Volumetric Functional Maps

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.

Figures (17)

• Figure 1: Visual representation of our pipeline. The eigenfunctions of the LBO for volume meshes (left) are used to compute a volumetric functional map (middle). Basis alignment is exploited for several tasks: volumetric correspondences, piece-wise linear maps, and volumetric segmentation transfer (right).
• Figure 2: The first 5 non-constant volumetric LBO eigenfunctions on two humanoid non-isometric shapes
• Figure 3: Our pipeline for extrapolating interior coordinates from surface correspondences. Given the surface map $\pi$ (first row), we approximate the spectral embedding of the surface coordinates of $\partial\mathcal{N}$ using the boundary restriction of the eigenfunctions of $\mathcal{M}$ (second row). Using the eigenfunctions on the entire volume, we reconstruct the interior coordinates from the spectral embedding (third row), and use these coordinates to transfer the inner connectivity (fourth row).
• Figure 4: Left: average geodesic error curves on the pairs from Su et al.su2019practical resulting from the application of FMaps nogneng2017informative, ZoomOut melzi2019zoomout, and Orthoprods maggioli2021orthogonalized. Right: runtime for the three methods plotted against the number of vertices in the source mesh.
• Figure 5: Coordinate transfer between two volumetric shapes using the Orthoprods basis. The normalized coordinates are treated as RGB channels for visualization (first column). For further validation, the transferred coordinates are also used for generating an error-sensitive procedural texture, which is visualized on different slices of the volume (second to fourth columns).