Eulerian models and algorithms for unbalanced optimal transport

Damiano Lombardi (1) , Emmanuel Maitre (2)

1 REO - Numerical simulation of biological flows
2 EDP - Equations aux Dérivées Partielles

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.

Mots clés: Optimal transport; Algorithm; Unbalanced; Tumor; Interpolation; Image

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Damiano Lombardi, Emmanuel Maitre. Eulerian models and algorithms for unbalanced optimal transport. 2013. ⟨hal-00976501v1⟩