Deep spectral features for 3D shape analysis

3D shape analysis tasks often involve characterizing a 3D object by an invariant, computationally efficient, and discriminative numerical representation, called shape descriptors. Among those, spectral-based shape descriptors have become increasingly widespread, since the spectrum is an isometry invariant, and thus is independent of the object’s representation including parametrization and spatial position[1]. However, large spectral decompositions and the choice of the most significant eigen-couples become computationally expensive for large set of data-points. We introduce a concise learning-based shape descriptor, computed through a Generalized Graph Neural Network (G-GNN) [2]. The G-GNN is an unsupervised graph neural network, leveraging spectral-based convolutional operators, derived from a learnable, energy-driven evolution process. Applied to a 3D polygonal mesh, the G-GNN allows to learn features acting as global shape descriptor of the 3D object. Using a 3D mesh related Dirichlet-like energy leads to a spectral and intrinsic shape descriptor, tied to the isometry invariant Laplace-Beltrami operator. Finally, by equipping the G-GNN with a suitable shape retrieval loss, the spectral shape descriptor can be employed in non-linear dimensionality reduction problems since it can define an optimal embedding, squeezing the latent information of a 3D model into a compact low-dimensional shape representation of the 3D object [1] Martin Reuter, Franz-Erich Wolter, Niklas Peinecke, Laplace–Beltrami spectra as ‘Shape-DNA’ of surfaces and solids, Computer-Aided Design, Volume 38, Issue 4, 2006, Pages 342-366, ISSN 0010-4485, https://doi.org/10.1016/j.cad.2005.10.011. [2] D. Lazzaro, S. Morigi, P. Zuzolo, Learning intrinsic shape representations via spectral mesh convolutions, Neurocomputing, Volume 598, 2024, 128152, ISSN 0925-2312, https://doi.org/10.1016/j.neucom.2024.128152.