用无网格方法高效计算隐式神经表示的高频拓扑损失，提升几何重建精度。
Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations
- 基于蒙特卡洛和自动微分，无需网格直接估计水平集的拓扑特征。
- 在2D/3D中误差仅1-3%，计算耗时仅3毫秒，比传统方法快250倍。
- 揭示四条关键设计准则，避免拓扑损失失效，适合几何建模与逆问题研究者。
连续神经场的上穿集的闵可夫斯基函数（面积、边界测度、欧拉示性数）可描述其水平集形态；欧拉示性数是获取拓扑信息最经济的方式。本文通过共面积公式与高斯-博内定理，仅用自动微分，在散点上推导出三者的平滑蒙特卡洛估计器：无需网格、无需复形、无需持久同调。该估计器在2D和3D中对精确拓扑的误差为1-3%，每次迭代耗时约3毫秒，而基于立方体网格的持久同调损失需650-1000毫秒，速度差距达250倍。我们提出四项设计准则，否则这些损失会无声失效：密集的水平梯度（不变量在参数远离跃迁处平坦）、$C^2$主干网络（ReLU网络在拐角隐藏曲率）、完整闵可夫斯基向量（仅欧拉示性数为交替和，易被碎片-孔洞抵消欺骗；边界项可关闭漏洞）、采样尺度覆盖。在2D中，该向量值损失是唯一在控制对比中同时修复拓扑（3/3种子成功）且保持保真度的方法——均匀平滑修复代价高达11-17倍，而仅用欧拉项则无法修复。但在3D神经SDF拟合中，一种我们认为普遍存在于采样软拓扑目标的失效模式出现：梯度下降将拓扑噪声主动隐藏在采样密度以下，使估计器不可见——伪特征计数在4倍采样下仍不变，关闭窗口需立方级点数，抹去成本优势。基于网格的持久同调基线（其复形等于评估分辨率）在相同基准上表现更优（4/9精确；中位 $b_1$ 误差1，而我们的超过 $10^4$）。目前，持久同调250倍的成本，正是无零空间的代价。我们已发布估计器、验证记录与基准测试。
原文摘要 · Abstract (English)
The Minkowski functionals of a field's excursion sets -- area, boundary measure, and Euler characteristic -- describe its level-set morphology; the Euler characteristic is the cheapest handle on topology. We derive smooth Monte-Carlo estimators for all three of a continuous neural field, evaluated at scattered points via the co-area formula and Gauss-Bonnet, using only autodiff: no grid, no complex, no persistence. The estimator is accurate to 1-3% against exact topology in 2D and 3D, and costs about 3 ms per iteration where a persistent-homology (PH) loss on a cubical grid costs 650-1000 ms -- a 250x gap. We establish four design rules without which these losses silently fail: a dense level ladder (invariants are flat in the parameters away from transitions), a $C^2$ backbone (ReLU nets hide curvature in kinks), the full Minkowski vector (Euler characteristic alone is an alternating sum, gamed by debris-hole cancellation; pricing perimeter closes the channel), and sampling-scale coverage. In 2D the vector-valued cap is the only method in a controlled comparison that both repairs topology (3/3 seeds) and preserves fidelity -- uniform smoothing repairs at 11-17x the fidelity cost, and the Euler term alone repairs nothing. In 3D neural-SDF fitting, however, a failure mode we believe general to any sampled soft topology objective appears: gradient descent adversarially hides topological noise below the sampling density, where the estimator is blind -- spurious-feature counts are invariant to 4x more samples, and closing the window needs cubically many points, erasing the cost advantage. A grid-based PH baseline, whose complex is the evaluation resolution, solves the same benchmark ($4/9$ exact; median $b_1$ error 1 vs. ours above $10^4$). The 250x cost of persistence is, at present, the price of having no null space. We release estimators, receipts, and benchmarks.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。