Restoration of images corrupted by mixed Gaussian-impulse noise by iterative soft-hard thresholding

We address the problem of restoration of images which have been affected by impulse or a combination of impulse and Gaussian noise. We propose a patch-based approach that exploits approximate sparse representations of image patches in learned dictionaries. For every patch, sparse representation in a dictionary is enforced by ℓ
₁
-norm penalty, and sparsity of the residual is enforced by ℓ
₀
-quasi-norm penalty. The obtained non-convex problem is solved iteratively by a combination of soft and hard thresholding, and a proof of convergence to a local minimum is given. Experimental evaluation suggests that the proposed approach can produce state-of-the-art results for some types of images, especially in terms of the structural similarity (SSIM) measure. In addition, the proposed iterative thresholding algorithm could possibly be applied to general inverse problems.