Variational Image Regularization with Euler's Elastica Using a Discrete Gradient Scheme

This paper concerns an optimization algorithm for unconstrained nonconvex 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 nonconvex 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.