Divergence-Free Shape Interpolation and Correspondence

TL;DR

This paper introduces a divergence-free shape interpolation method that simultaneously models deformation fields and correspondences, ensuring volume preservation and efficient computation across different resolutions.

Contribution

It proposes a novel divergence-free deformation framework using Karhunen-Loève expansion, eliminating the need for space discretization and enabling scalable, volume-preserving shape morphing.

Findings

Effective shape correspondence and registration demonstrated on TOSCA and FAUST datasets.

Volume-preserving deformations without discretizing the embedding space.

Scalable morphing applicable to various resolutions.

Abstract

We present a novel method to model and calculate deformation fields between shapes embedded in $\mathbb{R}^D$. Our framework combines naturally interpolating the two input shapes and calculating correspondences at the same time. The key idea is to compute a divergence-free deformation field represented in a coarse-to-fine basis using the Karhunen-Lo\`eve expansion. The advantages are that there is no need to discretize the embedding space and the deformation is volume-preserving. Furthermore, the optimization is done on downsampled versions of the shapes but the morphing can be applied to any resolution without a heavy increase in complexity. We show results for shape correspondence, registration, inter- and extrapolation on the TOSCA and FAUST data sets.