[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83002-en":3,"doc-seo-83002-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83002,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Level Crossing Density as a Mesh-Free High Frequency Auxiliary Loss for Implicit Neural Representations","Coordinate-MLP implicit neural representations suffer from spectral bias, learning low-frequency structure faster than high-frequency detail. The work proposes a spatial-domain auxiliary loss derived from the Rice level-crossing density in random-field theory. Using the co-area formula, level-crossing density becomes a differentiable functional of gradient magnitude concentrated on level sets and a closed-form proxy for high-frequency energy. The loss is computed pointwise on arbitrary, possibly scattered samples via a smoothed Monte-Carlo estimator—requiring no grid or FFT—and is matched to supervision crossing densities across multiple levels.","arXiv :2607 .058 15v 1 [ cs .LG] 7 Jul 2026  \nLevel-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations  \nGunner Levi Howe  \nIndependent Researcher  \n[gunnerlevihowe@gmail. com](gunnerlevihowe@gmail. com)  \nJuly 2026  \nAbstract  \nCoordinate-MLP implicit neural representations (INRs) exhibit spectral bias: they fit low-frequency structure quickly and high-frequency detail slowly or not at all. Existing remedies act in the frequency domain (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 on which a discrete Fourier transform is available. We revisit a classical object from random-field theory, the Rice 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 on level sets, and for stationary processes it is monotone in the ratio of spectral moments p λ2 /λ0—i.e., it is a spatial-domain, closed-form proxy for high-frequency content. We construct a smoothed Monte-Carlo estimator 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 signal across a set of levels. We validate the estimator against exact crossing counts and the Rice formula, then evaluate the loss under strict information parity against the Focal Frequency Loss, normalized Sobolev supervision, 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 auxiliary spectral loss is transformative (+2 .3–3.0dB over MSE-only on PE-MLP, +1 .4–1.8dB on SIREN); on an edge-dominated natural image the crossing-density loss matches the alternatives—within 0 .2–0.5dB of FFL on PSNR, matching or exceeding it on SSIM—without beating them, while on a statistically homogeneous texture, the regime closest to the stationary fields of the Rice theory, it overtakes FFL by 0.6dB. Its further distinctions are structural: it is the only loss requiring neither a sampling grid nor trustworthy pointwise gradient targets, and an oracle-gradient diagnostic shows it is uniquely insensitive to gradient-target quality. On dense regular grids, where auxiliary drives are unnecessary, finite-difference gradient targets actively hurt—a caution that applies equally to Sobolev training. We release code, raw results, and all experiment scripts.  \n1 Introduction  \nImplicit neural representations (INRs) parameterize a signal as a neural network fθ : Rd → Rm mapping coordinates to values, and have become a standard representation for images, signed distance fields, and radiance fields [16, 18 , 24 , 28] . Their best-known failure mode is spectral bias [3, 20]: trained with a plain reconstruction loss, coordinate MLPs learn low frequencies first and high frequencies slowly, blurring fine detail. The community’s responses fall into three families: (i) architectural fixes—Fourier feature encodings [25], periodic activations [24], their variable-periodic [14], wavelet [23], and Gaussian [21] relatives, multiplicative filter networks [6], band-limited architectures [13], and hash grids [17]; (ii) loss-based fixes—frequency-domain objectives such as the Focal Frequency Loss (FFL) [9] and gradient-domain (Sobolev) supervision [4, 30]; and (iii) curricula that schedule frequency content coarse-to-fine [8, 29] .  \nThe loss-based family has a structural blind spot. A discrete Fourier transform is defined on a regular grid. When supervision arrives as scattered samples—point clouds for neural SDFs, non-uniformly sampled sensor data, adaptively sampled renderings—a frequency-domain loss must first resample the scattered data onto a  \ngrid,","cbCaimdbf9Edlgt0","https://ap.wps.com/l/cbCaimdbf9Edlgt0","pdf",1043568,3,1,14,"English","en",105,"# Introduction\n## Spectral bias in coordinate-MLP INRs\n## Limitations of existing frequency- and gradient-based losses\n## Rice level-crossing density as an auxiliary objective\n## Differentiable Monte-Carlo estimator and loss matching","[{\"question\":\"What problem does the document address in implicit neural representations?\",\"answer\":\"It addresses spectral bias: coordinate-MLPs fit low-frequency structure quickly while high-frequency detail is learned slowly or not effectively under standard reconstruction losses.\"},{\"question\":\"How does the proposed auxiliary loss help with high-frequency learning?\",\"answer\":\"It uses Rice level-crossing density, linking crossing statistics to gradient magnitude on level sets, which serves as a spatial-domain proxy for high-frequency content. The loss matches an estimated crossing-density profile to the supervision signal at selected levels.\"},{\"question\":\"Why is the method considered mesh-free and grid-free?\",\"answer\":\"The estimator is evaluated pointwise at arbitrary training coordinates using automatic differentiation for exact gradients, and it avoids resampling onto a grid, FFTs, and mesh-based computations. This removes assumptions about regular grids and avoids grid-interpolation smearing of high-frequency information.\"}]",1784184589,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"level-crossing-density-as-a-mesh-free-high-frequency-auxiliary-loss-for-implicit-neural-representations","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/level-crossing-density-as-a-mesh-free-high-frequency-auxiliary-loss-for-implicit-neural-representations/83002/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the document address in implicit neural representations?","Question",{"text":75,"@type":76},"It addresses spectral bias: coordinate-MLPs fit low-frequency structure quickly while high-frequency detail is learned slowly or not effectively under standard reconstruction losses.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed auxiliary loss help with high-frequency learning?",{"text":80,"@type":76},"It uses Rice level-crossing density, linking crossing statistics to gradient magnitude on level sets, which serves as a spatial-domain proxy for high-frequency content. The loss matches an estimated crossing-density profile to the supervision signal at selected levels.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is the method considered mesh-free and grid-free?",{"text":84,"@type":76},"The estimator is evaluated pointwise at arbitrary training coordinates using automatic differentiation for exact gradients, and it avoids resampling onto a grid, FFTs, and mesh-based computations. 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