Typical reconstruction limits for distributed compressed sensing based on ℓ2,1-norm minimization and Bayesian optimal reconstruction

Abstract

The distributed compressed sensing framework provides an efficient compression scheme of multichannel signals that are sparse in some domains and highly correlated with one another. In particular, a signal model called the joint sparse model 2 (JSM-2) or multiple measurement vector problem, in which all sparse signals share their support, is important for dealing with practical problems such as magnetic resonance imaging and magnetoencephalography. In this paper, we investigate the typical reconstruction performance of JSM-2 problems for two schemes. One is ℓ2,1-norm minimization reconstruction and the other is Bayesian optimal reconstruction. Employing the replica method, we show that the reconstruction performance of both schemes which exploit the knowledge of the sharing of the signal support overcomes that of their corresponding approaches for the single-channel compressed sensing problem. We also develop a computationally feasible approximate algorithm for performing the Bayes optimal scheme to validate our theoretical estimation. Our replica-based analysis numerically indicates that the spinodal point of the Bayesian reconstruction disappears, which implies that a fundamental reconstruction limit can be achieved by the BP-based approximate algorithm in a practical amount of time when the number of channels is sufficiently large. The results of the numerical experiments of both reconstruction schemes agree excellently with the theoretical evaluation.