One-bit compressed sensing via total variation minimization method

Abstract

The goal of standard one-bit compressed sensing (1-bit CS) is to recover sparse signals with high fidelity from binary measurements that retain only their sign information. Besides sparsity, there are numerous signals consisting of other structures, such as signals consisting of piecewise constants (i.e., its gradient is sparse). This paper aims to address the recovery of such signals from one-bit measurements. Motivated by the superior performance of total variation (TV) minimization in conventional CS methods, this paper proposes the TV minimization in the 1-bit CS case. The proposed approaches can recover the direction of gradient sparse signals from ordinary one-bit measurements and the magnitude of gradient sparse signals from the thresholded one-bit measurements. We theoretically provide the upper bounds on the recovery errors, ensuring the effectiveness of the proposed methods. In practice, three algorithms: hard thresholding taut-string (HTTS), 1-bit total variation ADMM (TV-ADMM) and Second-Order Cone Programming (SOCP) are proposed to solve the proposed models. The promising performance of the new approach is supported by a series of simulated and real data examples.