Robust methods for sensing and reconstructing sparse signals

Compressed sensing (CS) is an emerging signal acquisition framework that goes against the traditional Nyquist sampling paradigm. CS demonstrates that a sparse, or compressible, signal can be acquired using a low rate acquisition process. Since noise is always present in practical data acquisition systems, sensing and reconstruction methods are developed assuming a Gaussian (light-tailed) model for the corrupting noise. However, when the underlying signal and/or the measurements are corrupted by impulsive noise, commonly employed linear sampling operators, coupled with Gaussian-derived reconstruction algorithms, fail to recover a close approximation of the signal. This dissertation develops robust sampling and reconstruction methods for sparse signals in the presence of impulsive noise. To achieve this objective, we make use of robust statistics theory to develop appropriate methods addressing the problem of impulsive noise in CS systems. We develop a generalized Cauchy distribution (GCD) based theoretical approach that allows challenging problems to be formulated in a robust fashion. Robust sampling operators, together with robust reconstruction strategies are developed using the introduced GCD framework. To solve the problem of impulsive noise embedded in the underlying signal prior the measurement process, we propose a robust nonlinear measurement operator based on the weighed myriad estimator. To recover sparse signals from impulsive noise introduced in the measurement process, a geometric optimization problem based on L1 minimization employing a Lorentzian norm constraint on the