C. Studer and R. G. Baraniuk, "Stable Restoration and Separation of Approximately Sparse Signals," preprint, 2011.
Abstract: This paper develops new theory and algorithms to recover signals that are approximately sparse in some general (i.e., basis, frame, over-complete, or incomplete)
dictionary but corrupted by a combination of measurement noise and interference having a sparse representation in a second general dictionary. Particular applications covered by our framework include the restoration of signals impaired by impulse noise, narrowband interference, or saturation, as well as image in-painting, super-resolution, and signal separation. We develop efficient recovery algorithms and deterministic conditions that guarantee stable restoration and separation. Two application examples demonstrate
the efficacy of our approach.
An example from the paper (Figure 3) showing the restoration of a scratched photograph. Clockwise from upper left: original image; scratched image; blind scratch removal that has no access to the scratch locations; scratch removal that has access to the scratch locations.