用曲率计算3D形状的拓扑属性，可直接用于深度学习模型。
Differentiable Topology Estimating from Curvatures for 3D Shapes
- 基于点云曲率与微分诺伊曼区域积分估计拓扑不变量。
- 在多个数据集上实现高精度、实时计算，支持GPU加速。
- 方法可微，适合嵌入生成模型等下游任务，无需网格重建。
在数据驱动的3D形状分析与生成领域，从点云、体素或神经隐式场等局部表示中估计全局拓扑特征是一项长期挑战。本文提出一种新型可微算法，精准估计3D形状的全局拓扑结构，突破了传统依赖网格重构与拓扑数据分析方法的局限。该方法首先高效计算点云的自伴随Weingarten映射，并推广至其他模态；进而提取曲率，并通过在切空间可微的Voronoi单元上积分，估算欧拉数和亏格等关键拓扑不变量。此外，引入自动优化机制，根据拓扑不变量的完整性动态调整局部运动框架与面积元素。实验表明，该方法在多个数据集上均表现优异。其鲁棒性与可微特性使其能无缝集成到深度学习框架中，为3D形状分析的下游任务提供了广阔应用前景。
原文摘要 · Abstract (English)
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.
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