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.

翻译：在数据驱动的三维形状分析与生成领域，从点云、体素和神经隐式场等局部化表示中估计全局拓扑特征是一个长期存在的挑战。本文提出了一种新颖的可微算法，旨在精确估计三维形状的全局拓扑，克服了传统基于网格重建和拓扑数据分析方法的局限性。该方法确保了高精度、高效率、GPU兼容的即时计算。算法首先高效计算点云的自伴随Weingarten映射，并适配至其他模态。随后提取曲率，并通过对切向可微Voronoi单元进行积分来估计关键拓扑不变量，包括欧拉数和亏格。此外，算法实现了一种自动优化机制，基于拓扑不变量的完整性来优化局部活动标架和面积元。实验结果表明，该方法在多个数据集上均表现出优越性能。算法的鲁棒性和可微性确保了其可无缝集成到深度学习框架中，为三维形状分析的下游任务提供了广阔的应用潜力。