Sparse image restoration using iterated linear expansion of thresholds

We focus on image restoration that consists in regularizing a quadratic data-fidelity term with the standard ℓ₁ sparse-enforcing norm. We propose a novel algorithmic approach to solve this optimization problem. Our idea amounts to approximating the result of the restoration as a linear sum of basic thresholds (e.g. soft-thresholds) weighted by unknown coefficients. The few coefficients of this expansion are obtained by minimizing the equivalent low-dimensional ℓ₁-norm regularized objective function, which can be solved efficiently with standard convex optimization techniques, e.g. iterative reweighted least square (IRLS). By iterating this process, we claim that we reach the global minimum of the objective function. Experimentally we discover that very few iterations are required before we reach the convergence.