Eulerian models and algorithms for unbalanced optimal transport

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.