Contents I. Introduction
The objective of image denoising is to estimate the original image from the corrupt image , where is the number of pixels in the image. The assumption is that is Additive White Gaussian Noise (AWGN). Recently proposed image denoising framework via sparse and redundant representations using learned dictionary has gained popularity among the researchers, and motivate many extensions [1]. The prior imposed in this denoising framework is that the signal of interest will have a sparse representation on a dictionary containing signal prototypes (or atoms), whereas the additive noise can not have sparse representation in any dictionary. Thus it extracts small overlapping image blocks of size from , where is a square (or any other appropriate block size), and estimates the sparse representation for each image block on a dictionary . That is $${\forall _{ij}}{\alpha _{ij}} = \arg \mathop {\min }\limits_\alpha \Vert \alpha {\Vert _0}\quad \hbox{such that}\quad \Vert {{\bf y}_{ij}} - {\bf D}\alpha\Vert _2^2 \leq {\epsilon ^2},$$where the estimation error bound depends on the noise variance . is an matrix that extracts a block from the columnized image starting from its 2D coordinate (). In order to reconstruct the global image, these individual sparse representations are aggregated as a closed form solution of a MAP formulation, $${\mathhat{\bf X}} = {\left({\lambda {\bf I} + \sum\limits_{ij} {\bf R}_{ij}^T{{\bf R}_{ij}}}\right)^{- 1}}\left({\lambda {\bf Y} + \sum\limits_{ij} {\bf R}_{ij}^T{\bf D}{\alpha _{ij}}} \right).$$For the details of the image denoising framework, we suggest the reader to refer to [1].
