Multigrid methods for total variation
Felipe Guerra, Tuomo Valkonen
TL;DR
The paper develops a two-grid forward-backward splitting framework to accelerate nonsmooth convex optimization with TV regularization by alternating fine- and coarse-grid updates. A key contribution is the nonsmooth coherence condition that guides the construction of coarse-grid problems from the fine-grid subgradient information, enabling descent directions on the fine grid. The authors derive a detailed dual formulation for TV and provide explicit constructions of the coarse proximal operator and the Fenchel conjugate for complex data terms, including MRI, as well as convergence guarantees for FBMG. Numerical results on TV denoising and MRI demonstrate meaningful speedups, particularly in MRI where the coarse grid reduces the dominance of expensive transforms, illustrating the practical impact of truly cheaper coarse problems in nonsmooth multigrid optimization.
Abstract
Based on a nonsmooth coherence condition, we construct and prove the convergence of a forward-backward splitting method that alternates between steps on a fine and a coarse grid. Our focus is a total variation regularised inverse imaging problems, specifically, their dual problems, for which we develop in detail the relevant coarse-grid problems. We demonstrate the performance of our method on total variation denoising and magnetic resonance imaging.