A New Approach for Sparse Signal Recovery in Compressed Sensing Based on Minimizing Composite Trigonometric Function

Abstract

Accurate signal recovery from an underdetermined system of linear equation (USLE) is a topic of considerable interest; such as compressed sensing (CS), recovery of low-rank matrix, blind source separation, and related fields. In order to improve the accuracy of signal recovery from an USLE in CS, we develop a new algorithm called composite trigonometric function null-space re-weighted approximate ℓ0-norm (CTNRAL0). The proposed algorithm deploys composite trigonometric function as a nonconvex penalty for sparsity which can better approximate ℓ0-norm and can yield more accurate solution. To solve nonconvex minimization formulation efficiently, we adopt a gradual nonconvexity method. In addition, the null space measurement matrix is applied in the CTNRAL0 algorithm, which reduces the dimension of the matrix. The proposed algorithm has been compared with the smoothed ℓ0-norm and null-space re-weighted approximate ℓ0-norm algorithm via numerical simulations to show its improved performance in the noise environment, while computation cost required is comparable. Furthermore, compared with the interior-point LP solvers and iteratively re-weighted least squares algorithm, the proposed algorithm computation cost can reduce by 1 or 2 orders of magnitude.

Keywords

• Composite trigonometric function
• iteratively re-weighted method
• null-space of the measurement matrix
• quasi-Newton algorithm
• smooth approximations of non-smooth functions
• unconstrained convex optimization