<p> The focus of this thesis is on the utilization of the sparsity concept in solving some challenging problems, e.g., finding a unique solution to the under-determined linear system of equations in which the number of equations is less than the number of unknowns. This concept is extended to the problem of solving the under-determined blind source separation (BSS) problem in which the number of source signals is greater than the number of sensors and both the mixing matrix and the source signals are unknowns. In this respect we study three problems: </p> <p> 1. Developing some algorithms for solving the under-determined linear system of equations for the case of a sparse solution vector. In this thesis we develop a new methodology for minimizing a class of non-convex (concave on the non-negative orthant) functions for solving the aforementioned problem. The proposed technique is based on locally replacing the original objective function by a quadratic convex function which is easily minimized. For a certain selection of the convex objective function, the existing class of algorithms called Iterative Re-weighted Least Squares (IRLS) can be derived from the proposed methodology. Thus the proposed algorithms are a generalization and unification of the previous methods. In this thesis we also propose a convex objective function that produces an algorithm that can converge to a sparse solution vector in significantly fewer iterations than the IRLS algorithms.</p> <p> 2. Solving the under-determined BSS problem by developing new clustering algorithms for estimating the mixing matrix. The under-determined BSS problem is usually solved by following a two-step approach, in which the mixing matrix is estimated in the first step, then the sources are estimated in the second step. For the case of sparse sources, the mixing matrix is usually estimated by clustering the columns of the observation matrix. In this thesis we develop three novel clustering algorithms that can efficiently estimate the mixing matrix, as well as the number of sources, which is usually unknown. Numerical simulations verify the efficiency of the proposed algorithms compared to some well known algorithms that are usually used for solving the same problem.</p> <p> 3. Extraction of a desired source signal from a linear mixture of hidden sources when prior information is available about the desired source signal. There are many situations in which one is interested in extracting a specific source signal. The a priori available information about the desired source signal could be temporal, spatial, or both. In this thesis we develop new algorithms for extracting a desired sparse source signal from a linear mixture of hidden sources. The information available about the desired source signal, as well as its sparsity, are incorporated in an optimization problem for extracting this source signal. Four different algorithms have been developed for solving this problem. Numerical simulations show that the proposed algorithms can be used successfully for removing different kind of artifacts from real electroencephalographic (EEG) data and for estimating the event related potential (ERP) signal from synthesized EEG data.</p> / Thesis / Doctor of Philosophy (PhD)

Cette thèse est dédiée à améliorer la compréhension neuro-scientifique à l'aide d'imagerie par résonance magnétique de diffusion (IRMd). Nous nous concentrons sur la modélisation du signal de diffusion et l'estimation par IRMd des biomarqueurs liés à la microstructure, appelé «Microstructure Imaging». Cette thèse est organisée en trois parties. Dans partie I nous commençons par la base de l'IRMd et un aperçu de l'anisotropie en diffusion. Puis nous examinons la plupart des modèles de microstructure utilisant PGSE, en mettant l'accent sur leurs hypothèses et limites, suivi par une validation par l'histologie de la moelle épinière de leur estimation. La partie II présente nos contributions à l'imagerie en 3D et à l’estimation de microstructure. Nous proposons une régularisation laplacienne de la base fonctionnelle MAP, ce qui nous permet d'estimer de façon robuste les indices d'espace q liés au tissu. Nous appliquons cette approche aux données du Human Connectome Project, où nous l'utilisons comme prétraitement pour d'autres modèles de microstructure. Enfin, nous comparons les biomarqueurs dans une étude ex-vivo de rats Alzheimer à différents âges. La partie III présente nos contributions au représentation de l’espace qt - variant sur l'espace q 3D et le temps de diffusion. Nous présentons une approche initiale qui se concentre sur l'estimation du diamètre de l'axone depuis l'espace qt. Nous terminons avec notre approche finale, où nous proposons une nouvelle base fonctionnelle régularisée pour représenter de façon robuste le signal qt, appelé qt-IRMd. Ce qui permet l'estimation des indices d’espace q dépendants du temps, quantifiant la dépendance temporelle du signal IRMd. / This thesis is dedicated to furthering neuroscientific understanding of the human brain using diffusion-sensitized Magnetic Resonance Imaging (dMRI). Within dMRI, we focus on the estimation and interpretation of microstructure-related markers, often referred to as ``Microstructure Imaging''. This thesis is organized in three parts. Part I focuses on understanding the state-of-the-art in Microstructure Imaging. We start with the basic of diffusion MRI and a brief overview of diffusion anisotropy. We then review and compare most state-of-the-art microstructure models in PGSE-based Microstructure Imaging, emphasizing model assumptions and limitations, as well as validating them using spinal cord data with registered ground truth histology. In Part II we present our contributions to 3D q-space imaging and microstructure recovery. We propose closed-form Laplacian regularization for the recent MAP functional basis, allowing robust estimation of tissue-related q-space indices. We also apply this approach to Human Connectome Project data, where we use it as a preprocessing for other microstructure models. Finally, we compare tissue biomarkers in a ex-vivo study of Alzheimer rats at different ages. In Part III, we present our contributions to representing the qt-space - varying over 3D q-space and diffusion time. We present an initial approach that focuses on 3D axon diameter estimation from the qt-space. We end with our final approach, where we propose a novel, regularized functional basis to represent the qt-signal, which we call qt-dMRI. Our approach allows for the estimation of time-dependent q-space indices, which quantify the time-dependence of the diffusion signal.