Generalized approximate message passing for cosparse analysis compressive sensing

In cosparse analysis compressive sensing (CS), one seeks to estimate a non-sparse signal vector from noisy sub-Nyquist linear measurements by exploiting the knowledge that a given linear transform of the signal is cosparse, i.e., has sufficiently many zeros. We propose a novel approach to cosparse analysis CS based on the generalized approximate message passing (GAMP) algorithm. Unlike other AMP-based approaches to this problem, ours works with a wide range of analysis operators and regularizers. In addition, we propose a novel ℓ₀-like soft-thresholder based on MMSE denoising for a spike-and-slab distribution with an infinite-variance slab. Numerical demonstrations on synthetic and practical datasets demonstrate advantages over existing AMP-based, greedy, and reweighted-ℓ₁ approaches.

1. Introduction

We consider the problem of recovering a signal (e.g., an -pixel image) from the possibly noisy linear measurements $$\begin{equation*} y=\Phi x+w\in \mathbb{R}^{M}, \tag{1} \end{equation*}$$ where represents a known linear measurement operator represents noise, and . We focus on the analysis compressive sensing (CS) problem [1], [2] where, for a given analysis operator , $$\begin{equation*} u\triangleq\Omega x\in \mathbb{R}^{D} \tag{2} \end{equation*}$$ is assumed to be cosparse (i.e., contain sufficiently many zero-valued coefficients). This differs from the synthesis CS problem, where itself is assumed to be sparse (i.e., contain sufficiently few non-zero coefficients). Although the two problems become interchangeable when is invertible, we are mainly interested in non-invertible , e.g., the “overcomplete” case where . For notational simplicity, we assume real-valued quantities in the sequel. However, our methods are easily generalized to complex-valued quantities, as we demonstrate in the numerical experiments.