Nonconvex-TV Based Image Restoration with Impulse Noise Removal

In this paper, we propose and study a nonconvex data fitting term and a total variation regularization term for image restoration with impulse noise removal. The proposed model is different from existing image restoration models where the data fitting term is based on the ℓ₁- or ℓ₂- norm, and the regularization term is based on the total variation, ℓ₁-norm, or some nonconvex functions. Theoretically, we analyze the properties of minimizers of the proposed objective function with the nonconvex data fitting term and the total variation regularization term. We show that minimizers can preserve piecewise constant regions or match with the data points perfectly. This property is particularly useful for impulse noise removal. The proposed image restoration model can be solved by the proximal linearized minimization algorithm, and the global convergence of the iterative algorithm can also be established according to Kurdyka-Łojasiewicz property. The performance of the proposed model is tested for image restoration with salt-and-pepper impulse noise or random-valued impulse noise. We demonstrate that the restored images by the proposed Nonconvex-TV model are better (in terms of PSNR and visual quality) than those by the other existing data fitting plus regularization models, including ℓ₁ plus total variation (L1TV) and ℓ₁ plus nonconvex (L1Nonconvex) methods.