A Survey of Compressive Sensing and Sparse Recovery - Barros, Benjamin (ProQuest Dissertations & Theses, 2018)
This thesis is expository in nature and provides an introductory overview of compressive sensing. Compressive sensing is a recently developed field of mathematics that leverages the fact that solutions are sparse, i.e., most of the coordinates are zero, in order to solve underdetermined linear systems. The standard problem in compressive sensing can be described as recovering a sparse vector x from the matrix equation y = Ax where y is the measurement vector and A is an m by N measurement matrix with m less than N. This thesis discusses a handful of sparse recovery algorithms along with sufficient conditions for their success. The first chapter introduces the basic idea of compressive sensing and establishes a preliminary result for the recovery of sparse vectors using a brute force search. The second chapter gives the definitions of the null space property, exact recovery condition, and coherence which are used to state the sufficient conditions for the success of the sparse recovery algorithms. The third chapter introduces the basis pursuit algorithm which performs sparse recovery by finding the L1-minimizer subject to the equation y = Ax. Linear programming using the simplex method is then presented as an implementation of basis pursuit. The fourth chapter examines a greedy algorithm called orthogonal matching pursuit (OMP). It will be shown that the number of iterations needed to recover a sparse vector using OMP is equal to its number of nonzero coordinates. The fifth chapter explains two Fourier-based sparse recovery methods that use the discrete Fourier transform (DFT) and the discrete cosine transform (DCT). The DFT method is an example of a sparse recovery algorithm in the complex numbers. The DCT method is a new adaptation of the DFT method in the real setting.