Symmetrized semi-discrete optimal transport

TL;DR

This paper introduces a symmetric approach to semi-discrete optimal transport using coupled power diagrams, enabling better approximation of transport maps and joint sampling of measures.

Contribution

It presents a fixed-point algorithm that enforces symmetry in semi-discrete optimal transport solutions via coupled power diagrams.

Findings

The method captures discontinuities in the transport map.

It allows linear interpolation of displacement through mesh vertices.

Coupled power diagrams enable joint measure sampling.

Abstract

Interpolating between measures supported by polygonal or polyhedral domains is a problem that has been recently addressed by the semi-discrete optimal transport framework. Within this framework, one of the domains is discretized with a set of samples, while the other one remains continuous. In this paper we present a method to introduce some symmetry into the solution using coupled power diagrams. This symmetry is key to capturing the discontinuities of the transport map reflected in the geometry of the power cells. We design our method as a fixed-point algorithm alternating between computations of semi-discrete transport maps and recentering of the sites. The resulting objects are coupled power diagrams with identical geometry, allowing us to approximate displacement interpolation through linear interpolation of the meshes vertices. Through these coupled power diagrams, we have a natural way of jointly sampling measures.