On the Systematic Measurement Matrix for Compressed Sensing in the Presence of Gross Errors

Abstract

Inspired by syndrome source coding using linear error-correcting codes, we explore a new form of measurement matrix for compressed sensing. The proposed matrix is constructed in the systematic form [A I], where A is a randomly generated submatrix with elements distributed according to i.i.d. Gaussian, and I is the identity matrix. In the noiseless setting, this systematic construction retains similar property as the conventional Gaussian ensemble achieves. However, in the noisy setting with gross errors of arbitrary magnitude, where Gaussian ensemble fails catastrophically, systematic construction displays strong stability. In this paper, we prove its stable reconstruction property. We further show its l1-norm sparsity recovery property by proving its restricted isometry property (RIP). We also demonstrate how the systematic matrix can be used to design a family of lossy-to-lossless compressed sensing schemes where the number of measurements trades off the reconstruction distortions.