A numerical algorithm for $L_2$ semi-discrete optimal transport in 3D

Abstract: This paper introduces a numerical algorithm to compute the $L_2$ optimal transport map between two measures $\mu$ and $\nu$, where $\mu$ derives from a density $\rho$ defined as a piecewise linear function (supported by a tetrahedral mesh), and where $\nu$ is a sum of Dirac masses. I first give an elementary presentation of some known results on optimal transport and then observe a relation with another problem (optimal sampling). This relation gives simple arguments to study the objective functions that characterize both problems. I then propose a practical algorithm to compute the optimal transport map between a piecewise linear density and a sum of Dirac masses in 3D. In this semi-discrete setting, Aurenhammer et.al [\emph{8th Symposium on Computational Geometry conf. proc.}, ACM (1992)] showed that the optimal transport map is determined by the weights of a power diagram. The optimal weights are computed by minimizing a convex objective function with a quasi-Newton method. To evaluate the value and gradient of this objective function, I propose an efficient and robust algorithm, that computes at each iteration the intersection between a power diagram and the tetrahedral mesh that defines the measure $\mu$. The numerical algorithm is experimented and evaluated on several datasets, with up to hundred thousands tetrahedra and one million Dirac masses.

A Numerical Algorithm for $L_2$ Semi-Discrete Optimal Transport in 3D

The paper "A Numerical Algorithm for $L_2$ Semi-Discrete Optimal Transport in 3D" by Bruno Levy explores a computational method for solving the optimal transport problem between two measures in a three-dimensional setting. Specifically, the contributions lie in handling the semi-discrete case where one measure ($\mu$) is represented by a continuous density supported on a tetrahedral mesh, while the other ($\nu$) is a discrete sum of Dirac masses.

The work builds on the foundation laid by Aurenhammer and others, who demonstrated that the optimal transport map in this semi-discrete setting can be characterized by the weights of a power diagram. The fundamental task, then, is to determine these optimal weights, which the paper achieves through the minimization of a convex objective function using a quasi-Newton method.

Key Methodological Innovations

1. Power Diagram Generation: The paper extends computational geometry principles by utilizing power diagrams, a generalization of Voronoi diagrams, to define transport regions for each Dirac mass.
2. Intersection Algorithms: At each iteration of the optimization process, the paper proposes an efficient algorithm to calculate the intersection of a power diagram with the tetrahedral mesh supporting $\mu$. This step is computationally intensive but crucial for evaluating the objective function and its gradient.
3. Convex Optimization: The methodological framework hinges on convex optimization techniques, specifically using a quasi-Newton solver which is combined with a multilevel optimization strategy to achieve numerical stability and efficiency.

Experimental Insights

The algorithm's performance is empirically validated through experiments on datasets with varying complexities, such as meshes with up to hundreds of thousands of tetrahedra and a target measure composed of up to a million Dirac masses. The results underscore the scalability of the approach and its applicability to complex three-dimensional transport problems.

Implications and Future Directions

This research offers significant progress in computational optimal transport, specifically for applications in computer graphics where the transformation of shapes is a pertinent problem. For instance, computing approximations of the Wasserstein distance or facilitating 3D morphing tasks become feasible through this numerical algorithm.

Theoretically, the work contributes to the understanding of semi-discrete optimal transport problems in higher dimensions, particularly through its innovative use of power diagrams matched to capacity constraints.

The study paves the way for several intriguing future research directions:

• Extending the algorithm to fully discrete or fully continuous settings.
• Exploring alternative numerical optimization methods that could further enhance computational efficiency.
• Investigating practical applications beyond computer graphics, such as geosciences or medical imaging, where significant shape transformations are essential.

In conclusion, the paper provides a robust computational strategy for tackling $L_2$ semi-discrete optimal transport in three dimensions, encouraging both theoretical exploration and practical application development in the realms where optimal transport is a critical component.