples (X; d; m) consisting of a complete separable metric space (X; d) and a Boral probability measure m on X, there are discussions about some distances between them. Among the three that we discuss, the transportation and distortion distances were introduced by Sturm. The later, denoted by 2, on the space X2 of all metric measure spaces having finite L2-size is the focus of the second part of this thesis. The second part is an exposition based on the work done by Sturm. Here we prove a number of results on the analytic and geometric properties of (X2; 2). Beginning by noting that (X2; 2) is a non-complete space, we try to understand its completion. Towards this end, the notion of a gauged measure space is useful. These are triples (X; f; m) where X is a Polish space, m a Boral probability measure on X and f a function, also called a gauge, on X X that is symmetric and square integral with respect to the product measure m2. We show that, Theorem 1. The completion of (X2; 2) consists of all gauged measure spaces where the gauges satisfy triangle inequality almost everywhere. We denote the space of all gauged measure spaces by Y. The space X2 can be embedded in Y and the transportation distance 2 extends easily from X2 to Y. These two spaces turn out to have similar geometric properties. On both these spaces 2 is a strictly intrinsic metric; i.e. any two members in them can be joined by a shortest path. But more importantly, using a description of the geodesics in these spaces, the following result is proved. Theorem 2. Both (X2; 2) and (Y; 2) have non-negative curvature in the sense of Alexandrov.

We present a method for extracting contours from digital images, using techniques from computational geometry. Our approach is different from traditional pixel-based methods in image processing. Instead of working directly with pixels, we extract a set of oriented feature points from the input digital images, then apply classical geometric techniques, such as clustering, linking, and simplification, to find contours among these points. Experiments on synthetic and natural images show that our method can effectively extract contours, even from images with considerable noise; moreover, the extracted contours have a very compact representation.

In this thesis, we conduct functional Magnetic Resonance Imaging (fMRI) data analysis with the aim of grouping the brain voxels depending on their responsiveness to a neural task. We mathematically treat the fMRI signals as the convolution of the neural stimulus with the hemodynamic response function (HRF). We first estimate a time series including HRFs for each of the observed fMRI signals from a given set and we cluster them in order to identify the groups of brain voxels. The HRF estimation problem is studied within the Bayesian framework through a blind deconvolution algorithm using MAP approach under completely unsupervised and model-free settings, i.e, stimulus is assumed to be unknown and also no particular shape is assumed for the HRF. Only using a given fMRI signal together with a weak Gaussian prior distribution imposed on HRF favoring &lsquo / smoothness&rsquo / , our method successfully estimates all the components of our framework: the HRF, the stimulus and the noise process. Then, we propose to use a modified version of Hausdorff distance to detect similarities within the space of HRFs, spectrally transform the data using Laplacian Eigenmaps and finally cluster them through EM clustering. According to our simulations, our method proves to be robust to lag, sampling jitter, quadratic drift and AWGN (Additive White Gaussian Noise). In particular, we obtained 100% sensitivity and specificity in terms of detecting active and passive voxels in our real data experiments. To conclude with, we propose a new framework for a mathematical treatment for voxel-based fMRI data analysis and our findings show that even when the HRF is unpredictable due to variability in cognitive processes, one can still obtain very high quality activation detection through the method proposed in this thesis.

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This diploma thesis deals with the creation and testing of optical measurement system. There are basic parts of computer vision. Some ways of image preprocessing and templates matching are discussed. Everything is directed to a particular practical task. Selected methods for templates matching are the Correlation Method, the Classical and Hybrid Hausdorff Distance, Radial and Circular Sampling Space. These methods are programmed in C++ and they are compared with function for searching templates from specific library.