A new numerical scheme for constrained total variation flows and its convergence

TL;DR

This paper introduces a novel numerical scheme for constrained total variation flows on Riemannian manifolds, addressing non-convexity issues and providing convergence analysis with numerical validation on spheres and rotation groups.

Contribution

It develops a new localization-based numerical method for constrained total variation flows on manifolds and proves its finite-time error estimate.

Findings

Successful implementation on $S^2$ and $SO(3)$ manifolds.

Proven finite-time error estimate for the scheme.

Numerical results demonstrate effectiveness and accuracy.

Abstract

In this paper, we propose a new numerical scheme for a spatially discrete model of constrained total variation flows, which are total variation flows whose values are constrained in a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex and it is hard to calculate the minimizer of the functional with the manifold constraint. We overcome this difficulty by "localization technique" using the exponential map and prove the finite-time error estimate in general situation. Finally, we show a few numerical results for the cases that the target manifolds are $S^2$ and $SO(3)$.