# Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations Source: [Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations]() Collection index: [llms.txt]() Data updated: 2026-07-09T05:00:01.225Z Authors: Gunner Levi Howe Published: 2026-07-07T04:23:23.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Scattered-domain fitting on Scattered-domain blobs sampling 256x256, N=8,192 (test)](): Scattered-domain fitting; Scattered-domain blobs sampling 256x256, N=8,192 (test); PSNR: 22.53. - [Scattered-domain fitting on Scattered-domain ramp sampling 256x256, N=8,192 (test)](): Scattered-domain fitting; Scattered-domain ramp sampling 256x256, N=8,192 (test); PSNR: 22.31. ## Original sources and implementation [Original paper]() [PDF]()