Sparse Signal Reconstruction–Introduction

Abstract

Sparse signals are characterized by a few nonzero values in one of their representation domains. The reconstruction of such signals is possible with a reduced set of measurements. The description and basic definitions of sparse signals, along with the conditions for their reconstruction, are discussed in the first part of this article. Among numerous reconstruction algorithms developed for the sparse signals reconstruction, three classes are reviewed. The first one is based on the principle of matching components. Analysis of noise and nonsparsity influence to the reconstruction is done. The second class of reconstruction algorithms is based on the problem reformulation into a constrained convex form when the linear programming optimization and regression methods can be used. The third class of algorithms to obtain the solution is based on the Bayesian approach. The examples of reconstruction using all of the considered approaches are presented, for a common transformation and observation matrices. Pseudo‐codes are provided for all considered reconstruction algorithms.

On the sparsity bound for the existence of a unique solution in compressive sensing by the Gershgorin theorem

“…In compressive sensing we are dealing with a reduced set of signal observations [1,2,3,4,5,6,7,8,9,10,11]. The reduced set of observations can be caused by a desire to acquire a signal with a low number of observations or by physical unavailability to measure the signal at all possible sampling positions and to get a complete set of samples [4,5].…”

“…In compressive sensing we are dealing with a reduced set of signal observations [1,2,3,4,5,6,7,8,9,10,11]. The reduced set of observations can be caused by a desire to acquire a signal with a low number of observations or by physical unavailability to measure the signal at all possible sampling positions and to get a complete set of samples [4,5]. In some applications, signal samples may be heavily corrupted at some arbitrary positions that their omission could be the best approach to their processing, when we are left with a reduced set of signal samples to reconstruct the signal [12,13,14].…”

Aspect Dependent-based Ghost Suppression for Extended Targets in Through-the-wall Radar Imaging under Compressive Sensing Framework

“…where φ 0 i is obtained from the measurement matrix, φ 0 , after removing columns with zero valued coefficients [35] such that…”

A Hybrid Orthogonal Forward-Backward Pursuit Algorithm for Partial Fourier Multiple Measurement Vectors Problem

“…However, it is also weak in distinguishing adjoining atoms. As stated in [24], the Bayesian-based CS reconstruction algorithms run in the opposite direction to GP algorithms.…”