Removal of Mixed Gaussian and Impulse Noise Using Directional Tensor Product Complex Tight Framelets

In this paper, we propose a frame-based iterative algorithm to restore images which are corrupted by mixed Gaussian and impulse noise, under the assumption that the image region corrupted by impulse noise is unknown. The removal of mixed Gaussian and impulse noise by our proposed algorithm is split into two subproblems which are solved alternatively and iteratively. With an initial guessed region of location for impulse noise, the first subproblem is to inpaint a corrupted image by solving a frame-based convex minimization scheme using the balanced approach, where sparse and redundant directional representations play a key role. Motivated by our recent work on frame-based image denoising and image inpainting, we shall employ the tight frame generated from the directional tensor product complex tight framelets in our balanced approach to remove the mixed Gaussian and impulse noise. Such tensor product complex tight framelets provide sparse directional representations for natural images and can capture the cartoon and texture parts of images very well. The second subproblem is to estimate the image region of locations where the pixels are corrupted by impulse noise. We solve the second subproblem using an $$l_0$$l0-minimization scheme. We consider both salt-and-pepper impulse noise and random-valued impulse noise. Numerical experiments show that our proposed algorithm compares favorably or often outperforms three well-known recent image-restoration methods employed for removing the mixed Gaussian and impulse noise.