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The most common form of a prior knowledge is the well-known smoothness based prior [2] which assumes that the image contents generally do not change abruptly in a neighborhood of space. Each pixel in the image is constrained to be relevant to its neighboring pixels. It is a generic image contextual constrain. These methods considering the image smoothness take advantage of the local structure of an image, leaving out the overall structure information of the image. Some non-local methods [3, 4, 5]thus have been widely studied. The non-local method assumes that some feature of an image does always repeat in different positions in the image. These methods always seek the similar image patches or exemplars meeting some specified measurements in the whole image space, which means they utilize the information of the entire image. In this paper, we propose a new form of prior for solving general inverse problems in image processing which combines the local total variation regularization with graph-based non-local variation regularization. This graph depicts the relationship among different image patches. This prior makes use of both local and non-local image information to get a better result. Because of the existing of non-local graph-based variation regularization, the problem to be solved in this paper turns to be non-convex. We, therefore, propose an iterative algorithm to work out the inverse problem using such a form of prior, which iteratively updates the image and the weight graph calculated from the current image till convergence. We also show the results when using this new framework in image denoising, image inpainting and image deblurring applications.

A Graph-based Iterative Algorithm for Inverse Problems with Local and Non-local Regularizations