用神经网络学习点到曲面的最短路径，提升点云重建精度与鲁棒性。
Neural Shortest Path for Surface Reconstruction from Point Clouds
- 将距离函数与梯度分解学习，通过变分分裂法联合优化。
- 在多个数据集上实现更高质量重建，对噪声和稀疏数据更鲁棒。
- 首次理论证明收敛性，适合高精度曲面重建研究者使用。
本文提出神经最短路径（NSP），一种向量值隐式神经表示（INR），用于逼近距离函数及其梯度。其核心是学习精确最短路径（ESP），引导任意点指向目标曲面上最近点。NSP被分解为模长与方向两部分，采用变分分裂方法分别逼近距离函数及其梯度。与以往直接学习距离函数的方法不同，NSP能同步恢复距离函数与梯度。我们从理论上证明了该分解表示在$H^1$范数下模长的收敛性。此外，设计了一种新型损失函数，确保全局最小值即为ESP。在多样数据集上的综合实验验证了其重建高质量曲面的能力，对噪声和数据稀疏具有强鲁棒性。数值结果表明，相比现有最优方法有显著提升，凸显学习距离函数与其梯度乘积（即ESP）对复杂曲面建模的重要性。
原文摘要 · Abstract (English)
In this paper, we propose the neural shortest path (NSP), a vector-valued implicit neural representation (INR) that approximates a distance function and its gradient. The key feature of NSP is to learn the exact shortest path (ESP), which directs an arbitrary point to its nearest point on the target surface. The NSP is decomposed into its magnitude and direction, and a variable splitting method is used that each decomposed component approximates a distance function and its gradient, respectively. Unlike to existing methods of learning the distance function itself, the NSP ensures the simultaneous recovery of the distance function and its gradient. We mathematically prove that the decomposed representation of NSP guarantees the convergence of the magnitude of NSP in the $H^1$ norm. Furthermore, we devise a novel loss function that enforces the property of ESP, demonstrating that its global minimum is the ESP. We evaluate the performance of the NSP through comprehensive experiments on diverse datasets, validating its capacity to reconstruct high-quality surfaces with the robustness to noise and data sparsity. The numerical results show substantial improvements over state-of-the-art methods, highlighting the importance of learning the ESP, the product of distance function and its gradient, for representing a wide variety of complex surfaces.
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