A relaxed approach for curve matching with elastic metrics

TL;DR

This paper introduces a unified framework for elastic curve matching using Riemannian metrics, employing varifold-based similarity and relaxed variational formulations to improve boundary constraints handling and include various shape transformations.

Contribution

It extends previous elastic metrics by integrating boundary condition handling with varifold-based similarity, enabling flexible shape matching without reparametrization optimization.

Findings

The method effectively computes geodesics with boundary conditions.

It accommodates transformations like translation, rotation, and scaling.

Numerical examples demonstrate advantages over existing shape registration methods.

Abstract

In this paper we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the family of $H^2$-metrics with constant coefficients and scale-invariant $H^2$-metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration.