Differentiable Topology Estimating from Curvatures for 3D Shapes

TL;DR

This paper presents a differentiable algorithm that accurately estimates the global topology of 3D shapes from point clouds and other modalities, enabling efficient integration into deep learning workflows.

Contribution

It introduces a novel, GPU-compatible method for topological estimation using curvature integration and auto-optimization, surpassing traditional mesh-based approaches.

Findings

High accuracy in topology estimation demonstrated on multiple datasets.

Efficient GPU implementation enables real-time processing.

Robustness confirmed across various 3D shape representations.

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.