abstract: Signal compressed using classical compression methods can be acquired using brute force (i.e. searching for non-zero entries in component-wise). However, sparse solutions require combinatorial searches of high computations. In this thesis, instead, two Bayesian approaches are considered to recover a sparse vector from underdetermined noisy measurements. The first is constructed using a Bernoulli-Gaussian (BG) prior distribution and is assumed to be the true generative model. The second is constructed using a Gamma-Normal (GN) prior distribution and is, therefore, a different (i.e. misspecified) model. To estimate the posterior distribution for the correctly specified scenario, an algorithm based on generalized approximated message passing (GAMP) is constructed, while an algorithm based on sparse Bayesian learning (SBL) is used for the misspecified scenario. Recovering sparse signal using Bayesian framework is one class of algorithms to solve the sparse problem. All classes of algorithms aim to get around the high computations associated with the combinatorial searches. Compressive sensing (CS) is a widely-used terminology attributed to optimize the sparse problem and its applications. Applications such as magnetic resonance imaging (MRI), image acquisition in radar imaging, and facial recognition. In CS literature, the target vector can be recovered either by optimizing an objective function using point estimation, or recovering a distribution of the sparse vector using Bayesian estimation. Although Bayesian framework provides an extra degree of freedom to assume a distribution that is directly applicable to the problem of interest, it is hard to find a theoretical guarantee of convergence. This limitation has shifted some of researches to use a non-Bayesian framework. This thesis tries to close this gab by proposing a Bayesian framework with a suggested theoretical bound for the assumed, not necessarily correct, distribution. In the simulation study, a general lower Bayesian Cram\'er-Rao bound (BCRB) bound is extracted along with misspecified Bayesian Cram\'er-Rao bound (MBCRB) for GN model. Both bounds are validated using mean square error (MSE) performances of the aforementioned algorithms. Also, a quantification of the performance in terms of gains versus losses is introduced as one main finding of this report. / Dissertation/Thesis / Masters Thesis Computer Engineering 2019

[pt] O problema de estimação de direção (DoA) de chegada é um importante tópico de pesquisa em áreas como radar, sonar, sismologia, vigilância eletrônica e comunicações sem fio. Este trabalho teve como principal resultado o desenvolvimento de um novo algoritmo que combina o método da máxima verossimilhança (ML) estocástica com o algoritmo ganancioso de busca ortogonal (OMP), comumente empregado em recuperação esparsa com Sensoriamento Compressivo (CS). Muito embora técnicas ML sejam consideradas ótimas em termos de erro médio quadrático, atingindo o limitante inferior de Cramér-Rao (CRLB), o menor limitante inferior para a variância do estimador, estas técnicas demandam de modo significante, às vezes inexequível, os recursos computacionais. Por outro lado, a partir de uma variante esparsa da equação de aquisição de dados, o problema de encontrar a solução mais esparsa possível de sistemas de equações subdeterminados com o algoritmo OMP tem sido empregado de modo satisfatório para encontrar as estimativas de direção, porém com muitas oportunidades de melhoria em casos com cenários sujeitos a condições severas. Por exemplo, cenários com acoplamento eletromagnético (EM), baixa razão sinal-ruído (SNR) e um número limitado de amostras temporais disponíveis. O estimador de direção proposto, baseado em coarranjo diferença, denominado OMP com Máxima Verossimilhança Baseado em Lista (LBML-OMP), apresentou uma melhora significativa no processo de estimação em comparação com técnicas tradicionais e modernas, tais como: OMP, Técnica de Limiar Iterativa (IHT) e Classificação de Múltiplos Sinais Espacialmente Suavizados (SS-MUSIC). A técnica proposta utiliza uma lista de candidatos gerada a partir da solução do algoritmo OMP original e decide pelo melhor a partir de uma busca limitada utilizando o estimador ML estocástico, o que justifica seu uso em cenários práticos. Para a amostragem dos sinais no ambiente, arranjos lineares não-uniformes clássicos e modernos foram empregados, tais como Arranjos Aninhados de Segunda Ordem (NAQ2), Arranjos Aninhados de Segunda Ordem Aperfeiçoados (SNAQ2), Arranjos de Redundância Mínima (MRA) e Arranjos Coprimos (CPA). Além disso, a estimação foi realizada considerando-se o efeito do acoplamento EM e ruído. Ainda, um novo modelo para estimação de direção em coarranjo diferença foi desenvolvido. Este modelo considera o número de amostras temporais finitas (não-assintótico) e mostrou melhora significativa quando do seu emprego no processo de estimação de direção de todos os algoritmos considerados, não apenas o LBML-OMP, evidenciando fontes secundárias de erro no modelo original estabelecido. De forma a complementar o trabalho, um algoritmo de atenuação de ruído chamado OMP aleatorizado (RandOMP) foi utilizado para aumentar a precisão da estimação em cenários com condições de ruído severas. Neste sentido, as contribuições deste trabalho estão relacionadas principalmente ao desenvolvimento de um novo algoritmo e um novo modelo de transformação em coarranjo diferença de modo a melhorar as estimativas de direção das fontes com arranjos lineares não-uniformes. Além disso, enfatiza-se o emprego de diferentes geometrias para as simulações, tornando-se evidente o impacto da posição dos sensores nas curvas de raiz quadrada do erro médio quadrático (RMSE). / [en] The Direction of Arrival (DoA) estimation or Direction Finding (DF) is a relevant topic for research in areas such as radar, sonar, seismology, electronic surveillance, and wireless communications. This thesis devises a new algorithm that combines a stochastic Maximum Likelihood (ML) method with the widely-known Orthogonal Matching Pursuit (OMP) greedy algorithm, commonly used in sparse recovery with Compressive Sensing (CS). Even though ML techniques are known to be optimal in the mean-squared error sense, achieving the Cramér-Rao Lower Bound (CRLB), the tighter lower bound on estimator variance, they demand a significant, sometimes infeasible, amount of computational resources. On the other hand, departing from a sparsified variant of the data acquisition equation, the problem of finding the sparsest solution of underdetermined systems of equations with OMP has been employed successfully to find the DoA estimates, but with many opportunities for improvement in cases of challenging scenarios. For instance, scenarios with electromagnetic (EM) coupling, low signal-to-noise ratio (SNR), and a limited number of available snapshots (time samples). The proposed difference coarray DoA estimator termed List-Based Maximum Likelihood OMP (LBML-OMP) has shown substantial improvements over traditional and modern techniques, such as OMP, Iterative Hard Thresholding (IHT), and Spatial Smoothing Multiple Signal Classification (SS-MUSIC). It uses a list of candidates generated from the OMP solution and decides for the best based on a limited search using the stochastic ML rule. Thus, it does not perform a grid search with the ML estimator, and this justifies its use in practical scenarios. For the sensing of space-time field, classic and modern non-uniform linear arrays are employed, such as 2-nd Order Nested Array (NAQ2), 2-nd Order Super Nested Array (SNAQ2), Minimum Redundancy Array (MRA), Minimum Hole Array (MHA), and Coprime Array (CPA). Additionally, the estimation is performed under the assumption of EM coupling and noise as disturbing side effects. Furthermore, a new model for difference coarray DoA estimation is developed. It accounts for the finite number of snapshots and has shown to increase the estimation accuracy for all the algorithms, not only LBML-OMP, evidencing secondary sources of error for the difference coarray transformation. To complement the work, a denoising algorithm called Randomized OMP (RandOMP) was applied to successfully increase the estimation accuracy for difference coarray estimators in scenarios with severe noisy conditions. The contributions of this work relate mainly to the development of a new algorithm and a new difference coarray transformation to improve the DoA estimation accuracy with non-uniform linear arrays. Also, it should be noticed the employment of different geometries for the numerical experiments, making evident the impact of the array sensors positions in the root mean square error (RMSE) curves.

[pt] Câmeras de pixel único consistem em reconstruir computacionalmente imagens em duas dimensões a partir de um conjunto de medidas feitas por um detector de único pixel. Para que se obtenha a informação espacial, um conjunto de padrões de modulação são aplicados à luz transmitida/refletida do objeto e essa informação é combinada com o sinal integral do detector. Primeiro, apresentamos uma visão geral desses sistemas e demonstramos a implementação de uma prova de conceito capaz de fazer aquisição de imagem usando três modos de operação: Varredura, escaneamento por base de Hadamard, e escaneamento por base de Hadamard com sensoriamento compreensivo. Segundo, discutimos como os diferentes parâmetros experimentais do sistema ótico afetam a aquisição. Finalmente, comparamos a performance dos três modos de operação quando usados para a aquisição de images com tamanhos entre (8px, 8px) e (128px, 128px). / [en] Single-pixel imaging consists in computationally reconstructing 2-dimensional images from a set of intensity measurements taken by a singlepoint detector. To derive the spatial information of a scene, a set of modulation patterns are applied to the transmitted/backscattered light from the object and combined with the integral signal on the detector. First, we present an overview of such optical systems and implement a proof of concept that can perform image acquisition using three different modes of operation: Raster scanning, Hadamard basis scanning, and Hadamard compressive sampling. Second, we explore how the different experimental parameters affect image acquisition. Finally, we compare how the three scanning mode perform for acquisition of images of sizes ranging from (8px, 8px) to (128px, 128px).

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In this dissertation, sparse processing of signals for directions-of-arrival (DoAs) estimation is addressed in the framework of Compressive Sensing (CS). In particular, DoAs estimation problem for different types of sources, systems, and applications are formulated in the CS paradigm. In addition, the fundamental conditions related to the ``Sparsity'' and ``Linearity'' are carefully exploited in order to apply confidently the CS-based methodologies. Moreover, innovative strategies for various systems and applications are developed, validated numerically, and analyzed extensively for different scenarios including signal to noise ratio (SNR), mutual coupling, and polarization loss. The more realistic data from electromagnetic (EM) simulators are often considered for various analysis to validate the potentialities of the proposed approaches. The performances of the proposed estimators are analyzed in terms of standard root-mean-square error (RMSE) with respect to different degrees-of-freedom (DoFs) of DoAs estimation problem including number of elements, number of signals, and signal properties. The outcomes reported in this thesis suggest that the proposed estimators are computationally efficient (i.e., appropriate for real time estimations), robust (i.e., appropriate for different heterogeneous scenarios), and versatile (i.e., easily adaptable for different systems).

Cette thèse est consacrée à l'élaboration de nouvelles méthodes d'acquisition et de traitement de données en IRM de diffusion (IRMd) afin de caractériser la diffusion des molécules d'eau dans les fibres de matière blanche à l'échelle d'un voxel. Plus particulièrement, nous travaillons sur un moyen de reconstruction précis de l'Ensemble Average Propagator (EAP), qui représente la fonction de probabilité de diffusion des molécules d'eau. Plusieurs modèles de diffusion tels que le tenseur de diffusion ou la fonction de distribution d'orientation sont très utilisés dans la communauté de l'IRMd afin de quantifier la diffusion des molécules d'eau dans le cerveau. Ces modèles sont des représentations partielles de l'EAP et ont été développés en raison du petit nombre de mesures nécessaires à leurs estimations. Cependant, il est important de pouvoir reconstruire précisément l'EAP afin d'acquérir une meilleure compréhension des mécanismes du cerveau et d'améliorer le diagnostique des troubles neurologiques. Une estimation correcte de l'EAP nécessite l'acquisition de nombreuses images de diffusion sensibilisées à des orientations différentes dans le q-space. Ceci rend son estimation trop longue pour être utilisée dans la plupart des scanners cliniques. Dans cette thèse, nous utilisons des techniques de reconstruction parcimonieuses et en particulier la technique connue sous le nom de Compressive Sensing (CS) afin d’accélérer le calcul de l'EAP. Les multiples aspects de la théorie du CS et de son application à l'IRMd sont présentés dans cette thèse. / This thesis is dedicated to the development of new acquisition and processing methods in diffusion MRI (dMRI) to characterize the diffusion of water molecules in white matter fiber bundles at the scale of a voxel. In particular, we focus our attention on the accurate recovery of the Ensemble Average Propagator (EAP), which represents the full 3D displacement of water molecule diffusion. Diffusion models such that the Diffusion Tensor or the Orientation Distribution Function (ODF) are largely used in the dMRI community in order to quantify water molecule diffusion. These models are partial EAP representations and have been developed due to the small number of measurement required for their estimations. It is thus of utmost importance to be able to accurately compute the EAP and order to acquire a better understanding of the brain mechanisms and to improve the diagnosis of neurological disorders. Estimating the full 3D EAP requires the acquisition of many diffusion images sensitized todifferent orientations in the q-space, which render the estimation of the EAP impossible in most of the clinical dMRI scanner. A surge of interest has been seen in order to decrease this time for acquisition. Some works focus on the development of new and efficient acquisition sequences. In this thesis, we use sparse coding techniques, and in particular Compressive Sensing (CS) to accelerate the computation of the EAP. Multiple aspects of the CS theory and its application to dMRI are presented in this thesis.

In the past decade, camera manufacturers have offered smaller form factors, smaller pixel sizes (leading to higher resolution images), and faster processing chips to increase the performance of consumer cameras. However, these conventional approaches have failed to capitalize on the spatio-temporal redundancy inherent in images, nor have they adequately provided a solution for finding $3$D point correspondences for cameras sampling different bands of the visible spectrum. In this thesis, we pose the following question---given the repetitious nature of image patches, and appropriate camera architectures, can statistical models be used to increase temporal, structural, or spectral resolution? While many techniques have been suggested to tackle individual aspects of this question, the proposed solutions either require prohibitively expensive hardware modifications and/or require overly simplistic assumptions about the geometry of the scene. We propose a two-stage solution to facilitate image reconstruction; 1) design a linear camera system that optically encodes scene information and 2) recover full scene information using prior models learned from statistics of natural images. By leveraging the tendency of small regions to repeat throughout an image or video, we are able to learn prior models from patches pulled from exemplar images. The quality of this approach will be demonstrated for two application domains, using low-speed video cameras for high-speed video acquisition and multi-spectral fusion using an array of cameras. We also investigate a conventional approach for finding 3D correspondence that enables a generalized assorted array of cameras to operate in multiple modalities, including multi-spectral, high dynamic range, and polarization imaging of dynamic scenes.

[pt] Neste trabalho, quatro métodos projetados especificamente para a estimação de sistemas esparsos são originalmente elaborados e apresentados. São eles: Encolhimentos Sucessivos, Expansões Sucessivas, Minimização da Norma l1 e Ajuste Automático do fator de regularização do Custo LS. Os quatro métodos propostos baseiam-se na técnica de estimação de sistemas lineares e invariantes no tempo pelo critério dos mínimos quadrados, universalmente conhecida por sua denominação em inglês - Least Squares (LS) Estimation, e incorporam técnicas relacionadas a otimização convexa e à teoria de compressive sensing. Os resultados obtidos em simulações mostram que os métodos em questão têm desempenho superior que a estimação LS convencional e que o algoritmo Recursive Least Squares (RLS) com regularização convexa denominado l1-RLS, em muitos casos alcançando o desempenho ótimo apresentado pelo método de estimação LS Oráculo, no qual o suporte da resposta ao impulso em tempo discreto do sistema estimado é conhecido a priori. Além disso, os métodos propostos apresentam custo computacional menor que do algoritmo l1-RLS. / [en] In this thesis, four methods specifically designed for sparse systems estimation are originally developed and presented, which were called here: Relaxations method, Successive Expansions method, l1-norm Minimization method and Automatic Adjustment of the Regularization Factor method. The four proposed methods are based on the Least Squares (LS) Estimation method and incorporate techniques related to convex optimization and to the theory of compressive sensing. The simulation results show that the proposed methods herein present superior performance than the ordinary LS estimation method and the Recursive Least Squares (RLS) with convex regularization method (l1-RLS), in many cases achieving the same optimal performance presented by the LS Oracle method. Furthermore, the proposed methods demand lower computational cost than the l1-RLS method.