Optimal transportation theory is a powerful tool to deal with image interpolation. This was first investigated by Benamou and Brenier [4] where an algorithm based on the minimization of a kinetic energy under a conservation of mass constraint was devised. By structure, this algorithm does not preserve image regions along the optimal interpolation path, and it is actually not very difficult to exhibit test cases where the algorithm produces a path of images where high density regions split at the beginning before merging back at its end. However, in some applications to image interpolation this behaviour is not physically realistic. Hence, this paper aims at studying how some physics can be added to the optimal transportation theory, how to construct algorithms to compute solutions to the corresponding optimization problems and how to apply the proposed methods to image interpolation.
The dynamical formulation of the optimal transport problem, introduced by J. D. Benamou and Y. Brenier [4], corresponds to the time-space search of a density and a momentum minimizing a transport energy between two densities. In order to solve this problem, an algorithm has been proposed to estimate a saddle point of a Lagrangian. We study the convergence of this algorithm in the most general case where initial and final densities vanish on some areas of the transportation domain. Under these conditions, the main difficulty of our study is the proof of existence of a saddle point and of uniqueness of the densitymomentum component, as it leads to deal with non-regular optimal transportation maps. For these reasons, a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space is required.
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