A fast patch-dictionary method for the whole image recovery,” UCLA

Abstract. Various algorithms have been proposed for dictionary learning. Among those for image processing, many use image patches to form dictionar-ies. This paper focuses on whole-image recovery from corrupted linear mea-surements. We address the open issue of representing an image by overlapping patches: the overlapping leads to an excessive number of dictionary coefficients to determine. With very few exceptions, this issue has limited the applications of image-patch methods to the “local ” kind of tasks such as denoising, inpaint-ing, cartoon-texture decomposition, super-resolution, and image deblurring, for which one can process a few patches at a time. Our focus is global imaging tasks such as compressive sensing and medical image recovery, where the whole image is encoded together, making it either impossible or very ineffective to update a few patches at a time. Our strategy is to divide the sparse recovery into multiple subproblems, each of which handles a subset of non-overlapping patches, and then the results of the subproblems are averaged to yield the final recovery. This simple strategy is surprisingly effective in terms of both quality and speed. In addition, we accelerate computation of the learned dictionary by apply-ing a recent block proximal-gradient method, which not only has a lower per-iteration complexity but also takes fewer iterations to converge, compared to the current state-of-the-art. We also establish that our algorithm globally con-verges to a stationary point. Numerical results on synthetic data demonstrate that our algorithm can recover a more faithful dictionary than two state-of-the-art methods. Combining our whole-image recovery and dictionary-learning methods, we numerically simulate image inpainting, compressive sensing recovery, and de-blurring. Our recovery is more faithful than those of a total variation method and a method based on overlapping patches. Our matlab code is competitive in terms of both speed and quality. 1. Introduction. Ou