Differentiable Topology Estimating from Curvatures for 3D Shapes

TL;DR

This work tackles the challenge of obtaining global topology from localized 3D representations by introducing a differentiable curvature-based pipeline. It builds moving frames and local tangent areas, estimates a self-adjoint Weingarten map to obtain principal curvatures, and integrates curvature over tangent Voronoi elements using a differentiable Gauss-Bonnet formulation to yield the Euler characteristic $\chi_M$ and genus $g(M)$. A self-optimization (integrity-well) loss refines normals and area elements to align topology with invariants, enabling stable end-to-end training. Empirical results demonstrate accurate curvature estimation, robust topology estimation across diverse datasets, and superior speed on GPUs, highlighting strong potential for integration into DL-based 3D shape analysis and generation.

Abstract

In the field of data-driven 3D shape analysis and generation, the estimation of global topological features from localized representations such as point clouds, voxels, and neural implicit fields is a longstanding challenge. This paper introduces a novel, differentiable algorithm tailored to accurately estimate the global topology of 3D shapes, overcoming the limitations of traditional methods rooted in mesh reconstruction and topological data analysis. The proposed method ensures high accuracy, efficiency, and instant computation with GPU compatibility. It begins with an efficient calculation of the self-adjoint Weingarten map for point clouds and its adaptations for other modalities. The curvatures are then extracted, and their integration over tangent differentiable Voronoi elements is utilized to estimate key topological invariants, including the Euler number and Genus. Additionally, an auto-optimization mechanism is implemented to refine the local moving frames and area elements based on the integrity of topological invariants. Experimental results demonstrate the method's superior performance across various datasets. The robustness and differentiability of the algorithm ensure its seamless integration into deep learning frameworks, offering vast potential for downstream tasks in 3D shape analysis.

Figures (10)

• Figure 1: Our method provides a novel approach to computing the Euler characteristic of 3D shapes in a differentiable manner by analyzing point clouds sampled from any 3D representations. The process begins by calculating Gaussian curvature. followed by a numerically stable integration step that yields an initial estimate of the Euler characteristic. The curvature is visualized as the color map on the 3D models. The ground truth Euler characteristics are $2, -10, -2, -3, -44, 0$ from left to right, and the estimated values are shown under each 3D object.
• Figure 2: Tangent Voronoi Diagram for estimating areas.
• Figure 3: Illustration of the Weingarten map for the manifold $M$.
• Figure 4: The estimation of the Weingarten map and the curvatures.
• Figure 5: Integrity-well Loss.