[pt] A estimação de direção de chegada (DoA) é uma importante área de processamento de arranjos de sensores que é encontrada em uma ampla gama de aplicações de engenharia. Este fato, juntamente com o desenvolvimento da área de Compressed Sensing (CS) nos últimos anos, são a principal motivação desta dissertação. Nesta dissertação, é apresentada uma formulação do problema de estimação de direção de chegada como um problema de representação esparsa da sinal e vários algoritmos de recuperação esparsa são derivados e investigados para resolver o problema atual. Os algoritmos propostos são baseados na incorporação da informação prévia sobre o sinal esparso no processo de estimativa. Na primeira parte, nos concentramos no desenvolvimento de dois algoritmos Bayesianos , que se baseiam principalmente no algoritmo iterative hard thresholding (IHT). Devido ao desempenho inferior dos algoritmos convencionais de estimação de chegada em cenários com fontes correlacionadas, nós prestamos atenção especial ao desempenho dos algoritmos propostos nesta condição. Na segunda parte, o problema de otimização baseados na minimização da norma l1 é apresentado e um algoritmo bayesiano é proposto para resolver o problema chamado basis pursuit denoising (BPDN). Os resultados da simulação mostram que os estimadores Bayesianos superam os estimadores não Bayesianos e que a incorporação do conhecimento prévio da distribuição do sinal melhorou substancialmente o desempenho dos algoritmos. / [en] Direction of arrival (DoA) estimation is a key area of sensor array processing which is encountered in a broad range of important engineering applications. This fact together with the development of the Compressed Sensing (CS) area in the last years are the principal motivation of this thesis. In this dissertation, a formulation of the source localization problem as a sparse signal representation problem is presented and several sparse recovery algorithms are derived and investigated for solving the current problem. The proposed algorithms are based on the incorporation of the prior information about the sparse signal in the estimation process. In the first part, we focus on the development of two Bayesian greedy algorithms which are principally based on the iterative hard thresholding (IHT) algorithm. Due to the inferior performance of the conventional DoA estimation algorithm in scenarios with correlated sources, we pay special attention to the performance of the proposed algorithms under this condition. In the second part, the optimization problem using a l1 penalty is introduced and a Bayesian algorithm for solving the basis pursuit denoising problem is presented. Simulation results shows that Bayesian estimators which take into account the prior knowledge of the signal distribution outperform and improve substantially the performance of the non-Bayesian estimators.

Biomedical optical imaging is capable of providing functional information of the soft bi-ological tissues, whose applications include imaging large tissues, such breastand brain in-vivo. Biomedical optical imaging uses near infrared light (600nm-900nm) as the probing media, givin ganaddedadvantageofbeingnon-ionizingimagingmodality. The tomographic technologies for imaging large tissues encompasses diffuse optical tomogra-phyandphotoacoustictomography. Traditional image reconstruction methods indiffuse optical tomographyemploysa �2-norm based regularization, which is known to remove high frequency no is either econstructed images and make the mappearsmooth. Hence as parsity based image reconstruction has been deployed for diffuse optical tomography, these sparserecov-ery methods utilize the �p-norm based regularization in the estimation problem with 0≤ p<1. These sparse recovery methods, along with an approximation to utilizethe �0-norm, have been used forther econstruction of diffus eopticaltomographic images.The comparison of these methods was performed by increasing the sparsityinthesolu-tion. Further a model resolution matrix based framework was proposed and shown to in-duceblurinthe�2-norm based regularization framework for diffuse optical tomography. This model-resolution matrix framework was utilized in the optical imaged econvolution framework. A basis pursuitdeconvolution based on Split AugmentedLagrangianShrink-ageAlgorithm(SALSA)algorithm was used along with the Tikhonovregularization step making the image reconstruction into a two-step procedure. This new two-step approach was found to be robust with no iseandwasabletobetterdelineatethestructureswhichwasevaluatedusingnumericalandgelatinphantom experiments. Modern diffuse optical imaging systems are multi-modalin nature, where diffuse optical imaging is combined with traditional imaging modalitiessuc has Magnetic Res-onanceImaging(MRI),or Computed Tomography(CT). Image-guided diffuse optical tomography has the advantage of reducingthetota lnumber of optical parameters beingreconstructedtothenumber of distinct tissue types identified by the traditional imaging modality, converting the optical image-reconstruction problem fromunder-determined innaturetoover-determined. In such cases, the minimum required measurements might be farless compared to those of the traditional diffuse optical imaging. An approach to choose these measurements optimally based on a data-resolution matrix is proposed, and it is shown that it drastically reduces the minimum required measurements (typicalcaseof240to6) without compromising the image reconstruction performance. In the last part of the work , a model-based image reconstruction approaches in pho-toacoustic tomography (which combines light and ultra sound) arestudied as it is know that these methods have a distinct advantage compared to traditionalanalytical methods in limited datacase. These model-based methods deployTikhonovbasedregularizationschemetoreconstruct the initial pressure from the boundary acoustic data. Again a model-resolution for these cases tend to represent the blurinduced by the regularization scheme. A method that utilizes this blurringmodelandper forms the basis pursuit econ-volution to improve the quantitative accuracy of the reconstructed photoacoustic image is proposed and shown to be superior compared to other traditional methods. Moreover, this deconvolution including the building of model-resolution matrixis achievedvia the Lanczosbidiagonalization (least-squares QR) making this approach computationally ef-ficient and deployable inreal-time. Keywords Medical imaging, biomedical optical imaging, diffuse optical tomography, photoacous-tictomography, multi-modalimaging, inverse problems,sparse recovery,computational methods inbiomedical optical imaging.

Fusion frames and block sparse recovery are of interest in signal processing and communication applications. In these applications it is required that the fusion frame have some desirable properties. One such requirement is that the fusion frame be tight and its subspaces form an optimal packing in a Grassmannian manifold. Such fusion frames are called Grassmannian fusion frames. Grassmannian frames are known to be optimal dictionaries for sparse recovery as they have minimum coherence. By analogy Grassmannian fusion frames are potential candidates as optimal dictionaries in block sparse processing. The present work intends to study fusion frames in finite dimensional vector spaces assuming a specific structure useful in block sparse signal processing. The main focus of our work is the design of Grassmannian fusion frames and their implication in block sparse recovery. We will consider burst error correction as an application of block sparsity and fusion frame concepts. We propose two new algebraic methods for designing Grassmannian fusion frames. The first method involves use of Fourier matrix and difference sets to obtain a partial Fourier matrix which forms a Grassmannian fusion frame. This fusion frame has a specific structure and the parameters of the fusion frame are determined by the type of difference set used. The second method involves constructing Grassmannian fusion frames from Grassmannian frames which meet the Welch bound. This method uses existing constructions of optimal Grassmannian frames. The method, while fairly general, requires that the dimension of the vector space be divisible by the dimension of the subspaces. A lower bound which is an analog of the Welch bound is derived for the block coherence of dictionaries along with conditions to be satisfied to meet the bound. From these results we conclude that the matrices constructed by us are optimal for block sparse recovery from block coherence viewpoint. There is a strong relation between sparse signal processing and error control coding. It is known that burst errors are block sparse in nature. So, here we attempt to solve the burst error correction problem using block sparse signal recovery methods. The use of Grassmannian fusion frames which we constructed as optimal dictionary allows correction of maximum possible number of errors, when used in conjunction with reconstruction algorithms which exploit block sparsity. We also suggest a modification to improve the applicability of the technique and point out relationship with a method which appeared previously in literature. As an application example, we consider the use of the burst error correction technique for impulse noise cancelation in OFDM system. Impulse noise is bursty in nature and severely degrades OFDM performance. The Grassmannian fusion frames constructed with Fourier matrix and difference sets is ideal for use in this application as it can be easily incorporated into the OFDM system.