Volumetric Functional Maps

TL;DR

This paper introduces a spectral volume mapping framework that extends surface-based functional maps to volumetric data, enabling high-quality signal transfer and improved shape matching accuracy.

Contribution

It pioneers the extension of functional maps to volumetric domains using the volumetric Laplace operator eigenfunctions, with validation on diverse datasets and applications.

Findings

Volumetric spectral methods outperform surface-only methods in shape matching.

The proposed approach enables effective signal transfer for segmentation and texturing.

Experimental validation on novel volumetric datasets demonstrates practical utility.

Abstract

Computing volumetric correspondences between 3D shapes is a prominent tool for medical and industrial applications. In this work, we pave the way for spectral volume mapping, extending for the first time the surface-based functional maps framework. We show that the eigenfunctions of the volumetric Laplace operator define a functional space that is suitable for high-quality signal transfer. We also experiment with various techniques that edit this functional space, porting them to volume domains. We validate our method on novel volumetric datasets and on tetrahedralizations of well established surface datasets, also showcasing practical applications involving both discrete and continuous signal mapping, for segmentation transfer, mesh connectivity transfer and solid texturing. Finally, we show that the volumetric spectrum greatly improves the accuracy for classical shape matching tasks among surfaces, consistently outperforming surface-only spectral methods.