ℓ ₁ - αℓ ₂ minimization methods for signal and image reconstruction with impulsive noise removal

In this paper, we study ℓ ₁ - αℓ ₂ (0 < α 1) minimization methods for signal and image reconstruction with impulsive noise removal. The data fitting term is based on ℓ ₁ fidelity between the reconstruction output and the observational data, and the regularization term is based on ℓ ₁ - αℓ 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.