In this article, we propose a novel variational model for restoring images in the presence of the mixture of Cauchy and Gaussian noise. The model involves a novel data-fidelity term that features the mixed noise as an infimal convolution of two noise distributions and total variation regularization. This data-fidelity term contributes to suitable separation of Cauchy noise and Gaussian noise components, facilitating simultaneous removal of the mixed noise. Besides, the total variation regularization enables adequate denoising in homogeneous regions while conserving edges. Despite the nonconvexity of the model, the existence of a solution is proven. By employing an alternating minimization approach and the alternating direction method of multipliers, we present an iterative algorithm for solving the proposed model. Experimental results validate the effectiveness of the proposed model compared to other existing models according to both visual quality and some image quality measurements.

In this article, we propose a novel group-based sparse representation (GSR) model for restoring color images in the presence of multiplicative noise. This model consists of a convex data-fidelity term, and two regularizations including GSR and saturation-value-based total variation (SVTV). The data-fidelity term is suitable for handling heavy multiplicative noise. GSR enables the retention of textures and details while sufficiently removing noise in smooth regions without producing the staircase artifacts engendered by total variation-based models. Furthermore, we introduce a multi-color channel-based GSR that involves coupling between three color channels. This avoids the generation of color artifacts caused by decoupled color channel-based methods. SVTV further improves the visual quality of restored images by diminishing certain artifacts induced by patch-based methods. To solve the proposed nonconvex model and its subproblem, we exploit the alternating direction method of multipliers, which contributes to an efficient iterative algorithm. Numerical results demonstrate the outstanding performance of the proposed model compared to other existing models regarding visual aspect and image quality evaluation values.

Color image denoising under the simultaneous presence of multiplicative and Gaussian noise is challenging due to the differing statistical properties of the two noise types. We propose a variational framework that integrates an infimal-convolution-based data-fidelity term with saturation-value total variation (SVTV) and group-based sparse representation (GSR) regularization. By explicitly decoupling the multiplicative and Gaussian noise components, the data-fidelity term enables effective suppression of mixed noise. The two regularizers play complementary roles: SVTV promotes piecewise-smooth reconstructions while preserving edges, whereas GSR enhances fine details and textures and mitigates the staircase artifacts induced by SVTV. The resulting nonconvex optimization problem is addressed using a proximal alternating minimization strategy, with the alternating direction method of multipliers employed to efficiently solve the subproblems. A convergence analysis of the proposed algorithm is provided. Numerical experiments demonstrate that the proposed method consistently outperforms existing approaches for denoising color images corrupted by mixed multiplicative and Gaussian noise.