Multi-physics optimal transportation and image interpolation

¹ Laboratoire Jean Kuntzmann, Grenoble University, Université Joseph Fourrier and CNRS, France.
romain.hug@imag.fr,emmanuel.maitre@imag.fr
² Institut de Mathématiques de Bordeaux, CNRS and Université Bordeaux 1, France.
nicolas.papadakis@math.u-bordeaux.fr

Received: 23 April 2015
Revised: 29 April 2015

Abstract

Optimal transportation theory is a powerful tool to deal with image interpolation. This was first investigated by [Benamou and Brenier, Numer. Math. 84 (2000) 375–393.] where an algorithm based on the minimization of a kinetic energy under a conservation of mass constraint was devised. By structure, this algorithm does not preserve image regions along the optimal interpolation path, and it is actually not very difficult to exhibit test cases where the algorithm produces a path of images where high density regions split at the beginning before merging back at its end. However, in some applications to image interpolation this behaviour is not physically realistic. Hence, this paper aims at studying how some physics can be added to the optimal transportation theory, how to construct algorithms to compute solutions to the corresponding optimization problems and how to apply the proposed methods to image interpolation.