Efficient and Robust Recovery of Signal and Image in Impulsive Noise via ${\ell }_1-\alpha {\ell }_2$ Minimization

Abstract

In this paper, we consider the efficient and robust reconstruction of signals and images via ${\ell }_1-\alpha {\ell }_2(0<\alpha \le 1)$ minimization in impulsive noise case. To achieve this goal, we introduce two new models: the ${\ell }_1-\alpha {\ell }_2$ minimization with ${\ell }_1$ constraint, which is called ${\ell }_1-\alpha {\ell }_2$-LAD, the ${\ell }_1-\alpha {\ell }_2$ minimization with Dantzig selector constraint, which is called ${\ell }_1-\alpha {\ell }_2$-DS. We first show that sparse signals or nearly sparse signals can be exactly or stably recovered via ${\ell }_1-\alpha {\ell }_2$ minimization under some conditions based on the restricted $1$-isometry property (${\ell }_1$-RIP). Second, for ${\ell }_1-\alpha {\ell }_2$-LAD model, we introduce unconstrained ${\ell }_1-\alpha {\ell }_2$ minimization model denoting ${\ell }_1-\alpha {\ell }_2$-PLAD and propose ${\ell }_1-\alpha {\ell }_2$LA algorithm to solve the ${\ell }_1-\alpha {\ell }_2$-PLAD. Last, numerical experiments %on success rates of sparse signal recovery demonstrate that when the sensing matrix is ill-conditioned (i.e., the coherence of the matrix is larger than 0.99), the ${\ell }_1-\alpha {\ell }_2$LA method is better than the existing convex and non-convex compressed sensing solvers for the recovery of sparse signals. And for the magnetic resonance imaging (MRI) reconstruction with impulsive noise, we show that the ${\ell }_1-\alpha {\ell }_2$LA method has better performance than state-of-the-art methods via numerical experiments.

arXiv admin note: text overlap with arXiv:1703.07952 by other authors