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 $\Gamma = \mathbb{S}^2$.

翻译：我们提出了一种体积公式化方法，用于计算在$\mathbb{R}^3$中曲面上定义的最优输运问题，该问题常见于光学、计算机图形学及计算方法学等领域。我们并非直接在曲面上处理原问题，而是在邻近曲面的薄管状区域$T_{\epsilon}$上定义一个新的最优输运问题。这一扩展为笛卡尔网格上的数值离散化提供了更高的灵活性和简便性。在$T_{\epsilon}$上计算得到的最优输运映射和势函数与原始曲面上的问题保持一致。我们证明，利用所提出的体积方法，能够采用简单直接的数值方法来求解$\Gamma = \mathbb{S}^2$上的最优输运问题。