D. Lombardi: Models and methods for unbalanced optimal transport
Abstract: The unbalanced optimal transport problem is considered, which is of interest when applying optimal transport principles to real data analysis. Indeed, real data are not, often, densities whose mass is conserved. In order to apply an optimal transport principle, the mass conservation constraint has to be relaxed by adding mass sources. Among the several possibilities, eulerian models of sources are investigated. A general functional form of the source is derived by requiring that the fundamental properties of optimal transport are satisfied and, when the mass is conserved, the problem collapse to the Benamou-Brenier formulation. As a result, the interpolation is not featured by an infinite speed of propagation. The numerical discretisation proposed is based on an explicit adaptation of the Benamou-Brenier method. Several test-cases are described in order to validate the approach and assess its numerical properties. Moreover, some examples on interpolation of medical images are proposed.