Eulerian models and algorithms for unbalanced optimal transport

¹ Equipe REO, INRIA Rocquencourt, France.
damiano.lombardi@inria.fr
² Laboratoire Jean Kuntzmann, Grenoble University and CNRS, France.
emmanuel.maitre@imag.fr

Received: 10 March 2015

Abstract

Benamou and Brenier formulation of Monge transportation problem [J.-D. Benamou and Y. Brenier, Numer. Math. 84 (2000) 375–393.] has proven to be of great interest in image processing to compute warpings and distances between pair of images [S. Agenent, S. Haker and A. Tannenbaum, SIAM J. Math. Anal. 35 (2003) 61–97]. One requirement for the algorithm to work is to interpolate densities of same mass. In most applications to image interpolation, this is a serious limitation. Existing approaches [J.-D. Benamou, ESAIM: M2AN 37 (2003) 851–868; B. Piccoli and F. Rossi, Arch. Rational Mech. Anal. 211 (2014) 335–358; B. Piccoli and F. Rossi, Preprint arXiv:1304.7014 (2014)]. to overcome this caveat are reviewed, and discussed. Due to the mix between transport and L² interpolation, these models can produce instantaneous motion at finite range. In this paper we propose new methods, parameter-free, for interpolating unbalanced densities. One of our motivations is the application to interpolation of growing tumor images.