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

Coordinate-MLP implicit neural representations (INRs) exhibit spectral bias: they fit low-frequencystructure quickly and high-frequency detail slowly or not at all. Existing remedies act in the frequencydomain (e.g., the Focal Frequency Loss), on activations and encodings (SIREN, Fourier features, FINER),or through coarse-to-fine curricula—and the loss-based remedies presuppose a regular sampling grid onwhich a discrete Fourier transform is available. We revisit a classical object from random-field theory, theRice level-crossing density, and turn it into a differentiable training objective. By the co-area formula,the density of level crossings of a field is a direct functional of its gradient magnitude concentrated onlevel sets, and for stationary processes it is monotone in the ratio of spectral moments √λ2/λ0—i.e., it isa spatial-domain, closed-form proxy for high-frequency content. We construct a smoothed Monte-Carloestimator of the crossing density that is evaluated pointwise at arbitrary sample locations, requires nogrid, no FFT, and no mesh, and match it to the empirical crossing density of the supervision signalacross a set of levels. We validate the estimator against exact crossing counts and the Rice formula, thenevaluate the loss under strict information parity against the Focal Frequency Loss, normalized Sobolevsupervision, and MSE-only training, on regular grids and on scattered non-uniform samples, and wereport the outcome without embellishment. Where supervision is scarce and irregular, every auxiliaryspectral loss is transformative (+2.3–3.0dB over MSE-only on PE-MLP, +1.4–1.8dB on SIREN); onan edge-dominated natural image the crossing-density loss matches the alternatives—within 0.2–0.5dBof FFL on PSNR, matching or exceeding it on SSIM—without beating them, while on a statisticallyhomogeneous texture, the regime closest to the stationary fields of the Rice theory, it overtakes FFL by0.6dB. Its further distinctions are structural: it is the only loss requiring neither a sampling grid nortrustworthy pointwise gradient targets, and an oracle-gradient diagnostic shows it is uniquely insensitiveto gradient-target quality. On dense regular grids, where auxiliary drives are unnecessary, finite-differencegradient targets actively hurt—a caution that applies equally to Sobolev training. We release code, rawresults, and all experiment scripts.