# CNNs on Surfaces using Rotation-Equivariant Features

arXiv: https://arxiv.org/abs/2006.01570 · HTML: https://mlsanity.com/paper/2006.01570/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jun 2020.

## Built on by (most cited first, 30 of 43)

- [NeuroMorph: Unsupervised Shape Interpolation and Correspondence in One Go](https://mlsanity.com/paper/2106.09431/) (2021)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [ShapeCrafter: A Recursive Text-Conditioned 3D Shape Generation Model](https://mlsanity.com/paper/2207.09446/) (2022)
- [A Survey of Non-Rigid 3D Registration](https://mlsanity.com/paper/2203.07858/) (2022)
- [DPFM: Deep Partial Functional Maps](https://mlsanity.com/paper/2110.09994/) (2021)
- [Unsupervised Learning of Robust Spectral Shape Matching](https://mlsanity.com/paper/2304.14419/) (2023)
- [Unsupervised Deep Multi-Shape Matching](https://mlsanity.com/paper/2207.09610/) (2022)
- [Deep Orientation-Aware Functional Maps: Tackling Symmetry Issues in Shape Matching](https://mlsanity.com/paper/2204.13453/) (2022)
- [Learning Multi-resolution Functional Maps with Spectral Attention for Robust Shape Matching](https://mlsanity.com/paper/2210.06373/) (2022)
- [Spatially and Spectrally Consistent Deep Functional Maps](https://mlsanity.com/paper/2308.08871/) (2023)
- [Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wall](https://mlsanity.com/paper/2212.05023/) (2022)
- [Multimodal Semi-Supervised Learning for 3D Objects](https://mlsanity.com/paper/2110.11601/) (2021)
- [Understanding and Improving Features Learned in Deep Functional Maps](https://mlsanity.com/paper/2303.16527/) (2023)
- [Field Convolutions for Surface CNNs](https://mlsanity.com/paper/2104.03916/) (2021)
- [Memory-Scalable and Simplified Functional Map Learning](https://mlsanity.com/paper/2404.00330/) (2024)
- [Learning Feature Aggregation for Deep 3D Morphable Models](https://mlsanity.com/paper/2105.02173/) (2021)
- [M"obius Convolutions for Spherical CNNs](https://mlsanity.com/paper/2201.12212/) (2022)
- [Implicit field supervision for robust non-rigid shape matching](https://mlsanity.com/paper/2203.07694/) (2022)
- [SRFeat: Learning Locally Accurate and Globally Consistent Non-Rigid Shape Correspondence](https://mlsanity.com/paper/2209.07806/) (2022)
- [Single Mesh Diffusion Models with Field Latents for Texture Generation](https://mlsanity.com/paper/2312.09250/) (2023)
- [DeltaConv: Anisotropic Operators for Geometric Deep Learning on Point Clouds](https://mlsanity.com/paper/2111.08799/) (2021)
- [G-MSM: Unsupervised Multi-Shape Matching with Graph-based Affinity Priors](https://mlsanity.com/paper/2212.02910/) (2022)
- [NeuralODF: Learning Omnidirectional Distance Fields for 3D Shape Representation](https://mlsanity.com/paper/2206.05837/) (2022)
- [DiscoMatch: Fast Discrete Optimisation for Geometrically Consistent 3D Shape Matching](https://mlsanity.com/paper/2310.08230/) (2023)
- [Generalizable Local Feature Pre-training for Deformable Shape Analysis](https://mlsanity.com/paper/2303.15104/) (2023)
- [PoissonNet: A Local-Global Approach for Learning on Surfaces](https://mlsanity.com/paper/2510.14146/) (2025)
- [AttWalk: Attentive Cross-Walks for Deep Mesh Analysis](https://mlsanity.com/paper/2104.11571/) (2021)
- [Fast Globally Optimal and Geometrically Consistent 3D Shape Matching](https://mlsanity.com/paper/2504.06385/) (2025)
- [Nonlinearities in Steerable SO(2)-Equivariant CNNs](https://mlsanity.com/paper/2109.06861/) (2021)
- [Interpolated SelectionConv for Spherical Images and Surfaces](https://mlsanity.com/paper/2210.10123/) (2022)

## Abstract (arXiv)

This paper is concerned with a fundamental problem in geometric deep learning that arises in the construction of convolutional neural networks on surfaces. Due to curvature, the transport of filter kernels on surfaces results in a rotational ambiguity, which prevents a uniform alignment of these kernels on the surface. We propose a network architecture for surfaces that consists of vector-valued, rotation-equivariant features. The equivariance property makes it possible to locally align features, which were computed in arbitrary coordinate systems, when aggregating features in a convolution layer. The resulting network is agnostic to the choices of coordinate systems for the tangent spaces on the surface. We implement our approach for triangle meshes. Based on circular harmonic functions, we introduce convolution filters for meshes that are rotation-equivariant at the discrete level. We evaluate the resulting networks on shape correspondence and shape classifications tasks and compare their performance to other approaches.