CanonNet: Canonical Ordering and Curvature Learning for Point Cloud Analysis

TL;DR

CanonNet introduces a lightweight pipeline that canonicalizes point clouds via spectral-graph methods to achieve permutation and orientation invariance, then learns local geometric structure from synthetic surfaces with known curvatures. The method uses a small MLP plus second-degree polynomial features to classify surface types and estimate Gaussian and mean curvature, operating on tiny 20-point patches. It achieves state-of-the-art mean curvature estimation on synthetic data and competitive descriptor retrieval on cross-domain benchmarks with about 100× fewer parameters than comparable methods, enabling efficient deployment. The combination of canonical preprocessing and curvature-aware training demonstrates that mathematical preprocessing can complement neural architectures for robust, resource-efficient point cloud analysis.

Abstract

Point cloud processing poses two fundamental challenges: establishing consistent point ordering and effectively learning fine-grained geometric features. Current architectures rely on complex operations that limit expressivity while struggling to capture detailed surface geometry. We present CanonNet, a lightweight neural network composed of two complementary components: (1) a preprocessing pipeline that creates a canonical point ordering and orientation, and (2) a geometric learning framework where networks learn from synthetic surfaces with precise curvature values. This modular approach eliminates the need for complex transformation-invariant architectures while effectively capturing local geometric properties. Our experiments demonstrate state-of-the-art performance in curvature estimation and competitive results in geometric descriptor tasks with significantly fewer parameters (\textbf{100X}) than comparable methods. CanonNet's efficiency makes it particularly suitable for real-world applications where computational resources are limited, demonstrating that mathematical preprocessing can effectively complement neural architectures for point cloud analysis. The code for the project is publicly available \hyperlink{https://benjyfri.github.io/CanonNet/}{https://benjyfri.github.io/CanonNet/}.

Figures (4)

• Figure 1: The different possible surfaces given the Gaussian (K) and mean (H) curvature as described in \ref{['sec:training']}. Note that up to rigid motion there are 4 different types of surfaces.
• Figure 2: The complete CanonNet pipeline: Synthetic data generation (LHS): We sample points from analytically defined surfaces with known principal curvatures ${{\kappa}_1},{{\kappa}_2}$. Processing and classification (RHS): The point cloud is transformed into canonical orientation and processed by an MLP that performs supervised classification into four geometric surface types (Saddle, Parabolic, Valley, Plane) using ground truth curvature labels.
• Figure 3: Illustration of the preprocessing pipeline for establishing canonical point cloud representation as described in \ref{['sec:preprocessing']}: (I) Input point cloud with arbitrary ordering and orientation. (II) Construction of fully connected graph with heat kernel weights and computation of normalized graph Laplacian. (III) Reordering points along a 1D axis based on Laplacian eigenvector values, ensuring consistency regardless of initial point indexing or spatial orientation and position. (IV) Identification of geometric landmarks: center of mass, 'M', and the point corresponding to the largest eigenvector value, 'A'. (V) First standardization rotation aligning center of mass with positive z-axis. (VI) Second standardization rotation placing 'A' in the XZ-plane with positive x-coordinate, completing the transformation pipeline that ensures both permutation and rigid-transformation invariance.
• Figure 4: (a) Impact of canonical preprocessing pipeline, showing consistent 10-17% accuracy improvements across architectures and noise levels (solid: baseline, dashed: with preprocessing). (b) Effect of second-degree polynomial features, yielding approximately 5% accuracy improvement across all tested architectures (solid: baseline, dashed: with polynomial features). (c) Impact of Laplacian eigenvalues, showing minimal differences (±0.2%), suggesting geometric information is already well-captured by existing features.

Theorems & Definitions (2)

• Theorem 1
• proof