L₁ -βL_q Minimization for Signal and Image Recovery

The nonconvex optimization method has attracted increasing attention due to its excellent ability of promoting sparsity in signal processing, image restoration, and machine learning. In this paper, we consider a new minimization method L₁ -β L_q ((β, q)∈[0, 1]×[1, ∞)\(1, 1)) and its applications in signal recovery and image reconstruction because L₁ -β L_q minimization provides an effective way to solve the q-ratio sparsity minimization model. Our main contributions are to establish a convex hull decomposition for L₁ -β L_q and investigate RIP-based conditions for stable signal recovery and image reconstruction by L₁ -β L_q minimization. For one-dimensional signal recovery, our derived RIP condition extends existing results. For two-dimensional image recovery under L₁ -β L_q minimization of image gradients, we provide the error estimate of the resulting optimal solutions in terms of sparsity and noise level, which is missing in the literature. Numerical results of the limited angle problem in computed tomography imaging and image deblurring are presented to validate the efficiency and superiority of the proposed minimization method among the state-of-art image recovery methods.