Projection matrix optimization for block-sparse compressive sensing

Traditionally, the projection matrix in compressive sensing (CS) is chosen as a random matrix. In recent years, we have seen that the performance of CS systems can be improved by using a carefully designed projection matrix rather than a random one. In particular, we can reduce the coherence between the columns of the equivalent dictionary thanks to a well-designed projection matrix. Then, we can get a lower reconstruction error and a higher successful reconstruction rate. In some applications, the signals of interest have nonzero entries occurring in clusters - i.e., block-sparse signals. In this paper, we use the equiangular tight frame (ETF) to approach the Gram matrix of equivalent dictionary rather than the identity matrix used in [1]. Then, we minimize a weighted sum of the subblock coherence and the interblock coherence of the equivalent dictionary. The simulation results show that our novel method for projection matrix optimization significantly improves the ability of block-sparse approximation techniques to reconstruct and classify signals than the method proposed by Lihi Zelnik-Manor (LZM) [1].

I. Introduction

CS declares that sparse signals can be exactly recovered from a number of linear projections of dimension lower than the number of samples required by Shannon-Nyquist Theorem [2]. Recent work has demonstrated that the performance of CS can be improved by using a carefully designed projection matrix rather than a random one. The goal of projection matrix optimization is to construct a projection matrix which can improve the recovery ability for a given sparsifying dictionary. In other words, the projection matrix is optimized to improve the ability of sparse apporximation algorithms such as BP and OMP to recover the sparsest representation from an underdetermined system.