Hybrid Riemannian Metrics for Diffeomorphic Shape Registration

TL;DR

This paper introduces a hybrid approach combining two Riemannian metric methods for shape registration, enhancing flexibility and performance in complex shape configurations through theoretical insights and experimental validation.

Contribution

It develops a novel hybrid Riemannian metric framework that merges parametrization-invariant Sobolev metrics with diffeomorphic group metrics, improving shape registration capabilities.

Findings

The hybrid method inherits advantages of both original approaches.

It demonstrates improved registration accuracy on complex shapes.

Experimental results validate the theoretical benefits.

Abstract

We consider the results of combining two approaches developed for the design of Riemannian metrics on curves and surfaces, namely parametrization-invariant metrics of the Sobolev type on spaces of immersions, and metrics derived through Riemannian submersions from right-invariant Sobolev metrics on groups of diffeomorphisms (the latter leading to the "large deformation diffeomorphic metric mapping" framework). We show that this quite simple approach inherits the advantages of both methods, both on the theoretical and experimental levels, and provide additional flexibility and modeling power, especially when dealing with complex configurations of shapes. Experimental results illustrating the method are provided for curve and surface registration.