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 ${L}^{2}$ 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.

DOI: 10.1051/m2an/2015025

Keywords: Optimal transport, image interpolation, numerical optimization

^{1}; Maitre, Emmanuel

^{2}

@article{M2AN_2015__49_6_1717_0, author = {Lombardi, Damiano and Maitre, Emmanuel}, title = {Eulerian models and algorithms for unbalanced optimal transport}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1717--1744}, publisher = {EDP-Sciences}, volume = {49}, number = {6}, year = {2015}, doi = {10.1051/m2an/2015025}, mrnumber = {3423273}, zbl = {1334.65112}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015025/} }

TY - JOUR AU - Lombardi, Damiano AU - Maitre, Emmanuel TI - Eulerian models and algorithms for unbalanced optimal transport JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1717 EP - 1744 VL - 49 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015025/ DO - 10.1051/m2an/2015025 LA - en ID - M2AN_2015__49_6_1717_0 ER -

%0 Journal Article %A Lombardi, Damiano %A Maitre, Emmanuel %T Eulerian models and algorithms for unbalanced optimal transport %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1717-1744 %V 49 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015025/ %R 10.1051/m2an/2015025 %G en %F M2AN_2015__49_6_1717_0

Lombardi, Damiano; Maitre, Emmanuel. Eulerian models and algorithms for unbalanced optimal transport. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 49 (2015) no. 6, pp. 1717-1744. doi : 10.1051/m2an/2015025. http://www.numdam.org/articles/10.1051/m2an/2015025/

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