Robust compressive sensing techniques

Abstract

Compressive Sensing theory details how a sparsely represented signal in a known basis can be reconstructed from an underdetermined linear measurements. However, in reality there is a mismatch between the assumed and the actual dictionary due to factors such as discretization of the parameter space defining basis components, sampling jitter in A/D conversion, and model errors. Due to this mismatch, a signal may not be sparse in the assumed basis, which causes signifi- cant performance degradation in sparse reconstruction algorithms. To eliminate the mismatch problem, this thesis presents two novel robust algorithm and an adaptive discretization framework that can obtain successful sparse representations. In the proposed techniques, the selected dictionary atoms are perturbed towards directions to decrease the orthogonal residual norm. The first algorithm named as Parameter Perturbed Orthogonal Matching Pursuit (PPOMP) targets the off-grid problem and the parameters of the selected dictionary atoms are perturbed. The second algorithm named as Perturbed Orthogonal Matching Pursuit (POMP) targets the unstructured basis mismatch problem and performs controlled rotation based perturbation of selected dictionary atoms. Based on detailed mathematical analysis, conditions for successful reconstruction are derived. Simulations show that robust results with much smaller reconstruction errors in the case of both parametric and unstructured basis mismatch problem can be obtained as compared to standard sparse reconstruction techniques. Different from the proposed perturbation approaches, the proposed adaptive framework discretizes the continuous parameter space depending on the estimated sparsity level. Once a provisional solution is obtained with a sparse solver, the framework recursively splits the problem into sparser sub-problems so that each sub-problem is exposed to less severe off-grid problem. In the presented recursive framework, any sparse reconstruction technique can be used. As illustrated over commonly used applications, the error in the estimated parameters of sparse signal components almost achieve the Cram´er-Rao lower bound in the proposed framework.