A Method of Reweighting the Sensing Matrix for Compressed Sensing

Abstract

In compressed sensing, a small enough restricted isometry constant (RIC) of the sensing matrix satisfying the restricted isometry property (RIP) is the powerful guarantee on the precise reconstruction of a sparse discrete signal. Under a certain condition, the RIC can be improved by weighting the sensing matrix so that its all nonzero singular values become the same, i.e., the condition number of the new sensing matrix is 1. In this article, by multiplying the linear system A <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> Ax = A <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> y by a matrix related to A <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> A and repeating the process many times, we propose a method of reweighting the sensing matrix to improve the condition number of sensing matrix so as to improve its RIC, prove that the condition number tends to 1 as the weighting times approaches infinity monotonically, and then obtain an RIC improvement model equivalent to the original CS model. For the improvement model, we use the algorithms of orthogonal matching pursuit, iterative hard thresholding and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> -regularisation to recover sparse signals, and verify the superiority of the proposed algorithms by using different existing sensing matrices for CS experiments.