Optimal transport using Helmholtz-Hodge decomposition and first-order primal-dual algorithms

This work deals with the resolution of the optimal transport problem between 2D images in the fluid mechanics framework of Benamou and Brenier formulation [1], which numerical resolution is still challenging even for medium-sized images. We develop a method using the Helmholtz-Hodge decomposition [2] in order to enforce the divergence-free constraint throughout the iterations. We then show how to use a first order primal-dual algorithm for convex problems of Chambolle and Pock [3] to solve the obtained problem, leading to a new algorithm easy to implement. Besides, numerical experiments demonstrate that this algorithm is faster than state of the art methods and efficient with real-sized images.