In this paper, we study 1 − α 2 (0 < α 1) minimization methods for signal and image reconstruction with impulsive noise removal. The data fitting term is based on 1 fidelity between the reconstruction output and the observational data, and the regularization term is based on 1 − α 2 nonconvex minimization of the reconstruction output or its total variation. Theoretically, we show that under the generalized restricted isometry property that the underlying signal or image can be recovered exactly. Numerical algorithms are also developed to solve the resulting optimization problems. Experimental results have shown that the proposed models and algorithms can recover signal or images under impulsive noise degradation, and their performance is better than that of the existing methods.

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We consider the block orthogonal multi-matching pursuit (BOMMP) algorithm for the recovery of block sparse signals. A sharp bound is obtained for the exact reconstruction of block K-sparse signals via the BOMMP algorithm in the noiseless case, based on the block restricted isometry constant (block-RIC). Moreover, we show that the sharp bound combining with an extra condition on the minimum ℓ 2 norm of nonzero blocks of block K−sparse signals is sufficient to recover the true support of block K-sparse signals by the BOMMP in the noise case. The significance of the results we obtain in this paper lies in the fact that making explicit use of block sparsity of block sparse signals can achieve better recovery performance than ignoring the additional structure in the problem as being in the conventional sense.Keywords: Compressed sensing, block sparse signal, block restricted isometry property, block orthogonal multi-matching pursuit.Mathematics Subject Classification (2010) 65D15, 65J22, 68W40 * W. Chen is with

The Orthogonal Least Squares (OLS) algorithm is an efficient sparse recovery algorithm that has received much attention in recent years. On one hand, this paper considers that the OLS algorithm recovers the supports of sparse signals in the noisy case. We show that the OLS algorithm exactly recovers the support of $K$-sparse signal $\boldsymbol{x}$ from $\boldsymbol{y}=\boldsymbol{\unicode[STIX]{x1D6F7}}\boldsymbol{x}+\boldsymbol{e}$ in $K$ iterations, provided that the sensing matrix $\boldsymbol{\unicode[STIX]{x1D6F7}}$ satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\unicode[STIX]{x1D6FF}_{K+1}<1/\sqrt{K+1}$, and the minimum magnitude of the nonzero elements of $\boldsymbol{x}$ satisfies some constraint. On the other hand, this paper demonstrates that the OLS algorithm exactly recovers the support of the best $K$-term approximation of an almost sparse signal $\boldsymbol{x}$ in the general perturbations case, which means both $\boldsymbol{y}$ and $\boldsymbol{\unicode[STIX]{x1D6F7}}$ are perturbed. We show that the support of the best $K$-term approximation of $\boldsymbol{x}$ can be recovered under reasonable conditions based on the restricted isometry property (RIP).

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