DiffusionNet: Discretization Agnostic Learning on Surfaces

Abstract

We introduce a new general-purpose approach to deep learning on three-dimensional surfaces based on the insight that a simple diffusion layer is highly effective for spatial communication. The resulting networks are automatically robust to changes in resolution and sampling of a surface—a basic property that is crucial for practical applications. Our networks can be discretized on various geometric representations, such as triangle meshes or point clouds, and can even be trained on one representation and then applied to another. We optimize the spatial support of diffusion as a continuous network parameter ranging from purely local to totally global, removing the burden of manually choosing neighborhood sizes. The only other ingredients in the method are a multi-layer perceptron applied independently at each point and spatial gradient features to support directional filters. The resulting networks are simple, robust, and efficient. Here, we focus primarily on triangle mesh surfaces and demonstrate state-of-the-art results for a variety of tasks, including surface classification, segmentation, and non-rigid correspondence.

Intrinsic Neural Fields: Learning Functions on Manifolds

“…They have been used in various applications, e.g., in shape matching [53,9,23], parallel transport [45], and robustness wrt. discretization [52,43]. Manifold kernels naturally consider the local and global geometry [5], and our approach follows in this direction by showing a natural extension of neural fields on manifolds.…”

“…5.3. To quantify the discretization dependence of intrinsic neural fields, we follow the procedure proposed by Sharp et al [43,Sec. 5.4] and rediscretize the meshes used in Sec.…”

“…We use the same datasets for the cat and human as in Sec. 5.1 but generate different discretizations of the meshes with the scripts from the Github project 8 by Sharp et al [43]. The hyperparameters can be found in Tab.…”

“…Discretization-agnostic intrinsic neural fields. We employ the procedure proposed by Sharp et al[43, Sec. 5.4] to generate different discretizations of the original meshes (orig): uniform isotropic remeshing (iso), densification around random vertices (dense), refinement and subsequent quadric error simplification[17] (qes), and point clouds sampled from the surfaces with more points than vertices (cloud ↑) and with fewer points (cloud ↓).…”

Neural Jacobian Fields: Learning Intrinsic Mappings of Arbitrary Meshes

“…In the context of deep-learning, Atlasnet-like works explored leveraging jacobians [Bednarik et al 2020], however this is to regulate their atlases and does not enable defining mappings of these surfaces. Networks that are agnostic to the discretization of surfaces have recently been studied in [Sharp et al 2022], however that work is focused on analysis of the surface by designing discretization-invariant diffusion operators, and less fitting for producing deformations of high-resolution surfaces (see also Fig 14 for a comparison of the two methods).…”