Fenchel duality based dictionary learning for restoration of noisy images

Dictionary learning based sparse modeling has been increasingly recognized as providing high performance in the restoration of noisy images. Although a number of dictionary learning algorithms have been developed, most of them attack this learning problem in its primal form, with little effort being devoted to exploring the advantage of solving this problem in a dual space. In this paper, a novel Fenchel duality based dictionary learning (FD-DL) algorithm has been proposed for the restoration of noise-corrupted images. With the restricted attention to the additive white Gaussian noise, the sparse image representation is formulated as an \ell-{2}\hbox{-}\ell₁ minimization problem, whose dual formulation is constructed using a generalization of Fenchel's duality theorem and solved under the augmented Lagrangian framework. The proposed algorithm has been compared with four state-of-the-art algorithms, including the local pixel grouping-principal component analysis, method of optimal directions, K-singular value decomposition, and beta process factor analysis, on grayscale natural images. Our results demonstrate that the FD-DL algorithm can effectively improve the image quality and its noisy image restoration ability is comparable or even superior to the abilities of the other four widely-used algorithms.