Algorithms for imaging inverse problems under sparsity regularization

This paper reviews our recent work on the application of a class of techniques known as ADMM (alternating direction method of multipliers, which belongs to the family of augmented Lagrangian methods) to several imaging inverse problems under sparsity-inducing regularization. After reviewing ADMM, a formulation that allows handling problems with more than two terms is introduced; this formulation is then applied to a variety of problems, namely: standard image restoration/reconstruction from linear observations (e.g., compressive sensing, deconvolution, inpainting) with Gaussian or Poisson noise, using either analysis or synthesis regularization, and unconstrained or constrained optimization. We also show how the proposed framework can be used to address hybrid analysis-synthesis regularization. In all these cases, the proposed approach inherits the theoretic convergence guarantees of ADMM and achieve state-of-the-art speed.