We propose a volumetric formulation for computing the Optimal Transport problem defined on surfaces in $\mathbb{R}^3$, found in disciplines like optics, computer graphics, and computational methodologies. Instead of directly tackling the original problem on the surface, we define a new Optimal Transport problem on a thin tubular region, $T_{\epsilon}$, adjacent to the surface. This extension offers enhanced flexibility and simplicity for numerical discretization on Cartesian grids. The Optimal Transport mapping and potential function computed on $T_{\epsilon}$ are consistent with the original problem on surfaces. We demonstrate that, with the proposed volumetric approach, it is possible to use simple and straightforward numerical methods to solve Optimal Transport for $\Gamma = \mathbb{S}^2$ and the $2$-torus.

翻译：我们提出了一种体积公式化方法来计算定义在$\mathbb{R}^3$中曲面上的最优传输问题，该问题常见于光学、计算机图形学及计算方法学等领域。为了绕过直接在曲面上处理原始问题的复杂性，我们在曲面邻近的薄管状区域$T_{\epsilon}$上定义了一个新的最优传输问题。这种延拓方法在笛卡尔网格上数值离散时具有更强的灵活性和简便性。在$T_{\epsilon}$上计算得到的最优传输映射与势函数，与原曲面问题具有一致性。研究表明，通过所提出的体积方法，可以采用简单直接的数值方案为$\Gamma = \mathbb{S}^2$（二维球面）及二维环面求解最优传输问题。