Image Restoration via Simultaneous Sparse Coding: Where Structured Sparsity Meets Gaussian Scale Mixture

In image processing, sparse coding has been known to be relevant to both variational and Bayesian approaches. The regularization parameter in variational image restoration is intrinsically connected with the shape parameter of sparse coefficients’ distribution in Bayesian methods. How to set those parameters in a principled yet spatially adaptive fashion turns out to be a challenging problem especially for the class of nonlocal image models. In this work, we propose a structured sparse coding framework to address this issue—more specifically, a nonlocal extension of Gaussian scale mixture (GSM) model is developed using simultaneous sparse coding (SSC) and its applications into image restoration are explored. It is shown that the variances of sparse coefficients (the field of scalar multipliers of Gaussians)—if treated as a latent variable—can be jointly estimated along with the unknown sparse coefficients via the method of alternating optimization. When applied to image restoration, our experimental results have shown that the proposed SSC–GSM technique can both preserve the sharpness of edges and suppress undesirable artifacts. Thanks to its capability of achieving a better spatial adaptation, SSC–GSM based image restoration often delivers reconstructed images with higher subjective/objective qualities than other competing approaches.