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Eulerian models and algorithms for unbalanced optimal transport

Damiano Lombardi 1 Emmanuel Maitre 2
1 REO - Numerical simulation of biological flows
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, UPMC - Université Pierre et Marie Curie - Paris 6
2 EDP [2007-2015] - Equations aux Dérivées Partielles [2007-2015]
LJK [2007-2015] - Laboratoire Jean Kuntzmann [2007-2015]
Abstract : Benamou and Brenier formulation of Monge transportation problem (Numer. Math. 84:375-393, 2000) has proven to be of great interest in image processing to compute warpings and distances between pair of images (SIAM J. Math. Analysis, 35:61-97, 2003). 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 to overcome this caveat are reviewed, and discussed. Due to the mix between transport and $L^2$ 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.
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Damiano Lombardi, Emmanuel Maitre. Eulerian models and algorithms for unbalanced optimal transport. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2015, Special Issue - Optimal Transport, 49 (6), pp.1717-1744. ⟨10.1051/m2an/2015025⟩. ⟨hal-00976501v3⟩