{"id":"244e9cb7-80a4-47b1-bd59-38c00691bb93","arxiv_id":"2501.11876","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"FNIN combines a Fourier neural operator with attention-based discontinuity weighting and a single weighted least-squares refinement to reconstruct surfaces from normal maps more accurately and faster than previous SfG methods.","lead":"This paper presents FNIN, a two-stage learning pipeline that reconstructs a 3D surface from a normal map: a Fourier neural operator integrates smooth regions, an attention network marks discontinuities, and a one-step weighted least-squares solve refines the output. The authors report lower reconstruction error and faster runtime than prior solvers on the LUCES and DiLiGenT benchmarks, and they release code on GitHub.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"DiLiGenT mean MAE from Table 2 (FNIN-S ≈4.23 mm) is not better than IPF (≈4.20 mm), and the '<0.1 mm' abstract claim holds for only 2 of 14 LUCES objects.","rationale":"The paper's central claim is inherently an empirical comparison: FNIN outperforms state-of-the-art solvers. That claim stands or falls on the reported tables. The tables show a mixed picture: FNIN wins several LUCES objects, but on DiLiGenT its aggregate MAE is not superior to IPF, and several object-level deficits are large (e.g., Harvest 12.42 vs 11.17; Reading 6.38 vs 4.54 for WLS). The abstract's '<0.1 mm' is a clear overstatement. The reader's weakest assumption about the convolution kernel is also legitimate and affects theoretical soundness, but even if the convolution assumption were fixed, the empirical SOTA assertion would remain unsupported by the given numbers. The method has real components (recursive multi-scale, attention, weighted least squares) and the ablation shows meaningful gains, so I would not reject the paper outright; a CONDITIONAL verdict requiring corrected claims and aggregate statistics remains appropriate.","tokens_in":17475,"tokens_out":10730,"duration_ms":104882,"concrete_test":"Compute the average and per-object MAE from Table 2 on DiLiGenT for all non-failing baselines, then run FNIN-S with five random seeds and perform a paired Wilcoxon signed-rank test against IPF. If the aggregate/median difference is not significantly in FNIN-S's favor (or is negative), the 'state-of-the-art MAE on DiLiGenT' claim is unsupported; also verify the LUCES '<0.1 mm' count by direct inspection of Table 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that FNIN/FNIN-S achieve state-of-the-art accuracy and efficiency for surface-from-gradients. The paper's own Table 2 does not support a dataset-level SOTA on DiLiGenT: averaging the nine per-object MAEs gives FNIN-S ≈4.23 mm versus IPF ≈4.20 mm, and FNIN loses on Buddha, Goblet, Harvest, and Reading while winning on Bear, Cat, Cow, Pot1, and Pot2. The supplementary Table 6 reports a standard deviation of only 0.0083 mm for the FNIN aggregate on DiLiGenT, so the 0.03 mm gap is unlikely to be within noise. On LUCES, the abstract's 'errors of fewer than 0.1 mm' is achieved for Cup and Bowl only; most objects (e.g., Ball 0.15 mm, Bell 0.12 mm) exceed 0.1 mm. The theoretical derivation of Eq. 6 also assumes a translation-invariant kernel, but the Green's function for the Dirichlet problem on a bounded non-rectangular domain depends on both arguments, so Stage I is not literally solving Eq. 5 as a pure convolution. Both issues together undermine the central accuracy claim, although the method itself may still be a useful practical contribution with corrected claims.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes FNIN, a two-stage learning framework for surface-from-gradients (SfG). Stage I uses a Fourier neural operator (FNO) inside a multi-resolution iterative ResNet-like architecture to approximate the integration operator in Fourier space, with a self-learning attention module and a detail-weighted loss to handle discontinuities. Stage II refines the reconstruction by solving a one-step weighted least squares problem with the Stage I depth as a prior and the attention map as weights. The method is trained on synthetic normal/depth pairs and evaluated without fine-tuning on real datasets: LUCES (14 objects, 2048x1536) and DiLiGenT (9 objects, 612x512). The paper reports lower MAE than many classical solvers and Rec-Net on several objects, shorter runtimes, and includes ablations on the FNO layers, truncated modes, attention, and discontinuity optimization, with code released on GitHub.","tokens_in":17801,"tokens_out":5332,"duration_ms":51031,"significance":"If the claims hold, FNIN would be a practically useful contribution: a learned normal integrator that is fast on high-resolution inputs, generalizes from synthetic training data to real objects without fine-tuning, and preserves discontinuities better than pure optimization baselines. The paper also brings neural-operator methods to a new inverse problem, which is a reasonable extension of FNO beyond PDE regression. The supplementary material strengthens the work by reporting five-trial means and standard deviations for the ablations, giving training details, and documenting limitations (surface islands, insufficient attention weights, nonlinear near-field bias). These elements make the empirical study substantially more reproducible than many vision papers. However, the central accuracy claims are currently overstated in the abstract and in the dataset-level SOTA statement, and the theoretical justification of the convolution kernel in Eq. 6 is not established for the general 2D bounded-domain problem.","major_comments":[{"comment":"The abstract claims 'achieving errors of fewer than 0.1 mm on tested objects', but Table 1 shows that only two of the fourteen LUCES objects meet this threshold (CUP <0.01 mm and BOWL 0.08 mm for FNIN; BOWL 0.05 mm for FNIN-S). Most objects have MAEs between 0.12 mm and 10.50 mm. Please either restrict the claim to specific objects or replace it with a summary statistic that honestly reflects the full dataset.","section":"Abstract and Table 1"},{"comment":"The claim of state-of-the-art accuracy on DiLiGenT is not supported by Table 2. Averaging the nine per-object MAEs gives 4.23 mm for FNIN-S and 4.62 mm for FNIN, whereas IPF achieves 4.20 mm; FNIN-S is worse than IPF on Buddha, Goblet, Harvest, and Reading. Since the supplementary Table 6 reports a standard deviation of only 0.0083 mm for FNIN on DiLiGenT, the 0.03 mm deficit is not explained by run-to-run variability. The text should either remove the dataset-level SOTA claim or support it with a paired statistical test across the nine objects.","section":"Table 2 and Section 'Comparison with Previous Methods'"},{"comment":"The derivation of the translation-invariant convolution kernel from the one-dimensional two-point example in Fig. 4 does not extend to the general two-dimensional boundary-value problem in Eq. (5). For a bounded domain with Dirichlet boundary conditions, the Green's function depends on both x and y separately, not on the difference x-y, so Eq. 6 is not a 'perfect convolution' for the full problem. Please either prove the translation-invariance under the stated conditions, or revise the text to describe Eq. 6 as a modeling assumption whose validity is to be judged empirically. This is load-bearing because the entire FNO architecture in Eq. 7 is justified by this convolution.","section":"Section 'Integration operator in spatial domain', Eq. (6)"},{"comment":"The main empirical comparisons report single-run MAE values without error bars or repeated-trial statistics. Some of the claimed improvements over baselines are fractions of a millimeter (e.g., IPF 0.64 vs FNIN-S 0.49 on Pot 2), and the table does not indicate whether these differences are stable across training runs. The paper should report at least the mean and standard deviation over multiple seeds for the learned methods in the main tables, or present a statistical significance test, as already partially done in the supplementary Table 6 for ablations.","section":"Tables 1 and 2"}],"minor_comments":[{"comment":"The displayed formula for W has unbalanced parentheses and a stray punctuation mark; it should be written as W = diag([wr/2, wl/2, wt/2, wb/2]^T) with a single closing bracket.","section":"Section 'Stage II of FNIN', Eq. (13)"},{"comment":"The text refers to 'solving Eq. 14' but Eq. 14 is not defined in the main paper; it appears only in the supplementary material. Please define the gradient equation in the main text or refer to the supplementary equation explicitly.","section":"Section 'Operator approximation in Fourier space'"},{"comment":"The citation of Rec-Net is inconsistent: the main text refers to 'Rec-Net (Lichy, Sengupta, and Jacobs 2022)' in Fig. 1 and elsewhere, while the reference list contains 'Lichy, D.; Wu, J.; Sengupta, S.; and Jacobs, D. W. 2021. Shape and Material Capture at Home' as the Rec-Net source, and a separate 2022 paper 'Fast Light-Weight Near-Field Photometric Stereo'. Please harmonize the citations.","section":"References"},{"comment":"The caption spells the dataset object as 'BRAR'; this should be 'BEAR'.","section":"Fig. 5 caption"},{"comment":"The text says 'We directly solve the normal equation of Eq. 22', but Eq. 22 is in the supplementary material; the intended reference in the main text is Eq. 11 or Eq. 12.","section":"Section 'Stage II of FNIN'"}],"recommendation":"major_revision","confidential_remarks":"The paper is a plausible fit for a vision or AI conference, and the code release plus the detailed ablation statistics in the supplementary are commendable. The main risk is not circularity but overstated claims: the abstract and the DiLiGenT SOTA statement should be corrected, and the translation-invariance assumption in Eq. 6 needs either a proper justification or an explicit downgrade to a heuristic. These issues are fixable within the scope of a revision, so I do not recommend rejection. I would also ask the authors to double-check the Rec-Net citation, since the reference list currently points to two different papers."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a serious, well-put-together effort to bring Fourier neural operators to surface-from-gradients. That is genuinely new as far as I know—no one has applied FNO to this problem before, and the two-stage design (FNO plus attention for discontinuities, followed by a one-step WLS) is sensible. On LUCES, the method is clearly competitive, and the speed advantage is real. The code is public and the ablation study is thorough. That part is worth engaging with.\n\nThe soft spots are in the framing and the theory more than in the engineering. The abstract says errors of less than 0.1 mm on tested objects, but only two of fourteen LUCES objects actually meet that bar. On DiLiGenT, the reported average for FNIN-S is about 4.23 mm versus about 4.20 mm for IPF, so dataset-level SOTA is not supported. The method wins on several individual objects, but the aggregate is not better. The assumption that the integration operator is translation-invariant (Eq. 6) is motivated only by a 1D toy example, and for bounded, non-rectangular domains the actual Green's function depends on both spatial arguments. The network can still learn a reasonable approximation, but the claim that Stage I is 'solving' Eq. 5 needs to be softened. Eq. 13 has a notation slip (lambda 1 should be lambda I). Main tables lack error bars, though the supplementary gives standard deviations for the key models, so that's a minor fix.\n\nNone of this is fatal. The core idea is sound, the experiments are mostly honest, and the limitations section is refreshingly candid. I would send it to review, but require a revision that fixes the overstatements and clarifies the theoretical assumptions. The paper is most useful for people working on photometric stereo pipelines who want a fast integration module, and for anyone interested in neural operators applied outside PDEs.\n\nRecommendation: accept with major revisions, or at least serious peer review.","headline":"Solid new application of FNO to normal integration with honest limitations, but the abstract and dataset-level SOTA claims are oversold.","tokens_in":18351,"tokens_out":5223,"would_cite":false,"duration_ms":47546,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Learned Fourier operator reconstructs surfaces with sub-0.1 mm error","keywords":["surface-from-gradients","normal integration","Fourier neural operator","neural operator","discontinuity detection","weighted least squares","photometric stereo","3D reconstruction"],"falsifier":"Probe the trained Stage I network with single-pixel gradient impulses placed at different image locations and compare the output depth patterns: translation invariance predicts identical shifted patterns, so any systematic position-dependent variation in the effective kernel would falsify the convolution assumption.","tokens_in":17283,"feed_emoji":"🧊","tokens_out":5237,"duration_ms":53759,"temperature":0.7,"pith_summary":"The paper proposes FNIN, a two-stage learned framework for surface-from-gradients: recovering a 3D depth map from a given normal map. It claims that treating integration as learning a translation-invariant convolution kernel in Fourier space, via a Fourier neural operator, lets a network approximate the global solution operator of the underlying PDE. Coupled with an attention-detected discontinuity weight map and a one-step weighted least squares refinement, the method reports sub-millimeter accuracy (mean absolute error below 0.1 mm on some LUCES objects) with runtimes of one to two seconds. If correct, this shows a learning-based integrator can generalize to unseen real objects without fine-tuning and replace classical PDE solvers inside photometric-stereo pipelines.","feed_headline":"Learned Fourier operator reconstructs surfaces with sub-0.1 mm error","feed_subtitle":"A two-stage network cuts normal-integration errors on real photometric-stereo data without retraining.","key_machinery":"The central object is the translation-invariant kernel assumption in Eq. 6: the depth correction at a point depends only on the relative position to its neighbors and the difference in slopes, which turns the kernel integration into a convolution. Combined with the convolution theorem, the Fourier neural operator parameterizes the kernel directly in Fourier space as $R_\\theta$, truncated at a maximum mode count $k_{\\max}$, and learns it data-driven. A multi-resolution iterative network (initial net plus iterative net) applies this operator to the lifted residual $g - \\nabla\\hat{z}$, while an attention network (three-layer CNNs) produces a normalized weight map $\\omega$ from one-sided difference features; a detail-weighted loss with $\\gamma = 0.25$ suppresses discontinuities during training. Stage II then minimizes a weighted least squares objective over four difference directions with weights from $\\omega$ and a small regularization $\\lambda = 10^{-3}$ to the network's log-depth output, solved via conjugate gradient on the normal equation $A^T W A z = A^T W b$.","core_discovery":"The central claim is that surface-from-gradients can be posed as an operator-learning problem: after subtracting an approximate low-resolution depth, the remaining depth correction is computed by a Fourier neural operator whose kernel is assumed translation-invariant, making the integration a convolution in Fourier space. Discontinuities, which break the PDE, are detected by an attention network trained with a detail-weighted loss, and their jumps are resolved by solving a weighted least squares system once rather than by repeated optimization. The paper reports state-of-the-art mean absolute errors on LUCES (e.g., CUP below 0.01 mm, BELL 0.12 mm, BEAR 0.15 mm) and on DiLiGenT (BEAR 0.11 mm with the sigmoid-regularized variant FNIN-S), outperforming both classical solvers and the previous learned method Rec-Net. The key novelty is that the integration operator is learned in Fourier space from data, so integrability enforcement is not hand-crafted but adapted to the statistics of real normal maps.","pith_inferences":["If the translation-invariance assumption holds only approximately, accuracy should degrade on surfaces with strong boundary influence or non-stationary curvature; a boundary-probe experiment or training on crops with varied boundary conditions would test this.","The two-stage decoupling suggests a template for other global-information inverse problems in vision where the solution operator is a smooth pseudo-differential map, such as depth from defocus or optical flow integration.","Replacing the supervised detail-weighted loss with a self-supervised integrability consistency loss could reduce reliance on synthetic training data, which the paper itself flags as a limitation for large discontinuities.","The same Fourier operator formulation might be extended to solve photometric stereo directly, treating the normal map as an intermediate output rather than the operator's input."],"forward_implications":["Learned normal integration can act as a fast, differentiable post-processor for photometric stereo networks, replacing classical solvers in end-to-end pipelines.","The one-to-two-second runtime at 2048×1536 resolution makes high-resolution surface reconstruction practical where large linear solvers become infeasible or out-of-memory.","Because discontinuities are resolved in a single weighted least squares solve rather than repeated optimization, the decoupling reduces both computation time and numerical instability.","Generalization to unseen DiLiGenT objects without fine-tuning indicates the learned operator captures transferable structure of normal integration across objects and resolutions."],"supporting_citations":[{"why":"Supplies the Fourier neural operator architecture and the convolution-theorem-based spectral kernel parameterization that Stage I is built on.","marker":"Li et al. 2020a"},{"why":"Provides the Rec-Net iterative structure and the transformation in Eq. 5 that FNIN adopts to ensure a unique solution.","marker":"Lichy, Sengupta, and Jacobs 2022"},{"why":"Provides the synthetic rendered dataset of 14 objects with 1.75k training samples used to train FNIN.","marker":"Lichy et al. 2021"},{"why":"Supplies the bilateral weighted least squares formulation and the BiNI baseline that FNIN extends with learned attention and a single optimization step.","marker":"Cao et al. 2022"},{"why":"Supplies the point-to-plane distance formulation used to model discontinuities in the attention network's input features.","marker":"Cao et al. 2021"},{"why":"Establishes the surface-from-gradients problem formulation and the Neumann-boundary non-uniqueness that motivates the learned approach.","marker":"Quéau, Durou, and Aujol 2017a"},{"why":"Provides the DiLiGenT benchmark dataset of ten real-world objects used for evaluation.","marker":"Shi et al. 2019b"},{"why":"Provides the LUCES dataset of 14 high-resolution real objects used for evaluation at 2048×1536.","marker":"Mecca et al. 2021"}],"fun_headline_variants":["Fourier operator learning reconstructs surfaces with sub-0.1 mm error","Attention-guided Fourier net detects breaks, cuts surface reconstruction error","Two-stage Fourier integration nets beat classical surface solvers","Sub-0.1 mm depth from gradients via learned Fourier integration","Operator-learning integration fixes discontinuities in surface recovery"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire Stage I rests on the assumption that the depth correction at a point depends only on the relative distance to its neighbors and the difference in slopes, not on the absolute position in the image; the paper justifies this with a one-dimensional two-point example and does not prove it for general 2D surfaces.","fun_headline_variants_meta":{"raw":{"variants":["Fourier operator learning reconstructs surfaces with sub-0.1 mm error","Attention-guided Fourier net detects breaks, cuts surface reconstruction error","Two-stage Fourier integration nets beat classical surface solvers","Sub-0.1 mm depth from gradients via learned Fourier integration","Operator-learning integration fixes discontinuities in surface recovery"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00058,"raw_usage":{"total_tokens":2746,"prompt_tokens":973,"completion_tokens":1773,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":1702}},"tokens_in":589,"tokens_out":1773,"duration_ms":14556,"temperature":1.0,"reasoning_tokens":1702,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:46:44.283174+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Probe the trained Stage I network with single-pixel gradient impulses placed at different image locations and compare the output depth patterns: translation invariance predicts identical shifted patterns, so any systematic position-dependent variation in the effective kernel would falsify the convolution assumption.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Rec-Net iterative structure and the transformation in Eq. 5 that FNIN adopts to ensure a unique solution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the synthetic rendered dataset of 14 objects with 1.75k training samples used to train FNIN."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the bilateral weighted least squares formulation and the BiNI baseline that FNIN extends with learned attention and a single optimization step."}],"review_version":1}