Spectrum Alignment for Robust 3D Point Cloud Correspondences Estimation

Estimating dense point-to-point correspondences between two isometric shapes represented as 3D point clouds is a fundamental problem in geometry processing, with applications in texture and motion transfer. However, this task becomes particularly challenging when the shapes undergo non-rigid transformations, as is often the case with approximately isometric point clouds. Most existing algorithms address this challenge by establishing correspondences between functions defined on the shapes, rather than directly between points, because function mappings admit a linear representation in the spectral domain. State-of-the-art methods compute this linear representation using the eigenfunctions of the Laplace-Beltrami Operator (LBO) along with a small set of initial corresponding functions between the shapes. However, for approximately isometric point clouds, two key issues arise: (1) the eigenfunctions of the LBO may become misaligned, and (2) the initial corresponding functions may include outliers, both of which degrade the quality of the resulting correspondences. In this work, we propose an efficient approach to align the spectra of the LBOs of the two shapes, enabling the eigenfunctions to remain compatible even for approximately isometric 3D point clouds. Additionally, we introduce a technique to make function correspondence estimation robust to outliers. We validate our approach by comparing it with state-of-the-art 3D shape-matching algorithms on benchmark datasets, demonstrating its effectiveness.