Fast approximate L0-norm deconvolution using structured wavelet domain priors

In this paper, we propose a new algorithm to solve linear inverse problems using an approximate l₀ penalty on overlapped groups of wavelet coefficients, and apply this to the deconvolution problem specifically. Prior work has shown the improvements gained from using group-sparse penalties over coefficient-sparse penalties for deconvolution and compressed sensing. Instead of minimizing an l₁-norm, as done in prior work, we instead design an overlapping group prior which utilizes the Gaussian scale mixture model, and use this to promote l₀ sparsity. Using a Bayesian argument, we derive a novel convex penalty function, which is a reweighted l₂ approximation to the l₀-norm that can be efficiently minimized. We show that the new group-sparse algorithm produces superior deconvolution results compared to the same algorithm utilizing an unstructured coefficient-sparse penalty.