{"id":"31210761-f414-4eb9-8cc6-f69b51cba525","arxiv_id":"2507.06075","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":5,"one_line_summary":"An exact-for-planar-patches, ray-based normal integration formulation with explicit discontinuity terms achieves state-of-the-art depth errors on DiLiGenT and extends to generic central cameras.","lead":"The paper introduces a new mathematical formulation for normal integration, recovering a 3D depth map from a surface normal map by assuming each local patch is flat and each depth jump is a step along the camera ray. The formulation is the first designed for generic central cameras (pinhole, distorted, fisheye) and reports lower reconstruction errors than prior state-of-the-art methods on the DiLiGenT benchmark.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The blanket claim that Eq. (2)-(3) 'more accurately approximates the relation between depth and surface normals' fails on curved surfaces: the per-edge z-offset model is not exact under curvature, and the paper's own buddha residual shows the advantage reverses (Ours 0.90 vs BiNI 0.46).","rationale":"The reader's weakest assumption identifies the same load-bearing concern I find: the local-planarity model with a single z-offset at the midpoint ray is inaccurate on strongly curved surfaces, and the paper's own buddha result confirms it. I agree that this does not overturn the benchmark claims, because the method still achieves state-of-the-art mean MADE on DiLiGenT and the algebraic derivation is sound. I would slightly sharpen the reader's interpretation of the oracle experiment: its evidential value depends on how the ground-truth discontinuity values were computed. If epsilon is obtained by inverting Eq. (2) from the ground-truth depth, the near-perfect oracle is a consistency check rather than a validation of the geometric model; if it is computed geometrically from the tangent-plane intersections, the oracle is genuine evidence. The paper does not specify this, and the ambiguity should be resolved in a revision. The conditional verdict is appropriate: the central geometric contribution is plausible and well-derived, but the generality of the accuracy claim is not fully established, and the absence of released code plus hyperparameter sensitivity (Table 5) keep reproducibility open. I keep the reader's verdict unchanged.","tokens_in":24621,"tokens_out":24660,"duration_ms":289596,"concrete_test":"Render a set of synthetic spheres with known radii (e.g., 50, 100, 200 mm) using a pinhole camera matching DiLiGenT intrinsics, at pixel spacings h in {1, 0.5, 0.25} px, with ground-truth depth and normal maps. (a) Compare, on the ground-truth data, the Ours residual |gamma(log z_a - log z_b) - gamma log(omega_{b->a})| against BiNI's residual |gamma(log z_a - log z_b) - delta| for all edge pairs. (b) Compute the physically defined discontinuity epsilon from the intersection of the two per-pixel tangent planes with the midpoint ray tau_m, and test whether Eq. (2) holds to machine precision with that epsilon. If r_ours exceeds r_BiNI on high-curvature spheres, or if the physical epsilon does not make Eq. (2) hold, the blanket accuracy claim fails on curved surfaces; if r_ours < r_BiNI across all curvatures and resolutions, and physical epsilon explains the oracle, the concern is dismissed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that Eq. (2)-(3) 'more accurately approximates the relation between depth and surface normals' rests on the geometric model in Fig. 2: at each neighbor pair, the two per-pixel tangent planes are assumed to intersect the midpoint ray tau_m at points separated purely along the camera z-axis. On a curved surface this is false, because the two intersection points differ in x and y as well as in z; relation (2) therefore carries a curvature-dependent error. The paper's own data show the limitation: on buddha, the highest-curvature object, Table 1 gives Ours a mean absolute residual of 0.90 +/- 0.91, worse than BiNI's 0.457 +/- 0.921, and the relative residual tables in Appendix G show Ours with larger standard deviation on buddha and pot1. The benchmark MADE results still hold on average, but the blanket accuracy claim is not uniform; it is driven by low-curvature objects. Because the method's largest MADE gains are on discontinuous objects (harvest, goblet), the contribution of the proposed smooth-surface relation versus the discontinuity mechanism remains entangled. The oracle experiment in Sec. 4.3 is strong only if the ground-truth epsilon values are computed geometrically from the true tangent-plane intersections; the paper does not specify this, and if epsilon were instead defined by inverting (2), the oracle would be near-tautological. This unresolved curvature regime is the load-bearing soft spot of the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a discontinuity-aware normal integration method for generic central cameras. The core idea is a local planarity assumption: for neighboring pixels a and b and a subpixel location m between them, the surface is modeled as two plane segments perpendicular to the local normals, and the two tangent-plane intersections with the midpoint ray tau_m are assumed to be separated by a scalar offset eps_b->a along the camera z-axis. Solving the resulting 6x6 system of ray-direction and perpendicularity equations yields the pairwise depth relation z_a = omega_eps^a * eps_b->a + omega_b->a * z_b with coefficients omega_eps^a = n_az/(n_a^T tau_a) and omega_b->a = (n_a^T tau_m * n_b^T tau_b)/(n_a^T tau_a * n_b^T tau_m), which is exact for piecewise-planar scenes. Depth and discontinuities are recovered by iterating conjugate-gradient solves of a weighted least-squares problem in log-depth variables, using BiNI-style bilateral weights and a sigmoidal activation beta that selectively enables relative discontinuities alpha_b->a. The paper validates the approach on DiLiGenT: the formulation residual on ground-truth depth is one to two orders of magnitude smaller than BiNI's on 8 of 9 objects, MADE is best or second best on all 9 objects, and an oracle row with known discontinuities reaches 0.01-0.34 mm. Experiments with Brown-Conrady lens distortion, real-world normal maps, and DiLiGenT-MV, plus extensive ablations, are also reported.","tokens_in":24954,"tokens_out":19981,"duration_ms":199176,"significance":"I view the contribution as solid and, conditional on the points below, publishable. The derivation of (2)-(3) is transparent, parameter-free, and correct; the concise alternative derivation in Appendix A provides an independent check. The oracle experiment, once its protocol is specified, would provide strong validation of the geometric model. The experimental protocol is careful: the BiNI baseline is rerun with its official code, Kim et al.'s numbers are quoted from the original paper, and the paper is unusually honest about hyperparameter sensitivity (Appendix E), outlier preprocessing requirements (Appendix H), and the limited scope of the distorted-camera comparison (Sec. 4.4). The gamma_b->a analysis in Appendix B, separating the focal-length and n^T tau effects, is a genuine insight beyond the main result. The main risks are the scope of the accuracy claim and the attribution of benchmark gains to the smooth relation versus the discontinuity mechanism; both are addressable in revision without new algorithmic development.","major_comments":[{"comment":"The 'Ours with known discontinuity values' row is the evidence for the claim in Sec. 3.2 that, given ground-truth discontinuities, the optimization would recover ground-truth depth 'with close-to-perfect accuracy,' but the paper never specifies how the ground-truth eps_b->a (or alpha_b->a) values are computed. Please state explicitly whether they are computed geometrically from the ground-truth mesh as the z-offset between the intersections of the two per-pixel tangent planes with the midpoint ray tau_m, or obtained by inverting (2) from ground-truth depths. Only the former protocol validates the geometric model; the latter would make the near-perfect oracle recovery (0.01-0.34 mm) a tautology by construction. The small but nonzero errors on harvest (0.34 mm), goblet (0.11 mm), and buddha (0.10 mm) suggest a geometric computation, but without a stated protocol the experiment cannot be interpreted.","section":"Sec. 4.3, Table 2"},{"comment":"The blanket claim that Eq. (2)-(3) 'more accurately approximates the relation between depth and surface normals' is not uniform across the benchmark. On buddha, the highest-curvature object, the mean absolute residual is 0.90 +/- 0.91 versus BiNI's 0.457 +/- 0.921 (Table 1), and the relative-residual tables in Appendix G show larger standard deviations for the proposed method on buddha and pot1. The text in Sec. 4.2 acknowledges 'all but one object,' but the abstract, introduction, and conclusion repeat the unqualified claim, and curvature sensitivity is absent from the limitations list in Appendix J. I ask for (i) a qualified claim 'on 8 of 9 DiLiGenT objects,' and (ii) an analysis of the model error against local surface curvature (e.g., a synthetic curved-surface experiment or a residual-versus-curvature scatter plot) so the reader can see where the per-edge z-offset model degrades.","section":"Abstract, Sec. 4.2, Table 1"},{"comment":"The headline gains on the two discontinuous objects come almost entirely from the discontinuity mechanism rather than the smooth-surface relation. With alpha_b->a identically 0 at 1200 iterations, the method is worse than BiNI on harvest (4.83 vs 1.84 mm) and only moderately better on goblet (6.56 vs 9.00 mm); with alpha computation, the same rows become 0.73 and 4.72 mm. Since the abstract attributes the state-of-the-art results to a more accurate approximation of the depth-normal relation, the main text should attribute the gains explicitly: the smooth relation provides broad but modest improvements on low-curvature objects, while the explicit eps model is what delivers the large improvement on discontinuous objects. The rows of Table 2 already contain this information; what is missing is an explicit statement in the text.","section":"Sec. 4.3, Table 2"}],"minor_comments":[{"comment":"There is a sign inconsistency in the first paragraph: the terms n_a^T tau_m and n_b^T tau_m are said to be 'strictly positive for all but a single pixel,' whereas the visibility discussion in Sec. 3.1 requires n^T tau < 0 for these dot products, and the subsequent argument that omega_b->a is positive relies on the same terms being negative. The text should read 'strictly negative.'","section":"Appendix C"},{"comment":"The generalization gamma_b->a = ||u_b-u_a||/||tau_b-tau_a|| * n_a^T tau_a is derived for the pinhole case from the identity ||Delta u||/||Delta tau|| = f, but for generic central cameras it is a heuristic. Because gamma changes the effective weighting (Appendix B, Table 3), the distorted-camera experiments would benefit from a brief justification or an ablation of alternative normalizations.","section":"Eq. (13), Sec. 3.3"},{"comment":"Labeling the two findings as 'Proposition 1' and 'Proposition 2' overstates their status; they are empirical observations supported by the ablations in Table 3, and should be relabeled as observations or empirical findings.","section":"Appendix B"},{"comment":"The residual comparison is performed only against BiNI; since Kim et al. is presented as a co-leading method, the text should state explicitly that no public implementation was available and that the quantitative 'more accurate than existing methods' phrasing in the abstract is therefore based on a comparison with BiNI only.","section":"Sec. 4.2"},{"comment":"The paper does not mention code availability; given the number of hyperparameters (q, rho, k, lambda_m, eps_out) and the iterative scheme, releasing the implementation would substantially support reproducibility.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"For the editor: this is a solid methods paper with a correct derivation, a careful benchmark protocol (including rerunning BiNI's official code), and an unusually complete set of ablations. My major-revision recommendation rests on three points that affect the interpretation of the paper's central claims: (1) the oracle experiment does not specify how ground-truth discontinuity values are computed, which determines whether it validates the geometric model or is near-tautological; (2) the accuracy claim is stated without the curvature qualification that the paper's own Table 1 requires; (3) the attribution of benchmark gains between the smooth relation and the discontinuity mechanism needs an explicit statement. None of these requires new algorithmic work. I also noticed a sign inconsistency in Appendix C ('strictly positive' where the visibility condition requires 'strictly negative') that should be corrected. The novelty relative to BiNI is moderate in the smooth regime, but the explicit eps-modeling and the central-camera extension are genuine; the paper is honest about its limitations, which is a point in its favor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a solid paper, and I would send it to review. The core contribution is the planar-pair relation (2)-(3): a parameter-free, closed-form coupling between depth at neighboring pixels, normals, and ray directions, derived from local planarity rather than from discretizing a PDE. I checked the algebra in Appendix A; it is correct. The oracle experiment with ground-truth discontinuities gives near-perfect depth, which is a strong sanity check. The benchmark is run honestly: official BiNI code, matched iteration counts, per-object numbers reported. The gains on discontinuous objects are large and credible.\n\nThe main soft spot is curvature. Relation (2) assumes the two tangent planes intersect the midpoint ray at points separated purely along z. On a curved surface that assumption is false, so the relation carries a curvature-dependent error. The paper's own Table 1 shows it: on buddha, the highest-curvature object, the mean residual is 0.90 versus BiNI's 0.46. So the claim that (2) 'more accurately approximates the relation between depth and surface normals' is not uniform. It holds on planar and low-curvature regions, and most of the benchmark gains come from discontinuous objects. Interestingly, the reconstructed MADE on buddha is still better (0.24 vs 1.06), so the practical implication is not as stark as the residual comparison suggests, but the blanket claim in the abstract should be qualified.\n\nTwo smaller concerns. The oracle experiment does not specify how the ground-truth epsilon values are computed. If they are derived from the true surface geometry, the experiment is meaningful; if they are obtained by inverting (2), it borders on circular. The paper should state this explicitly. And the headline SOTA depends on the q, rho activation hyperparameters; Table 5 shows object-specific sensitivity. The default q=50 is not best for every object. That is minor, but readers should know.\n\nThe 'first generic central camera' claim is demonstrated only on a distorted pinhole, not on an actual fisheye or catadioptric camera. The formulation is ray-based and plausible, but the evidence is thin. Also, no code is released, which is a real reproducibility gap for a method whose value is largely in the derivation and benchmark numbers.\n\nThe derivation is the genuine contribution: new, non-circular, and mostly well-supported. The paper is honest in the supplement about limitations. I would accept it into review; the authors should be asked to qualify the curvature claim and clarify the oracle computation.\n\nFor a reading group, it could spark discussion about benchmark tuning and the gap between formulation residuals and final reconstructions.","headline":"A genuinely new planar-pair relation for normal integration with honest benchmarks, but the blanket accuracy claim does not survive curved surfaces.","tokens_in":25556,"tokens_out":3730,"would_cite":true,"duration_ms":39487,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Normal integration now handles depth jumps and any central camera","keywords":["normal integration","depth discontinuities","central cameras","photometric stereo","local planarity","ray directions","DiLiGenT benchmark"],"falsifier":"Render a known smooth object (e.g., a sphere of varying radius) with a pinhole camera, compute normals from the mesh, then measure the mean absolute residual of equation (2) with $\\alpha=0$ on the true depth map and compare it against the residual of the BiNI PDE (8) on the same data; the paper's claim predicts the new residual should be lower or comparable across curvatures, while a fixed planarity-identity residual that grows faster with curvature would falsify the claim. A second decisive check: run the method on a scene with a known step discontinuity of known height and verify that the estimated $\\varepsilon_{b\\to a}$ converges to the true jump rather than to a jump that merely minimizes depth error.","tokens_in":24391,"feed_emoji":"📷","tokens_out":7104,"duration_ms":78376,"temperature":0.7,"pith_summary":"The paper tries to establish that normal integration—recovering a depth map from a per-pixel normal map—can be recast as a system of pairwise linear constraints derived from a local planarity assumption, with explicit scalar terms that model depth jumps, and that this recasting holds for any central camera, not just orthographic or ideal-pinhole models. The central identity couples the depths of neighbouring pixels through ray directions and normals, plus one discontinuity offset per pair. The paper reports that this relation approximates the ground-truth depth-to-normal link one to two orders of magnitude more accurately than the leading PDE-based formulation on most DiLiGenT objects, and that the resulting optimization reaches the best or second-best mean absolute depth error on eight of nine benchmark objects. If the claim holds, normal integration becomes a camera-agnostic linear problem whose only hard part is estimating where and how much the surface jumps, which is exactly the part the method makes explicit.","feed_headline":"Normal integration now handles depth jumps and any central camera","feed_subtitle":"A ray-based planarity coupling beats prior PDE methods on DiLiGenT and works beyond ideal pinhole cameras","key_machinery":"The load-bearing object is the pairwise identity (2)--(3), built from the ray-direction map $\\tau(u)=(\\tau_x(u),\\tau_y(u),1)^T$ of a central camera and the surface normal $n$. The identity follows from solving a $6\\times 6$ linear system whose six unknowns are the displacement components from the subpixel midpoint $m$ to the two pixel points; the two non-trivial equations impose that the local planes are perpendicular to $n_a$ and $n_b$ and that they differ only by a scalar $z$-offset $\\varepsilon_{b\\to a}$. Around this identity the paper constructs an optimization: a log-depth change of variables (7), BiNI-style bilateral weights (10), a discontinuity-activation term $\\beta^{(t)}_{b\\to a}$ (16) that switches on the $\\varepsilon$ terms only where the bilateral weights signal a one-sided jump, and a generalized multiplicative factor $\\gamma_{b\\to a} = \\|u_b-u_a\\|/\\|\\tau_b-\\tau_a\\| \\, n_a^T\\tau_a$ that the paper analyses and shows controls both equation weighting and the sharpness of the bilateral sigmoid.","core_discovery":"The paper's central claim is the depth-coupling relation $z_a = \\omega^\\varepsilon_a \\, \\varepsilon_{b\\to a} + \\omega_{b\\to a}\\,z_b$ with $\\omega^\\varepsilon_a = n_{az}/(n_a^T\\tau_a)$ and $\\omega_{b\\to a} = (n_a^T\\tau_m \\, n_b^T\\tau_b)/(n_a^T\\tau_a \\, n_b^T\\tau_m)$, derived from asking that the surface around each of two neighbouring pixels be a plane perpendicular to its normal and that the two planes differ only by a z-axis offset $\\varepsilon_{b\\to a}$ at a subpixel midpoint along the connecting ray. This equation turns normal integration into a large sparse linear system $Az=b$ in the depth unknowns, with the discontinuity terms in $b$ recovered by an iterative bilateral scheme. The paper claims this formulation is the first to directly handle generic central cameras, that it approximates the true depth-normal relationship more accurately than PDE discretizations, and that it reaches state-of-the-art mean absolute depth error on the DiLiGenT benchmark (for instance 4.72 mm on goblet versus 8.53 mm for BiNI), with near-perfect reconstruction when the true discontinuities are supplied.","pith_inferences":["The failure on the high-curvature buddha object is an implicit boundary of the planarity assumption; an extension could compose this linear coupling with a curvature-adaptive smoothness term, using the residual of (2) itself as a curvature detector.","Because the relation depends only on per-pixel ray directions, one can test it with non-central ray sets (e.g., ray maps of axial or catadioptric cameras) even though the paper explicitly excludes them; if the identity still holds with per-pixel $\\tau$, the method would generalize in a straightforward way.","The $\\tau_m$ degree of freedom, currently fixed at the midpoint, could be tuned per pixel pair as a denoising mechanism; the paper shows outlier spikes correlate with $\\omega_{b\\to a}$ far from 1 and that re-sampling $\\lambda_m$ reduces them, which suggests a confidence-weighted variant.","The formulation accuracy metric (residual on ground-truth depth) could be repurposed as a benchmark signal for learning-based normal estimators: a normal map is geometrically consistent with some piecewise-planar scene exactly when this residual is low."],"forward_implications":["One algorithm covers orthographic, ideal-pinhole, and lens-distorted central cameras; previous leading methods needed separate or undistorted formulations.","Explicit discontinuity variables let the optimizer estimate jumps instead of only hedging against them with robust weights, and the oracle experiment shows that if those jumps are known, depth is recovered to roughly 0.01--0.11 mm MADE.","The generalized $\\gamma_{b\\to a}$ analysis transfers the earlier BiNI factor from a numerical device to a geometrically meaningful weight that down-weights pixel pairs near occluding boundaries.","On the standard benchmark the method reports best or second-best mean absolute depth error on eight of nine objects, including a four-fold reduction on the discontinuous goblet object.","The formulation also yields a practical pre-check for input normals: pixels with $n^T\\tau > 0$ are physically invalid and can be filtered before integration, which the noise experiments show removes most spike artifacts."],"supporting_citations":[{"why":"Supplies the bilateral weighting scheme and iterative discontinuity optimization that the paper adapts, and serves as the primary PDE baseline for comparison.","marker":"[7]"},{"why":"The other leading discontinuity-preserving integration method and the second baseline in the DiLiGenT comparison.","marker":"[24]"},{"why":"The DiLiGenT benchmark provides the ground-truth normal and depth maps on which the central accuracy claims are evaluated.","marker":"[31]"},{"why":"BiNI's supplementary material introduces the log-depth variable and the multiplicative gamma factor whose role the paper generalizes and analyses.","marker":"[9]"},{"why":"Recent statement of the visibility/orthogonality relation between normals and ray directions that the paper uses for sign and positivity conditions.","marker":"[4]"},{"why":"The classic observation that visible surfaces satisfy a specific angle condition between normals and ray directions, which grounds the interpretation of n^T tau < 0.","marker":"[26]"}],"fun_headline_variants":["Normal integration with explicit depth jumps for any central camera","Ray-planarity coupling beats prior PDE methods on normal integration","First direct normal integration for generic central cameras","Depth jumps and non-pinhole cameras handled in normal integration","State-of-the-art normal integration via ray-based planarity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The identity assumes every pixel neighbourhood is exactly a plane perpendicular to the normal, and that the two planes at neighbouring pixels cut a single subpixel ray with only one z-axis offset; strongly curved surfaces violate this, and the paper's own measurements show the accuracy advantage disappears on the high-curvature buddha object (residual 0.90 vs 0.46 for BiNI).","fun_headline_variants_meta":{"raw":{"variants":["Normal integration with explicit depth jumps for any central camera","Ray-planarity coupling beats prior PDE methods on normal integration","First direct normal integration for generic central cameras","Depth jumps and non-pinhole cameras handled in normal integration","State-of-the-art normal integration via ray-based planarity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000188,"raw_usage":{"total_tokens":1327,"prompt_tokens":938,"completion_tokens":389,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":311}},"tokens_in":554,"tokens_out":389,"duration_ms":4425,"temperature":1.0,"reasoning_tokens":311,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:13:24.029095+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Render a known smooth object (e.g., a sphere of varying radius) with a pinhole camera, compute normals from the mesh, then measure the mean absolute residual of equation (2) with $\\alpha=0$ on the true depth map and compare it against the residual of the BiNI PDE (8) on the same data; the paper's claim predicts the new residual should be lower or comparable across curvatures, while a fixed planarity-identity residual that grows faster with curvature would falsify the claim. A second decisive check: run the method on a scene with a known step discontinuity of known height and verify that the estimated $\\varepsilon_{b\\to a}$ converges to the true jump rather than to a jump that merely minimizes depth error.","supporting_citations":[{"cited_title":"Bilateral Normal Integration","cited_arxiv_id":null,"evidence_quote":"Supplies the bilateral weighting scheme and iterative discontinuity optimization that the paper adapts, and serves as the primary PDE baseline for comparison."},{"cited_title":"Discontinuity-preserving Normal Integration with Auxiliary Edges","cited_arxiv_id":null,"evidence_quote":"The other leading discontinuity-preserving integration method and the second baseline in the DiLiGenT comparison."},{"cited_title":"A Benchmark Dataset and Evaluation for Non-Lambertian and Uncalibrated Photometric Stereo","cited_arxiv_id":null,"evidence_quote":"The DiLiGenT benchmark provides the ground-truth normal and depth maps on which the central accuracy claims are evaluated."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Recent statement of the visibility/orthogonality relation between normals and ray directions that the paper uses for sign and positivity conditions."},{"cited_title":"Analysis of occluding contour","cited_arxiv_id":null,"evidence_quote":"The classic observation that visible surfaces satisfy a specific angle condition between normals and ray directions, which grounds the interpretation of n^T tau < 0."}],"review_version":1}