Humans perceive the world in three dimensions, but many sensing capabilities only display two-dimensional information to users by way of images. In this work we develop two novel reconstruction techniques utilizing synthetic aperture radar (SAR) data in three dimensions given sparse amounts of available data. We additionally leverage a hybrid joint-sparsity and sparsity approach to remove a-priori influences on the environment and instead explore general imaging properties in our reconstructions. We evaluate the required sampling rates for our techniques and a thorough analysis of the accuracy of our methods. The results presented in this thesis suggest a solution to sparse three-dimensional object reconstruction that effectively uses a substantially less amount of phase history data (PHD) while still extracting critical features off an object of interest.

Group sparsity reveals underlying sparsity patterns and contains rich structural information in data. Hence, exploiting group sparsity will facilitate more efficient techniques for recovering large and complicated data in applications such as compressive sensing, statistics, signal and image processing, machine learning and computer vision. This thesis develops efficient algorithms for solving a class of optimization problems with group sparse solutions, where arbitrary group configurations are allowed and the mixed L21-regularization is used to promote group sparsity. Such optimization problems can be quite challenging to solve due to the mixed-norm structure and possible grouping irregularities. We derive algorithms based on a variable splitting strategy and the alternating direction methodology. Extensive numerical results are presented to demonstrate the efficiency, stability and robustness of these algorithms, in comparison with the previously known state-of-the-art algorithms. We also extend the existing global convergence theory to allow more generality.

COOPERATIVE WIDEBAND SPECTRUM SENSING BASED ON JOINT SPARSITY By Ghazaleh Jowkar, Master of Science A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University Virginia Commonwealth University 2017 Major Director: Dr. Ruixin Niu, Associate Professor of Department of Electrical and Computer Engineering In this thesis, the problem of wideband spectrum sensing in cognitive radio (CR) networks using sub-Nyquist sampling and sparse signal processing techniques is investigated. To mitigate multi-path fading, it is assumed that a group of spatially dispersed SUs collaborate for wideband spectrum sensing, to determine whether or not a channel is occupied by a primary user (PU). Due to the underutilization of the spectrum by the PUs, the spectrum matrix has only a small number of non-zero rows. In existing state-of-the-art approaches, the spectrum sensing problem was solved using the low-rank matrix completion technique involving matrix nuclear-norm minimization. Motivated by the fact that the spectrum matrix is not only low-rank, but also sparse, a spectrum sensing approach is proposed based on minimizing a mixed-norm of the spectrum matrix instead of low-rank matrix completion to promote the joint sparsity among the column vectors of the spectrum matrix. Simulation results are obtained, which demonstrate that the proposed mixed-norm minimization approach outperforms the low-rank matrix completion based approach, in terms of the PU detection performance. Further we used mixed-norm minimization model in multi time frame detection. Simulation results shows that increasing the number of time frames will increase the detection performance, however, by increasing the number of time frames after a number of times the performance decrease dramatically.

La microscopie à éclairements structurés(structured illumination microscopy, SIM) permet de dépasser la limite de résolution en microscopie optique due à la diffraction, en éclairant l’objet avec un ensemble de motifs périodiques parfaitement connus. Cependant, il s’avère difficile de contrôler exactement la forme des motifs éclairants. Qui plus est, de fortes distorsions de la grille de lumière peuvent être générées par l’échantillon lui-même dans le volume d’étude, ce qui peut provoquer de forts artefacts dans les images reconstruites. Récemment, des approches dites blind-SIM ont été proposées, où les images sont acquises à partir de motifs d’éclairement inconnus, non-périodiques, de type speckle,bien plus faciles à générer en pratique. Le pouvoir de super résolution de ces méthodes a été observé, sans forcément être bien compris théoriquement. Cette thèse présente deux nouvelles méthodes de reconstruction en microscopie à éclairements structurés inconnus (blind speckle-SIM) : une approche conjointe et une approche marginale. Dans l’approche conjointe, nous estimons conjointement l’objet et les motifs d’éclairement au moyen d’un modèle de type Basis Pursuit DeNoising (BPDN) avec une régularisation en norme lp,q où p=>1 et 0<q<=1. La norme lp,q est introduite afin de prendre en compte une hypothèse de parcimonie sur l’objet. Dans l’approche marginale, nous reconstruisons uniquement l’objet et les motifs d’éclairement sont traités comme des paramètres de nuisance. Notre contribution est double. Premièrement, une analyse théorique démontre que l’exploitation des statistiques d’ordre deux des données permet d’accéder à un facteur de super résolution de deux, lorsque le support de la densité spectrale du speckle correspond au support fréquentiel de la fonction de transfert du microscope. Ensuite, nous abordons le problème du calcul numérique de la solution. Afin de réduire à la fois le coût de calcul et les ressources en mémoire, nous proposons un estimateur marginal à base de patches. L’élément clé de cette méthode à patches est de négliger l’information de corrélation entre les pixels appartenant à différents patches. Des résultats de simulations et en application à des données réelles démontrent la capacité de super résolution de nos méthodes. De plus, celles-ci peuvent être appliquées aussi bien sur des problèmes de reconstruction 2D d’échantillons fins, mais également sur des problèmes d’imagerie 3D d’objets plus épais. / Conventional structured illumination microscopy (SIM) can surpass the resolution limit inoptical microscopy caused by the diffraction effect, through illuminating the object with a set of perfectly known harmonic patterns. However, controlling the illumination patterns is a difficult task. Even worse, strongdistortions of the light grid can be induced by the sample within the investigated volume, which may give rise to strong artifacts in SIM reconstructed images. Recently, blind-SIM strategies were proposed, whereimages are acquired through unknown, non-harmonic,speckle illumination patterns, which are much easier to generate in practice. The super-resolution capacity of such approaches was observed, although it was not well understood theoretically. This thesis presents two new reconstruction methods in SIM using unknown speckle patterns (blind-speckle-SIM): one joint reconstruction approach and one marginal reconstruction approach. In the joint reconstruction approach, we estimate the object and the speckle patterns together by considering a basis pursuit denoising (BPDN) model with lp,q-norm regularization, with p=>1 and 0<q<=1. The lp,q-norm is introduced based on the sparsity assumption of the object. In the marginal approach, we only reconstruct the object, while the unknown speckle patterns are considered as nuisance parameters. Our contribution is two fold. First, a theoretical analysis demonstrates that using the second order statistics of the data, blind-speckle-SIM yields a super-resolution factor of two, provided that the support of the speckle spectral density equals the frequency support of the microscope point spread function. Then, numerical implementation is addressed. In order to reduce the computational burden and the memory requirement of the marginal approach, a patch-based marginal estimator is proposed. The key idea behind the patch-based estimator consists of neglecting the correlation information between pixels from different patches. Simulation results and experiments with real data demonstrate the super-resolution capacity of our methods. Moreover, our proposed methods can not only be applied in 2D super-resolution problems with thin samples, but are also compatible with 3D imaging problems of thick samples.