Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations

Researchers derived smooth Monte-Carlo estimators for Minkowski functionals of neural fields, achieving 1-3% accuracy in 2D and 3D topology estimation at 250x speedup over persistent homology. However, in 3D neural-SDF fitting, gradient descent hides topological noise below sampling density, making the estimator blind and requiring cubically many points to close the gap. The grid-based PH baseline remains more reliable for 3D benchmarks.

arXiv:2607.05815v1 Announce Type: new Abstract: 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.