Learning Spectral Descriptors for Shape Analysis

The Laplace-Beltrami Operator (LBO) is a cornerstone of geometry processing, valued for invariance under isometric transformations, a discrete orthonormal spectrum, and its ability to encode intrinsic manifold geometry via geodesic distances, proving effective for point-to-point correspondence on recent benchmarks. Yet it is as fragile as it is powerful: even near-isometric deformations can significantly alter its eigenbasis, and few methods backpropagate loss through it, given its complexity. This thesis introduces a learning-based framework that constructs mesh operators by modifying the system matrices governing the LBO eigenbasis, yielding a data-adaptive LBO robust to near- and non-isometric deformations. The work unfolds in three stages: a neuroevolutionary approach probing mass-matrix weighting; LBONet, a learnable architecture adjusting the system matrices for task-dependent spectral behaviour; and LBONet embedded unsupervised, via functional maps and ZoomOut, yielding a fully intrinsic correspondence framework without supervision. Operating mainly in the intrinsic domain, with rotation-invariant extrinsic measures at the input, the pipeline is largely unaffected by the embedding and invariant to rigid transformations, properties coordinate-based approaches typically approximate via data augmentation. Inverted orientations may thus lead such models to misread features as noise, whereas intrinsic methods offer a more stable representation, motivating a notion of shape understanding invariant to embedding and rigid transformations. Custom sparse routines accelerate backpropagation through the eigenbasis, and combined with growing Graphics Processing Unit (GPU) memory on consumer hardware, make the framework practical and broadly applicable. Experiments on global tasks (SHREC*, SHAPENET) and local tasks (SHAPESEG, FAUST, SMAL, DT4D) show consistent improvements over existing methods.