Weighted Schatten $p$-Norm Minimization for Image Denoising with Local and Nonlocal Regularization

This paper presents a patch-wise low-rank based image denoising method with constrained variational model involving local and nonlocal regularization. On one hand, recent patch-wise methods can be represented as a low-rank matrix approximation problem whose convex relaxation usually depends on nuclear norm minimization (NNM). Here, we extend the NNM to the nonconvex schatten p-norm minimization with additional weights assigned to different singular values, which is referred to as the Weighted Schatten p-Norm Minimization (WSNM). An efficient algorithm is also proposed to solve the WSNM problem. The proposed WSNM not only gives better approximation to the original low-rank assumption, but also considers physical meanings of different data components. On the other hand, due to the naive aggregation schema which integrates all the denoised patches into a whole image, current patch-wise denoising methods always produce various degree of artifacts in denoised results. Therefore, to further reduce artifacts, a data-driven regularizer called Steering Total Variation (STV) combined with nonlocal TV is derived for a variational model, which imposes local and nonlocal consistency constraints on the patch-wise denoised image. A highly simple but efficient algorithm is proposed to solve this variational model with convergence guarantee. Both WSNM and local & nonlocal consistent regularization are integrated into an iterative restoration framework to produce final results. Extensive experimental testing shows, both qualitatively and quantitatively, that the proposed method can effectively remove noise, as well as reduce artifacts compared with state-of-the-art methods.