  This paper develops new theory and algorithms to recover signals that are
approximately sparse in some general dictionary (i.e., a basis, frame, or
over-/incomplete matrix) but corrupted by a combination of interference having
a sparse representation in a second general dictionary and measurement noise.
The algorithms and analytical recovery conditions consider varying degrees of
signal and interference support-set knowledge. Particular applications covered
by the proposed framework include the restoration of signals impaired by
impulse noise, narrowband interference, or saturation/clipping, as well as
image in-painting, super-resolution, and signal separation. Two application
examples for audio and image restoration demonstrate the efficacy of the
approach.
