Looking beyond Pixels - Theory, Algorithms and Applications of Continuous Sparse Recovery

Sparse recovery is a powerful tool that plays a central role in many applications, including source estimation in radio astronomy, direction of arrival estimation in acoustics or radar, superresolution microscopy, and X-ray crystallography. Conventional approaches usually resort to discretization, where the sparse signals are estimated on a pre-defined grid. However, the sparse signals do not line up conveniently on any grid in reality. While the discrete setup usually leads to a simple optimization problem that can be solved with standard tools, e.g., 1-based sparse recovery, there are two noticeable drawbacks as a consequence of discretization: (i) Because of the model mismatch, the effective noise level in the measurements is increased; (ii) The minimum resolution a discrete algorithm can reach is limited by the artificially imposed grid step-size, which may not be sufficient in an imaging system, e.g., microscopy. Because of the limitations of the conventional grid-based methods, it is essential to develop a technique that estimates sparse signals in the continuous-domain—in essence looking beyond pixels. The aims of this thesis are (i) to further develop a continuous-domain sparse recovery framework based on finite rate of innovation (FRI) sampling on both theoretical and algorithmic aspects; (ii) adapt the proposed generic sparse recovery technique to several applications, namely radio astronomy point source estimation, direction of arrival estimation in acoustics, and single image up-sampling; (iii) show that the continuous-domain sparse recovery approach can surpass the instrument resolution limit and achieve super-resolution. We propose a continuous-domain sparse recovery technique by generalizing the FRI sampling framework to cases with non-uniform measurements. We achieve this by identifying a set of unknown uniform sinusoidal samples (which are related to the sparse signal parameters to be estimated) and the linear transformation that links the uniform samples of sinusoids to the measurements. It is shown that the continuous-domain sparsity constraint can be equivalently enforced with a discrete convolution equation of these sinusoidal samples. Then, the sparse signal is reconstructed by minimizing the fitting error between the given and the re-synthesized measurements (based on the estimated sparse signal parameters) subject to the sparsity constraint. Further, we develop a multi-dimensional sampling framework for Diracs in two or higher dimensions with linear sample complexity. This is a significant improvement over previous methods, which have a complexity that increases exponentially with space dimension. An efficient algorithm has been proposed to find a valid solution to the continuous-domain sparse recovery problem such that the reconstruction (i) satisfies the sparsity constraint; and (ii) fits the given measurements (up to the noise level). We validate the flexibility and robustness of the FRI-based continuous-domain sparse recovery in both simulations and experiments with real data. Specifically, we show that the proposed method surpasses the diffraction limit of radio telescopes, which is inversely proportional to the maximum baseline, with both realistic simulation and real data from the LOFAR radio telescope. In addition, FRI-based sparse reconstruction requires fewer measurements and smaller baselines to reach a similar reconstruction quality compared with conventional methods. Next, we apply