Sparse Representation-based Image Deconvolution by Iterative Thresholding

Image deconvolution algorithms with overcomplete sparse representations and fast iterative thresholding methods are presented. The image to be recovered is assumed to be sparsely represented in a redundant dictionary of transforms. These transforms are chosen to offer a wider range of generating atoms; allowing more flexibility in image representation and adaptativity to its morphological content. The deconvolution inverse problem is formulated as the minimization of an energy functional with a sparsity-promoting regularization (e.g. l1 norm of the image representation coefficients). As opposed to quadratic programming solvers based on the interior point method, here, recent advances in fast solution algorithms of such problems, i.e. Stagewise Iterative Thresholding, are exploited to solve the optimization problem and provide fast and good image recovery results. Some theoretical aspects as well as computational and practical issues are investigated. Illustrations are provided for potential applicability of the method to astronomical data.