Volumetric Functional Maps (2506.13212v1)

Abstract: The computation of volumetric correspondences between 3D shapes has great potential for medical and industrial applications. In this work, we pave the way for spectral volume mapping, extending for the first time the functional maps framework from the surface setting to the volumetric domain. 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 from the surface to the volume setting. 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. Last but not least, we show that considering the volumetric spectrum greatly improves the accuracy for classical shape matching tasks among surfaces, consistently outperforming existing surface-only spectral methods.