# ZerNet: Convolutional Neural Networks on Arbitrary Surfaces via Zernike Local Tangent Space Estimation

arXiv: https://arxiv.org/abs/1812.01082 · HTML: https://mlsanity.com/paper/1812.01082/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2018.

## Built on by (most cited first, 8 of 8)

- [A Survey of Non-Rigid 3D Registration](https://mlsanity.com/paper/2203.07858/) (2022)
- [Challenges and opportunities for machine learning in multiscale computational modeling](https://mlsanity.com/paper/2303.12261/) (2023)
- [PoissonNet: A Local-Global Approach for Learning on Surfaces](https://mlsanity.com/paper/2510.14146/) (2025)
- [MomentsNeRF: Leveraging Orthogonal Moments for Few-Shot Neural Rendering](https://mlsanity.com/paper/2407.02668/) (2024)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [Noise-to-Norm Reconstruction for Industrial Anomaly Detection and Localization](https://mlsanity.com/paper/2307.02836/) (2023)
- [SOMA: Efficient Multi-turn LLM Serving via Small Language Model](https://mlsanity.com/paper/2605.11317/) (2026)
- [Intrinsic and Triangulation-Agnostic Attention: A Simple and Powerful Approach for Learning on Meshes](https://mlsanity.com/paper/2607.24954/) (2026)

## Abstract (arXiv)

In this paper, we propose a novel formulation to extend CNNs to two-dimensional (2D) manifolds using orthogonal basis functions, called Zernike polynomials. In many areas, geometric features play a key role in understanding scientific phenomena. Thus, an ability to codify geometric features into a mathematical quantity can be critical. Recently, convolutional neural networks (CNNs) have demonstrated the promising capability of extracting and codifying features from visual information. However, the progress has been concentrated in computer vision applications where there exists an inherent grid-like structure. In contrast, many geometry processing problems are defined on curved surfaces, and the generalization of CNNs is not quite trivial. The difficulties are rooted in the lack of key ingredients such as the canonical grid-like representation, the notion of consistent orientation, and a compatible local topology across the domain. In this paper, we prove that the convolution of two functions can be represented as a simple dot product between Zernike polynomial coefficients; and the rotation of a convolution kernel is essentially a set of 2-by-2 rotation matrices applied to the coefficients. As such, the key contribution of this work resides in a concise but rigorous mathematical generalization of the CNN building blocks.