The Square Root Normal Field Distance and Unbalanced Optimal Transport

TL;DR

This paper establishes a connection between shape analysis and unbalanced optimal transport by characterizing the SRNF shape distance as a pullback of the WFR distance, and introduces a new algorithm for its computation.

Contribution

It provides a novel theoretical link between shape analysis and unbalanced optimal transport and proposes an efficient algorithm for computing the WFR distance.

Findings

The SRNF shape distance is characterized as a pullback of the WFR distance.

The new algorithm effectively computes the WFR distance.

Insights into the degeneracy of the SRNF shape distance.

Abstract

This paper explores a novel connection between two areas: shape analysis of surfaces and unbalanced optimal transport. Specifically, we characterize the square root normal field (SRNF) shape distance as the pullback of the Wasserstein-Fisher-Rao (WFR) unbalanced optimal transport distance. In addition, we propose a new algorithm for computing the WFR distance and present numerical results that highlight the effectiveness of this algorithm. As a consequence of our results we obtain a precise method for computing the SRNF shape distance directly on piecewise linear surfaces and gain new insights about the degeneracy of this distance.