Conformal Wasserstein distances: comparing surfaces in polynomial time

TL;DR

This paper introduces a polynomial-time method for comparing surfaces by solving a mass-transportation problem on conformal densities, invariant under M"{o}bius transformations, with detailed focus on disk-like surfaces.

Contribution

It proposes a novel, efficient surface comparison technique based on conformal Wasserstein distances that accounts for global M"{o}bius invariance, extending to various surface types.

Findings

Polynomial-time algorithm for surface comparison

Invariant under global M"{o}bius transformations

Applicable to disk-like and other surfaces

Abstract

We present a constructive approach to surface comparison realizable by a polynomial-time algorithm. We determine the "similarity" of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global M\"{o}bius transformations. We present in detail the case where the surfaces to compare are disk-like; we also sketch how the approach can be generalized to other types of surfaces.