Variational image regularization with Euler's elastica using a discrete gradient scheme

TL;DR

This paper introduces a novel optimization algorithm using a discrete gradient scheme for non-convex problems, demonstrating improved convergence in image regularization tasks like Euler's elastica.

Contribution

It presents a dissipation-preserving numerical integrator-based algorithm with proven convergence rates, especially effective for problems with sparse connections.

Findings

Proves convergence rate estimates for the proposed algorithm.

Shows improved convergence for problems with sparse connections.

Demonstrates effectiveness in Euler's elastica image regularization.

Abstract

This paper concerns an optimization algorithm for unconstrained non-convex problems where the objective function has sparse connections between the unknowns. The algorithm is based on applying a dissipation preserving numerical integrator, the Itoh--Abe discrete gradient scheme, to the gradient flow of an objective function, guaranteeing energy decrease regardless of step size. We introduce the algorithm, prove a convergence rate estimate for non-convex problems with Lipschitz continuous gradients, and show an improved convergence rate if the objective function has sparse connections between unknowns. The algorithm is presented in serial and parallel versions. Numerical tests show its use in Euler's elastica regularized imaging problems and its convergence rate and compare the execution time of the method to that of the iPiano algorithm and the gradient descent and Heavy-ball algorithms.