Robustness of Spectral–Geometric Invariants Under Realistic 3D Acquisition Distortions: A Critical Review, Taxonomy, and Comparative Evaluation Protocol

Spectral–geometric invariants are central to 3D shape analysis because they encode intrinsic structure via differential operators while being insensitive to rigid motion and often stable under near-isometric deformation. Yet practical 3D pipelines—photogrammetry, LiDAR, structured-light scanning, medical reconstruction, and virtual modeling—rarely produce ideal watertight meshes. Instead, real data contains irregular sampling, anisotropic remeshing, noise, holes, partial observations, and occasional topology defects that modify the computational domain and destabilize discrete operators. This article provides an analytical, distortion-aware review of spectral–geometric invariants used for shape retrieval, classification, and comparison. We contribute (i) a structured taxonomy that groups invariants by generating operator (Laplace–Beltrami spectrum, heat/diffusion kernels, wave kernels), descriptor scope (global, pointwise, functional/operator), and invariance target (rigid, scale, near-isometric, partiality, topology tolerance); (ii) a unified comparative evaluation protocol—the Robustness Envelope—that measures descriptor drift, discriminative margin collapse, and task-level degradation as explicit functions of distortion severity across controlled distortion families; and (iii) a synthesis of failure modes and deployment-oriented recommendations. We emphasize reproducibility by specifying implementation details that must be reported (discrete operator, boundary handling, normalization, eigenpair truncation, solver tolerances) and by providing tables that map distortions to expected robustness profiles. Finally, we connect robustness needs to geometric modeling and virtual environments, where stable shape descriptors support reconstruction and comparison under real acquisition variability. The results indicate that diffusion signatures often provide the best robustness–discriminativity tradeoff under moderate noise and resolution changes, while eigenvalue fingerprints are compact but brittle under domain change. Wave signatures can be highly discriminative yet inherit eigenbasis fragility, and functional-map pipelines excel for correspondence when constraints remain reliable under partial overlap.