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 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
Lombardi, Damiano 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},
year = {2015},
publisher = {EDP Sciences},
volume = {49},
number = {6},
doi = {10.1051/m2an/2015025},
mrnumber = {3423273},
zbl = {1334.65112},
language = {en},
url = {https://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 - https://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 https://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 , Optimal Transport, Tome 49 (2015) no. 6, pp. 1717-1744. doi: 10.1051/m2an/2015025
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