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Surface Reconstruction from Range Scans Under Visibility Constraints

Gabriel Lenain, Edmond Boyer, Jean Ponce

arXiv:2610.18005v1cs.CV

TL;DR

Range-scan reconstruction is difficult because observations leave geometry underdetermined and neural methods often discard visibility from acquisition rays. This paper fits first-hit depths with a robust, visibility-aware surface formulation. It reports competitive clean-data performance and improved results under noise and outliers, alongside a fidelity–robustness trade-off with model capacity.

  • Problem

    Surface samples underdetermine geometry, and most neural implicit methods discard sensor-provided visibility information.

    Source: §1 · p. 2

  • Method

    The method robustly regresses predicted first surface intersections along acquisition rays against measured depths, using a 1-Lipschitz implicit field.

    Source: §1 · p. 2

  • Results

    The method retains competitive results on clean data and improves performance under noise and outliers.

    Source: §1 · p. 2, §6 · pp. 9–10

  • Takeaways & Limitations

    Increasing positional-encoding level L improves clean-data fitting but increases sensitivity to corruption, with L = 2 providing a good intermediate value across the experiments.

    Source: §1 · p. 2, §5 · p. 9

    Finite first-hit measurements do not uniquely determine the underlying surface.

    Source: §6 · p. 10

Abstract

from arXiv · show

Surface reconstruction from range scans is challenging in the presence of noise and outliers, as fidelity to observations must be balanced against sensitivity to measurement errors. Yet, most neural approaches discard the sensor positions and corresponding visibility information, compensating for this loss with geometric priors or regularization terms. We instead formulate surface reconstruction as robust regression of the predicted first surface intersection along each acquisition ray against its measured depth, which penalizes predicted surface intersections within the segment between the sensor and the observed point. We represent the surface as the zero set of a 1-Lipschitz multi-layer perceptron, allowing the use of linearly convergent sphere tracing to compute the first hit, and use a robust Huber loss for regression. Our formulation requires neither oriented normals, sampled off-surface points, nor an eikonal loss. Comparisons with the state of the art on synthetic and real data with varying levels of noise and outlier contamination demonstrate competitive performance on clean data and much improved fidelity and robustness under even moderate corruption. Our analysis further confirms a trade-off between fidelity and robustness: greater fitting capacity improves data fidelity in the clean case but increases sensitivity to the measurement errors inherent in real acquisitions, with a sweet spot typically emerging at intermediate capacity.

1 INTRODUCTION

Range-scan reconstruction is underdetermined from surface samples, while most neural implicit methods discard acquisition visibility. This work directly regresses first surface intersections along acquisition rays and examines the resulting fidelity–robustness trade-off.

1 INTRODUCTION

  • Surface samples leave topology, inside–outside orientation, and geometry between or beyond observations underdetermined, and most neural implicit methods discard acquisition visibility.Prior approaches address ambiguity with implicit-surface methods, geometric priors, occupancy reasoning, or learned reconstruction priors.Source: §1 · p. 2
  • The method directly regresses each acquisition ray’s first surface intersection against its measured depth.The authors describe this as, to their knowledge, the first from-scratch implicit-surface fitting method for range scans using this objective.Source: §1 · p. 2
  • The single-loss formulation needs neither auxiliary geometric supervision nor regularization, retains competitive clean-data results, and improves robustness under noise and outliers.Source: §1 · p. 2
  • Increasing positional encoding level L exposes a fidelity–robustness trade-off and provides a simple control over sensitivity to corruption.Source: §1 · p. 2

2 RELATED WORK

Prior surface-reconstruction methods constrain implicit fields using point samples, learned priors, geometric mechanisms, or visibility evidence. This work’s related context also includes architectures constrained to be 1-Lipschitz.

2 RELATED WORK

  • Point-cloud methods range from fitting implicit functions and signed-distance fields to learned priors and per-cloud neural-field fitting with geometric constraints.Poisson methods use oriented samples; SPSR adds interpolation constraints, while neural methods include DeepSDF, POCO, Shape As Points, IGR, SAL, Neural-Pull, DiGS, and DiffCD.Source: §2 · pp. 2–3
  • Visibility-aware reconstruction uses sensor-known free space through classical volumetric reasoning, learned ray-based features, or losses on sampled points and volume rendering.The cited methods include Delaunay-based approaches, graph neural networks, Neural Poisson, and neural RGB-D mapping.Source: §2 · p. 3
  • Lipschitz-constrained networks have been developed for sensitivity control, certification, and Wasserstein objectives, with scalable architectures including CPL, Sandwich, and SLL layers.Coiffier and Béthune (2024) fit 1-Lipschitz neural distance fields to point clouds with normals or triangle soups.Source: §2 · p. 3

3 METHOD

The method fits a continuous implicit surface by regressing first-hit depths along sensor rays. Incidence weighting, sphere tracing, a 1-Lipschitz field, and practical variants support this formulation.

3.1 PROBLEM STATEMENT

  • The reconstruction input is a set of sensor–point pairs, and the unknown surface is represented as the zero level set of an MLP field.The formulation first considers noise along the acquisition ray and treats fitting as robust regression of noisy first-hit depths.Source: §3.1 · p. 3

3.2 INCIDENCE-WEIGHTED FIRST-HIT REGRESSION

  • Incidence weighting converts depth residuals toward surface-distance residuals, while a Huber penalty limits the influence of gross outliers.The objective keeps the weighting fixed during differentiation so the Huber threshold retains a consistent geometric meaning across incidence angles.Source: §3.2 · pp. 3–4

3.3 DIFFERENTIABLE FIRST-HIT TRACING

  • For a 1-Lipschitz field, sphere tracing locates the first ray intersection with linear convergence at a non-grazing root; implicit differentiation avoids backpropagating through every tracing iteration.The method filters for finite, non-grazing intersections before applying implicit differentiation.Source: §3.3 · pp. 4–5

3.4 1-LIPSCHITZ MODEL

  • The model composes 1-Lipschitz functions to ensure sphere-tracing convergence, with the final field constrained to remain 1-Lipschitz.The experiments use Convex Potential Layers and MaxMin activations.Source: §3.4 · p. 5

3.5 PRACTICAL VARIANTS

  • Rescaling addresses slack in the Lipschitz bound, but its sampled estimate is not formally certified; Fourier positional encoding increases fitting capacity and also removes the formal 1-Lipschitz guarantee.The positional-encoding variant is empirically rescaled for sphere tracing.Source: §3.5 · p. 5

4 EXPERIMENTS

Experiments show a fidelity–robustness trade-off across synthetic corruption: smaller L is more robust, while larger L fits clean observations more closely. Evaluation spans synthetic benchmarks and real LiDAR, using reference-based depth metrics for ARKitScenes and qualitative results for KITTI.

4.1 ARCHITECTURE AND OPTIMIZATION.

  • Experiments use an 8-layer, width-256 CPL network, with PE level L, δ set by corruption detection, and 5k ShapeNet versus 25k other optimization steps.Sphere tracing uses K = 192; the field is initialized as an approximate SDF of an enclosing sphere.Source: §4.1 · p. 5

4.2 SYNTHETIC DATASETS AND EVALUATION

  • SRB and 13-category ShapeNet scans retain acquisition rays; evaluation compares neural and classical baselines using global metrics plus observation-supported CDobs and NCobs.The observation-supported metrics retain predicted and reference samples within 2h0 of the corrupted and clean acquisitions, respectively.Source: §4.2 · p. 6

4.3 ROBUSTNESS TO MEASUREMENT ERRORS

  • At λ = 1, Ours (L = 0) beats DiGS globally (0.310 vs. 0.610 CD1) and on observed geometry (0.226 vs. 0.488 CDobs), while larger L degrades faster.The scanner-noise table reports SRB results; the text notes the degradation also occurs in observed regions, indicating larger L increasingly fits corrupted measurements.Source: Table 1 · p. 6, §4.3 · p. 7
  • On clean scans, Ours (L = 6) improves observation-supported CD1 over DiGS (0.064 vs. 0.071) despite worse global CD1 (0.133 vs. 0.108), concentrated in weakly observed regions.Stronger geometric priors can improve global metrics in weakly observed regions by extrapolating beyond measurements.Source: §4.3 · pp. 6–7
  • On ShapeNet, robustness gains persist under isotropic and ray-aligned Gaussian noise.The evaluation uses 3K input points and considers ray-aligned range noise as a first-order model of depth uncertainty in direct time-of-flight sensors.Source: §4.3 · p. 7, Figure 6 · p. 8
  • At α = 0.2, Ours (L = 2) retains clean-data performance and has lower CD1 than Labatut at every tested outlier level; neural baselines deteriorate sharply.The authors attribute the behavior to first-hit visibility and Huber loss: many inconsistent outlier rays yield no valid hit, while the loss limits the influence of remaining outliers.Source: Table 3 · p. 8, §4.3 · p. 8

4.4 REAL DATA EXPERIMENTS

  • ARKitScenes evaluation uses 2.34M iPad LiDAR points and compares rendered first-hit depths with FARO depth maps; KITTI has no reference surface and is evaluated qualitatively.ARKitScenes is scored across 36 FARO frames; KITTI uses one urban driving sequence and a static urban crop.Source: Table 4 · p. 8, §4.4 · pp. 8–9

5 DISCUSSION

The discussion frames model capacity as a fidelity–robustness control and examines the roles of visibility, loss design, and computational cost. Across experiments, L = 2 is a useful intermediate setting.

Fidelity–robustness trade-off

  • L = 2 provides a good intermediate value across experiments, balancing closer fitting at larger L against greater robustness to corruption at smaller L.The paper notes that L can in principle be set using known or calibrated sensor noise characteristics.Source: §5 · p. 9

Role of visibility

  • Random sensor reassignment makes most ray–point pairs inconsistent and causes loss of valid first hits, while moderate pose perturbations have little effect.The contrast indicates that acquisition rays carry geometric information.Source: §5 · p. 9

Ablations

  • Robust depth penalties matter mainly under outliers: ℓ1 remains stable while ℓ2 degrades, and a miss penalty is harmful because it forces the surface through outliers.Incidence weighting primarily improves outlier robustness, while empirical rescaling improves performance across all three regimes.Source: §5 · p. 9, Table 5 · p. 9

Computational cost

  • Full-budget training takes about 1h40min versus up to 40min for baselines, but Ours (L = 6) reaches within 2.8% of final CD1 after 40% of training.Peak memory is 3× below DiGS and DiffCD; each sphere-tracing iteration requires a network evaluation, while implicit differentiation keeps the backward pass efficient.Source: §5 · p. 9

6 CONCLUSION

The conclusion presents visibility-aware first-hit depth regression as the paper's reconstruction formulation and reports competitive clean-data results with gains under corruption. The appendices document acquisition protocols and additional DTU reconstructions.

6 CONCLUSION

  • The formulation regresses robust first-hit depths along acquisition rays, using those rays to constrain surface position and preceding free space without auxiliary geometric supervision or regularization losses.Experiments show competitive clean-data results and consistent gains under noise and outliers.Source: §6 · pp. 9–10
  • Finite first-hit measurements do not uniquely determine the underlying surface, leaving recoverable detail in relation to sampling density, surface regularity, and model capacity as future work.Sphere tracing increases optimization cost, although most accuracy is reached early in the step budget.Source: §6 · p. 10

A.1 ACQUISITION PROTOCOLS

  • The SRB protocol places scanner poses on S(r), casts rays through a uniform grid, and retains their first surface intersections.Figure 8 illustrates the protocol.Source: Figure 8 · p. 13, §A.1 · p. 13
  • The ShapeNet protocol uses virtual sensors on two concentric spheres, with uniformly sampled targets on an enclosing sphere defining ray directions.The first surface intersections are retained; Figure 9 illustrates the geometry.Source: Figure 9 · p. 13, §A.1 · p. 13

A.2 ADDITIONAL REAL-WORLD ACQUISITIONS

  • Additional real-world evaluation uses four DTU scenes and point clouds formed by fusing depth maps estimated with ACMMP.The scenes are 37, 69, 105, and 122; Figure 10 shows Ours (SW L = 6) using Sandwich layers.Source: §A.2 · p. 13, Figure 10 · p. 14

A.3 ARCHITECTURE AND IMPLEMENTATION

The appendix details architecture and implementation choices, evaluation metrics, and supporting mathematical arguments for Lipschitz sphere tracing and incidence-weighted residuals.

A.3 ARCHITECTURE AND IMPLEMENTATION

  • The CPL residual mapping is described as 1-Lipschitz, with CPL chosen as a practical compromise among expressivity, memory, speed, and certifiable Lipschitz control.One power-iteration step per gradient step estimates the spectral norm with median relative error 3 × 10^-13% and worst-case error 0.06% across CPL layers in 49 final-checkpoint SRB models.Source: §A.3 · p. 14

A.4 BASELINE DETAILS

  • The visibility-supervised NDF adaptation replaces normal-derived winding-number supervision with visibility constraints, labeling points before measurements outside and points immediately behind them inside.It also samples free-space points along sensor-to-measurement segments while retaining the original SLL architecture and hKR objective.Source: §A.4 · p. 15

A.5 EVALUATION METRICS

  • The evaluation reports symmetric Chamfer distance and observation-supported CDobs and NCobs, computed on surface samples supported by the acquisition; β is 2 unless stated otherwise.Source: §A.5 · pp. 15–16

B.1 LIPSCHITZ PRELIMINARIES

  • A function is C-Lipschitz when its Lipschitz constant is at most C.The appendix uses the 1-Lipschitz constraint in its sphere-tracing convergence argument.Source: §B.1 · p. 17, §B.2 · p. 17

B.2 SPHERE TRACING CONVERGENCE

  • Sphere tracing converges to the first root at a local linear rate under the stated non-grazing condition.The iterates increase to the first-root distance, and the local error contracts by a factor of at most 1 − κ.Source: §B.2 · pp. 17–18

B.3 INCIDENCE-WEIGHTED RESIDUAL

  • The incidence-weighted residual analysis relates ray-depth error to distance from the predicted surface, with a local quadratic remainder when the surface is C2.For a CPWL field affine near the hit, the local surface graph is flat.Source: §B.3 · p. 18

B.4 VOLUMETRIC FREE SPACE INDUCED BY RAY SUPERVISION

The appendix extends ray-based free-space certification beyond acquisition rays and describes how certified volume is evaluated within the initialization domain.

B.4 VOLUMETRIC FREE SPACE INDUCED BY RAY SUPERVISION

  • Although certification begins only along finitely many acquisition rays, the 1-Lipschitz constraint extends the guarantee to a three-dimensional neighborhood.The appendix constructs certified free space from pre-hit sphere-tracing iterates and intersects their union with a sub-ball of the initialization domain.Source: §B.4 · pp. 18–19
  • Certified free-space volume is normalized by the ground-truth free-space volume within a concentric sub-ball of the initialization domain.Table 6 reports CFVτ (%) for Ours (L = 0, no rescale) across such sub-balls, with τ = 1 denoting the full initialization ball.Source: §B.4 · pp. 18–19, Table 6 · p. 19

C.1 OBJECTIVE ABLATIONS

The objective ablations compare the proposed setup with architectural and loss variants, while the Huber-scale sensitivity study varies the scale relative to its default.

C.1 OBJECTIVE ABLATIONS

  • The objective-ablation results average over five SRB shapes but use a single seed.Source: Table 7 · p. 20
  • ReLU with eikonal regularization consistently underperforms Ours (L = 0), so eikonal regularization alone does not recover the architectural Lipschitz constraint’s benefits.The miss-penalty ablation additionally penalizes rays for which sphere tracing finds no intersection.Source: §C.1 · p. 20, Table 7 · p. 20

C.2 HUBER-SCALE SENSITIVITY

  • The Huber-scale study compares normalized scale settings against ℓ1 and ℓ2 losses, using δ0 = 0.05 on clean data and δ0 = 0.005 with outliers (α = 0.2).Source: Figure 15 · p. 20

C.3 BACKBONE ARCHITECTURE

The appendix tests an alternative 1-Lipschitz backbone, examines sensitivity to sensor provenance and fitting capacity, and reports computational costs and extended benchmark settings.

C.3 BACKBONE ARCHITECTURE

  • The first-hit formulation remains effective when the CPL backbone is replaced with Sandwich layers, with the training protocol unchanged.The comparison is preliminary and uses a single seed.Source: §C.3 · p. 20, Table 8 · p. 21

C.7 SENSITIVITY TO SENSOR PROVENANCE

  • Performance is nearly unchanged up to 3.7% sensor-position error relative to scanner standoff, and reconstruction still succeeds at 7.4%.The perturbations are deliberately substantial and are not intended as a precise model of scanner calibration errors.Source: §C.7 · p. 22, Table 11 · p. 22
  • Randomly shuffled sensor origins leave 62–65% of rays occluded before their measured endpoints, and all 15 tested fits lose valid traced hits after a few hundred steps.Source: §C.7 · p. 22

C.8 FIDELITY–ROBUSTNESS ANALYSIS

  • The fidelity–robustness analysis reports clean-input distance to the reconstructed surface and plots mean CD1 and CDobs1 across fitting levels, corruption levels, and a weakly observed concavity.The clean-SRB error maps normalize distance to ground truth by object radius.Source: §C.8 · p. 22, Table 12 · p. 22, Figure 18 · p. 23

C.10 COMPUTATIONAL COST

  • The computational-cost results report wall-clock training time, peak GPU memory, intermediate-training CD1, and sphere-tracing workload.The timing and memory comparison is for one SRB shape; the workload uses K = 192 tracing steps.Source: Table 14 · p. 25, §C.10 · p. 25, Table 15 · p. 25

D.1 SRB SCANNER NOISE; D.2 SRB OUTLIERS; D.3 RAY-ALIGNED OUTLIERS

  • The extended SRB results cover scanner noise, outlier fractions, and ray-aligned range corruption, with the latter using 10% corrupted rays.The corresponding tables report per-shape reconstruction and observation-supported metrics for noise and outlier settings.Source: Table 16 · p. 26, Table 17 · p. 27, Table 18 · p. 28

D.4 SHAPENET EXTENDED RESULTS

The ShapeNet extended results report category-wise metrics under isotropic and ray-aligned Gaussian noise, using 3K input points.

D.4 SHAPENET EXTENDED RESULTS

  • The category-wise ShapeNet CD2 results cover isotropic and ray-aligned Gaussian noise with 3K input points.Source: Table 21 · p. 30
  • The category-wise ShapeNet normal-consistency results cover isotropic and ray-aligned Gaussian noise with 3K input points.Source: Table 22 · p. 31
  • The category-wise ShapeNet IoU results cover isotropic and ray-aligned Gaussian noise with 3K input points.Source: Table 23 · p. 32

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