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Pré-Publication, Document De Travail Année : 2019

Primal-dual formulation of the Dynamic Optimal Transport using Helmholtz-Hodge decomposition

Résumé

This work deals with the resolution of the dynamic optimal transport (OT) problem between 1D or 2D images in the fluid mechanics framework of Benamou-Brenier [6]. The numerical resolution of this dynamic formulation of OT, despite the successful application of proximal methods [36] is still computationally demanding. This is partly due to a space-time Laplace operator to be solved at each iteration, to project back to a divergence free space. In this paper, we develop a method using the Helmholtz-Hodge decomposition [23] in order to enforce the divergence-free constraint throughout the iterations. We prove that the functional we consider has better convexity properties on the set of constraints. In particular we explain that in 1D+time, this formulation is equivalent to the resolution of a minimal surface equation. We then adapt the first order primal-dual algorithm for convex problems of Chambolle and Pock [12] to solve this new problem, leading to an algorithm easy to implement. Besides, numerical experiments demonstrate that this algorithm is faster than state of the art methods for dynamic optimal transport [36] and efficient with real-sized images.
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Dates et versions

hal-02087017 , version 1 (28-04-2019)

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  • HAL Id : hal-02087017 , version 1

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Morgane Henry, Emmanuel Maitre, Valérie Perrier. Primal-dual formulation of the Dynamic Optimal Transport using Helmholtz-Hodge decomposition. 2019. ⟨hal-02087017⟩
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