Block Matching Local SVD Operator Based Sparsity and TV Regularization for Image Denoising

Abstract

We propose a denoising method by integrating group sparsity and TV regularization based on self-similarity of the image blocks. By using the block matching technique, we introduce some local SVD operators to get a good sparsity representation for the groups of the image blocks. The sparsity regularization and TV are unified in a variational problem and each of the subproblems can be efficiently optimized by splitting schemes. The proposed algorithm mainly contains the following four steps: block matching, basis vectors updating, sparsity regularization and TV smoothing. The self-similarity information of the image is assembled by the block matching step. By concatenating all columns of the similar image block together, we get redundancy matrices whose column vectors are highly correlated and should have sparse coefficients after a proper transformation. In contrast with many transformation based denoising methods such as BM3D with fixed basis vectors, we update local basis vectors derived from the SVD to enforce the sparsity representation. This step is equivalent to a dictionary learning procedure. With the sparsity regularization step, one can remove the noise efficiently and keep the texture well. The TV regularization step can help us to reduced the artifacts caused by the image block stacking. Besides, we mathematically show the convergence of the algorithms when the proposed model is convex (with $p=1$) and the bases are fixed. This implies the iteration adopted in BM3D is converged, which was not mathematically shown in the BM3D method. Numerical experiments show that the proposed method is very competitive and outperforms state-of-the-art denoising methods such as BM3D.