HodgeNet

Abstract

Fig. 1. Mesh segmentation results on the full-resolution MIT animation dataset. Each mesh in the dataset contains 20,000 faces (10,000 vertices). We show an example ground truth segmentation in the bottom-left. In contrast to previous works, which downsample each mesh by more than 10×, we efficiently process dense meshes both at train and test time.Constrained by the limitations of learning toolkits engineered for other applications, such as those in image processing, many mesh-based learning algorithms employ data flows that would be atypical from the perspective of conventional geometry processing. As an alternative, we present a technique for learning from meshes built from standard geometry processing modules and operations. We show that low-order eigenvalue/eigenvector computation from operators parameterized using discrete exterior calculus is amenable to efficient approximate backpropagation, yielding spectral per-element or per-mesh features with similar formulas to classical descriptors like the heat/wave kernel signatures. Our model uses few parameters, generalizes to high-resolution meshes, and exhibits performance and time complexity on par with past work.

MeT: A graph transformer for semantic segmentation of 3D meshes

“…Other approaches, like [44], [45], [46] propose alternative solutions to the segmentation task. MeshWalker [44] represents mesh's geometry and topology by a set of random walks along the surface; these walks are fed to a recurrent neural network.…”

“…MeshWalker [44] represents mesh's geometry and topology by a set of random walks along the surface; these walks are fed to a recurrent neural network. HodgeNet [45], instead, tackles the problem relying on spectral geometry, and proposes parallelizable algorithms for differentiating eigencomputation, including approximate backpropagation without sparse computation. Finally, Diffu-sionNet [46] introduces a general-purpose approach to deep learning on 3D surfaces, using a simple diffusion layer to agnostically represent any mesh.…”

Spectral Coarsening with Hodge Laplacians

“…Laplacians make frequent appearances across geometry processing, machine learning and computational topology. Specific definitions and flavours vary widely across discrete exterior calculus [Crane et al 2013;Desbrun et al 2005], vector-field processing [de Goes et al 2016;Poelke and Polthier 2016;Vaxman et al 2016;Wardetzky 2020;Zhao et al 2019b], fluid simulation [Liu et al 2015], mesh segmentation and editing [Khan et al 2020;Lai et al 2008;Sorkine et al 2004], topological signal processing [Barbarossa and Sardellitti 2020], random walks [Lahav and Tal 2020;Schaub et al 2020], clustering and learning [Ebli et al 2020;Ebli and Spreemann 2019;Keros et al 2022;Nascimento and De Carvalho 2011;Smirnov and Solomon 2021;Su et al 2022]. Their ability to effectively capture salient geometric, topological, and dynamic information makes their spectrum a versatile basis.…”

MeshFormer: High‐resolution Mesh Segmentation with Graph Transformer

“…1(e), once high‐frequency information in simplified meshes is overlooked, shape information will suffer severe distortion, e.g., flattened segmentation boundaries, potentially resulting in more segmentation errors. Recently, Sharp et al [SACO22] and Smirnov et al [SS21] presented their solutions to tackle these two issues. However, these solutions cannot eliminate high‐frequency information loss in such processing.…”