{"id":"f48671ef-0afd-462a-81c8-dcd225ca1589","arxiv_id":"2603.19834","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":6,"one_line_summary":"Planar Fourier-boundary surfels let a single radiance-field model render at continuous levels of detail by truncating coefficients, with STE gradients and HYDRA densification enabling stable training.","lead":"Fourier Splatting replaces fixed Gaussian or disc surfels with planar primitives whose boundaries are Fourier series, so one trained model can drop quality at runtime by truncating coefficients. This gives bandwidth-friendly level-of-detail without pruning whole primitives, matching or beating other planar splatters on standard novel-view benchmarks.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper's central claim is empirical and scoped carefully: a planar Fourier-boundary surfel that is inherently LoD-scalable by coefficient truncation, enabled by STE + HYDRA, and competitive on standard NVS benchmarks. The reader correctly identified the optimization-stability assumption as the softest point; the manuscript already tests it via ablation and LoD analysis, so the assumption does not undermine the claim. No hidden mathematical inconsistency (normalization, ray-plane intersection inheritance from 2DGS, MCMC relocation formulas) is present. Novelty is incremental but real within the planar-splatting line; correctness risk remains low. Therefore the ACCEPT / HIGH verdict stands without adjustment.","tokens_in":18806,"tokens_out":513,"duration_ms":5058,"concrete_test":"Re-train one outdoor Mip-NeRF 360 scene (e.g., bicycle) with the full pipeline, then at test time render the identical checkpoint at K_active = 1,2,...,6 and measure PSNR/SSIM/LPIPS against the full-K render; if the quality gap between K=1 and K=6 is <0.15 dB or the intermediate steps are non-monotonic, the expressiveness/truncation claim would be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (STE + HYDRA sufficiency for unlocking non-circular Fourier boundaries) is already the right soft spot, but the paper supplies direct empirical checks that keep it from being load-bearing against the central claim. Ablation Table 2 on Tanks & Temples shows clear drops when either component is removed (w/o STE: -0.34 dB PSNR; w/o lobe decomposition: -0.22 dB), and the LoD curves (Fig. 7) plus qualitative truncation comparisons (Fig. 5) demonstrate that higher-order coefficients are active and that progressive truncation produces the claimed graceful degradation rather than collapse to discs. Amplitude normalization (Eq. 5) and the hard power window (Eq. 7) are consistent with the forward model; the STE (Eq. 8) is a standard gradient-extension device whose fixed hyperparameters are reported. No internal contradiction appears between the claimed runtime truncation property and the optimization machinery that realizes it. The SOTA-among-planar and competitive-with-volumetric numbers in Table 1 are therefore on solid ground for an empirical methods paper.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces Fourier Splatting, a planar surfel primitive whose boundary is parameterized by a Fourier series (Eq. 4) with amplitude normalization (Eq. 5) and a power-window opacity (Eq. 7). A single trained model can be rendered at continuous levels of detail by truncating coefficients at runtime. Optimization uses a straight-through estimator (Eq. 8) to supply exterior gradients and HYDRA, an MCMC-compatible densification procedure that either geometrically splits or MLP-decomposes multi-lobed primitives. On Mip-NeRF 360 and Tanks & Temples the method reports state-of-the-art metrics among planar primitives and competitive LPIPS versus volumetric baselines; ablations isolate the STE and HYDRA contributions, and qualitative/quantitative LoD curves illustrate graceful degradation under truncation.","tokens_in":19133,"tokens_out":882,"duration_ms":18492,"significance":"If the empirical claims hold, the work supplies the first radiance-field primitive whose expressiveness itself is continuously scalable, decoupling fidelity from primitive count. This is a genuine conceptual advance over pruning- or hierarchy-based LoD schemes and is directly useful for bandwidth-constrained delivery. The paper supplies concrete supporting evidence: public-benchmark tables under a shared protocol, component ablations (Table 2), DTU surface metrics in the supplement, and explicit truncation experiments (Figs. 5 and 7). The Fourier orthonormality argument for truncation is classical and parameter-free, strengthening the scalability claim.","major_comments":[{"comment":"§5.3 and Fig. 7 demonstrate monotonic quality loss under coefficient truncation for the authors’ own model, and Fig. 5 offers a qualitative side-by-side with Octree-GS. However, no quantitative rate–distortion comparison (PSNR/LPIPS versus bits or versus equal primitive count) is provided against any pruning- or hierarchy-based baseline. Without such a curve the claim of “more graceful” degradation remains only partially substantiated; a single equal-budget or equal-bitrate plot would make the central scalability advantage load-bearing rather than suggestive.","section":null},{"comment":"Table 3 reports arithmetic operations per primitive but the manuscript never states measured FPS or peak memory on the same hardware used for the quality tables. Because the abstract and conclusion advertise a “versatile solution for bandwidth-constrained high-fidelity rendering,” the absence of wall-clock numbers leaves open whether the extra 8 ops per frequency (and the STE/HYDRA machinery) remain real-time once K > 1. A short runtime column or paragraph would close this gap.","section":null}],"minor_comments":[{"comment":"Several concatenated words appear in the supplied text (“fidelitystrictlytothenumber”, “density-drivenoptimization”, etc.); these are presumably paste artifacts but should be cleaned for the camera-ready version.","section":null},{"comment":"Eq. (8) introduces β and γ without stating that they are frozen at 3.0 / 0.5 until the supplement; a one-sentence note in the main text would help reproducibility.","section":null},{"comment":"The HYDRA MLP architecture (2×2048) and its training objective are described only in the supplement; a brief pointer in §4.3 would improve self-containment.","section":null},{"comment":"Fig. 1 caption claims “progressively improved rendering quality” but the figure itself shows only primitive shapes; a small inset of the corresponding rendered patch would make the visual argument clearer.","section":null},{"comment":"In the related-work discussion of surface primitives, BBSplat is cited for textured billboards; a short remark on how Fourier boundaries differ from learned alpha masks would sharpen the positioning.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The work is a solid empirical methods contribution that cleanly occupies a previously empty niche (per-primitive rather than per-scene LoD). The two major comments are easily addressable with existing experimental infrastructure and do not threaten the core claims. I would be comfortable accepting after a light revision cycle."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that they actually deliver a primitive whose expressiveness can be dialed at runtime by truncating Fourier coefficients, without retraining or pruning. That is new relative to 2DGS, Triangle Splatting, BBSplat, and the various LoD-GS / Octree-GS lines, which all scale by adding or removing whole primitives.\n\nWhat works: the boundary is a classical Fourier series on a planar surfel (Eq. 4), amplitude-normalized so it stays inside the circumradius (Eq. 5). Truncation is therefore free and L2-optimal by orthonormality; they show the expected graceful degradation against Octree-GS (Figs. 5, 7). They get the hard power-window opacity to train by a straight-through estimator that leaks gradient outside the boundary, and they keep multi-lobed shapes from stalling densification with HYDRA (learned lobe split inside MCMC). Ablations on Tanks & Temples are clean: drop STE and you lose 0.34 dB; drop lobe decomposition and you lose 0.22 dB. Table 1 puts them first among planar methods on Mip-NeRF 360 and competitive on LPIPS with the volumetric pack. Supplement DTU numbers stay close to 2DGS, so the planarity constraint is not just marketing. Math is standard, citations are on-point, free parameters are listed and fixed.\n\nSoft spots are real but secondary. K=6, STE betas, death thresholds, and per-scene budgets are free knobs; the paper does not claim otherwise. Bitrate is not optimized (they say so in Limitations), and the per-pixel op count is a bit higher than 2DGS. None of that undercuts the central claim.\n\nThis is for people who already work on real-time radiance fields and care about bandwidth or continuous LoD. It is a clear, well-executed methods paper, not a paradigm shift. I would send it to referees without hesitation; the evidence matches the claims and the weakest assumption (STE+HYDRA actually unlock complex shapes) is checked by ablation and LoD curves. Worth reading and, if you are in the planar-splatting lane, worth citing.","headline":"Solid planar-splatting paper: Fourier-boundary surfels give real runtime LoD by coefficient truncation, with clean ablations and SOTA-among-planar numbers.","tokens_in":19756,"tokens_out":557,"would_cite":true,"duration_ms":6496,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"A single trained model of Fourier-boundary surfels can be rendered at continuous levels of detail by truncating coefficients at runtime.","keywords":["novel view synthesis","radiance fields","splatting","Fourier primitives","level of detail","planar surfels","scalable rendering"],"falsifier":"Train a full-K model, then measure image metrics and visual quality while successively truncating coefficients to K=1; if quality does not degrade more gracefully than an equal-budget Octree-GS or pruning baseline, or if the learned boundaries remain near-circular, the central scalability claim fails.","tokens_in":19675,"feed_emoji":"🌀","tokens_out":601,"duration_ms":6543,"temperature":0.7,"pith_summary":"Existing real-time radiance-field methods improve quality only by adding more primitives and reduce quality only by pruning them. This paper claims that the primitive itself can be made scalable: each planar surfel is given a closed boundary described by a Fourier series whose coefficients control shape complexity. At the lowest order the primitive is a disc; each extra frequency lets it approximate more complex closed curves. Because the Fourier basis is orthogonal, discarding higher coefficients at render time yields a coarser but still valid shape without retraining. The authors show that, with a straight-through estimator to keep gradients alive outside the hard boundary and a densification routine that splits multi-lobed primitives, the representation reaches state-of-the-art quality among planar methods and stays competitive with volumetric ones, while quality degrades more gracefully under coefficient truncation than under primitive pruning.","feed_headline":"One model, many detail levels: truncate Fourier coefficients","feed_subtitle":"Planar Fourier surfels let quality scale without pruning or retraining, matching top planar methods.","key_machinery":"The Fourier-encoded boundary (Eq. 4): a closed polar curve whose radius is the modulus of a truncated Fourier polynomial of the polar angle, normalized so the curve never exceeds a learned circumradius. Truncation of higher coefficients is the LoD operator; a straight-through estimator and the HYDRA lobe-decomposition densifier keep the coefficients learnable under a hard power-window opacity.","core_discovery":"Fourier Splatting is the first inherently scalable primitive for real-time radiance fields: planar surfels whose boundaries are Fourier-encoded can be trained once and then rendered at any continuous level of detail simply by truncating the active frequency coefficients, decoupling reconstruction fidelity from primitive count.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Train once, render any LoD: truncate Fourier surfel coeffs","Fourier-encoded planar surfels scale detail without pruning","One model, continuous quality via runtime Fourier truncation","Scalable radiance fields from Fourier-parameterized primitives","Truncate coefficients for multi-LoD Fourier splatting"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The straight-through estimator and the learned lobe-decomposition densifier must actually unlock complex, non-circular boundaries; if they fail, the method collapses back to ordinary disc primitives whose only scalability is pruning.","fun_headline_variants_meta":{"raw":{"variants":["Train once, render any LoD: truncate Fourier surfel coeffs","Fourier-encoded planar surfels scale detail without pruning","One model, continuous quality via runtime Fourier truncation","Scalable radiance fields from Fourier-parameterized primitives","Truncate coefficients for multi-LoD Fourier splatting"]},"model":"grok-4.5","effort":"low","cost_usd":0.004292,"raw_usage":{"total_tokens":1244,"prompt_tokens":697,"num_sources_used":0,"completion_tokens":80,"cost_in_usd_ticks":42920000,"prompt_tokens_details":{"text_tokens":697,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":467,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":697,"tokens_out":80,"duration_ms":4446,"temperature":1.0,"reasoning_tokens":467,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T21:53:48.733952+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Train a full-K model, then measure image metrics and visual quality while successively truncating coefficients to K=1; if quality does not degrade more gracefully than an equal-budget Octree-GS or pruning baseline, or if the learned boundaries remain near-circular, the central scalability claim fails.","supporting_citations":[],"review_version":1}