Sparse Image Deconvolution with Message Passing

We introduce an approximate message passing (AMP) algorithm for the problem of image deconvolution. The recovery problem is formulated in Bayesian terms, and uses sparse statistical priors for estimating the minimum-mean-squared-error solution. Our setting differs from previous investigations where AMP was considered for sparse signal recovery from random or Fourier measurements. AMP is incompatible with the original structure of the system matrices involved in deconvolution problems. We propose to recast the problem into a more suitable form, exploiting the fact that the convolution operator is diagonalized in the Fourier domain. This provides a remarkably effective deconvolution technique, which achieves significant improvement over `1based methods, both in speed and quality.