A Volumetric Approach to Monge's Optimal Transport on Surfaces

TL;DR

This work introduces a volumetric reformulation of Monge's Optimal Transport on smooth surfaces by solving an equivalent problem in a thin tubular neighborhood $T_ε$ around the surface $Γ$. By extending densities to $T_ε$ and augmenting the cost with a penalty term $c_σ$, the authors derive a PDE-based OT formulation on Euclidean space, whose solution recovers the surface OT map via projection, while preserving Wasserstein energy. They prove equivalence between the surface and volumetric problems and develop a straightforward Cartesian-grid discretization that leverages standard finite differences and a Jacobi-type solver. The framework is demonstrated on the unit sphere, hemisphere, and torus for multiple cost functions, with extensive computational experiments validating accuracy, convergence, and applicability to optics-inspired problems such as reflector design and mesh moving. Overall, the volumetric extension enables robust, flexible, and efficient computation of OT on surfaces without surface parameterization, broadening practical impact in graphics, geometry processing, and related fields.

Abstract

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, $T_ε$, 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 $T_ε$ 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 $Γ= \mathbb{S}^2$ and the $2$-torus.

Figures (15)

• Figure 1: New Optimal Transport problems are defined in $T_\epsilon$ to have solutions to the Optimal Transport problems on $\Gamma$.
• Figure 2: Different boundary conditions are applied for different parts of the boundary. For $\partial T_{\epsilon}^{1} \cap \partial T_{\epsilon}$, we apply the boundary condition $u(\mathbf{z}) = u(P_{\Gamma}\mathbf{z})$ (depicted in red). For the rest of the boundary, we extend the natural zero-Neumann condition on the boundary of $\Gamma$ to zero-Neumann conditions on the corresponding part of $\partial T_{\epsilon}^{1} \cap \partial Q_{\epsilon}$ (depicted in green).
• Figure 3: (a) A schematic of the grid nodes in $\mathcal{T}^{h}_{\epsilon}$ (red); (b) A horizontal cross-section of the computational grid nodes for for the unit sphere, showing the interior computational points in red and the boundary points $\mathcal{B}^h$ in black.
• Figure 4: (a) Convergence of the residual and (b) $L^{\infty}$ error.
• Figure 5: Visualization of the changes in the local density changes via a set of anchor points. The anchor points are connected to form a mesh over the sphere. (a) The initial distribution of anchor points; (b) The distribution of the transported anchor points. We highlight four connected vertices in red for comparison. The mass density in this computational example is required to increase in the northern hemisphere and decrease in the southern hemisphere. Details of this example can be found in Section \ref{['sec:analytical']}

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Theorem 3
• Remark
• Theorem 4
• Corollary 5
• proof
• Lemma 6
• proof
• Definition 7