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Projective Normal Fields: A Convex Optimization Method for Constructing Smooth UDFs

Jiayi Kong, Chen Zong, Fei Hou, Junhui Hou, Wenping Wang, Ying He

arXiv:2609.34784v1cs.CV

TL;DR

Raw point clouds lack connectivity and consistently oriented normals, while direct scalar UDF learning must handle a non-differentiable surface target and weak supervision away from samples. PNF estimates orientation-free normal axes through convex soft-tensor optimization, then uses confidence-guided diffusion and integration; experiments report reduced neighborhood sensitivity, competitive reconstruction under noise and outliers, and improved accuracy near evaluated non-manifold junctions.

  • Problem

    Raw point clouds provide neither surface connectivity nor consistently oriented normals, and direct scalar UDF learning faces a non-differentiable surface target and weak supervision away from samples.

    Source: §1 · p. 1

  • Method

    PNF relaxes normal-axis projectors to soft tensors and jointly estimates a coherent field on a fixed graph before confidence-guided directional diffusion and scalar integration.

    Source: §1 · pp. 1–2, §4.1 · p. 3, §4.3 · p. 4

  • Results

    Experiments show reduced sensitivity to neighborhood size, competitive reconstruction under noise and outliers, and improved accuracy near evaluated non-manifold junctions.

    Source: §6 · p. 6

  • Takeaways & Limitations

    The orientation-free normal-field formulation separates local normal-axis estimation from global UDF construction.

    Source: §6 · p. 6

    The global-optimality guarantee covers fixed-graph soft-tensor optimization, not exact surface recovery; extraction may retain a double-layered mesh, and a suitable single-layer representation requires post-processing.

    Source: §F.5 · p. 13

Abstract

from arXiv · show

Constructing a smooth approximation of an unsigned distance field (UDF) from a raw point cloud is challenging because the input provides neither surface connectivity nor consistently oriented normals. Methods that directly learn a scalar UDF must also handle its non-differentiability on the zero level set and weak supervision away from the samples, which can lead to unstable optimization and spatial artifacts. We introduce Projective Normal Fields (PNFs), an orientation-free representation and convex optimization framework for estimating bidirectional normals from point positions alone. Each normal axis is encoded by a rank-one projector, which is invariant to normal reversal. We relax the non-convex set of hard projectors to its convex hull: the symmetric positive-semidefinite matrices with unit trace. Each soft tensor defines a local quadratic distance model and retains the relative weights of candidate normal axes. We estimate a coherent PNF by combining local tangent-plane fitting, soft-PCA anchoring, and overlap regularization on a fixed neighborhood graph. With positive anchoring weights, the objective is strongly convex and admits a unique global minimizer. Principal eigenvectors provide bidirectional normals, while the corresponding eigengaps provide spectral confidence indicators. We use these indicators to select and weight directional sources for heat diffusion, followed by Poisson integration to construct a regularized UDF approximation. By separating local geometry estimation from scalar-field construction, PNF avoids directly fitting the non-differentiable UDF. Experiments demonstrate reduced sensitivity to neighborhood size, competitive reconstruction under noise and outliers, and improved accuracy near non-manifold junctions. The project page is available at https://anonymous17777367.github.io/PNF-page/

1 Introduction

Projective Normal Fields represent unoriented normal axes with sign-invariant tensors and estimate them before constructing a global unsigned distance field. Their convex formulation combines local geometry and neighborhood consistency, while a confidence-guided second stage constructs the UDF.

1 Introduction

  • Raw point samples lack surface connectivity and consistently oriented normals, while direct scalar UDF learning must handle a non-differentiable surface target and weak supervision away from samples.These challenges can lead to unstable optimization, inaccurate gradients, and spatial artifacts.Source: §1 · p. 1
  • PNF represents bidirectional normals with sign-invariant tensors, relaxing hard normal-axis projectors so neighboring estimates can retain ambiguous directional evidence.The approach estimates local geometry before constructing the global scalar field.Source: §1 · p. 1
  • PNF estimates a coherent normal field from point positions, then uses confidence-guided heat diffusion and scalar integration to construct a regularized UDF approximation.Figure 1 shows a corrupted Goldfish reconstruction and UDF slices; the caption reports preserved thin fins, open boundaries, and non-manifold junctions, while competing reconstructions show holes, irregularities, or spurious fragments.Source: §1 · p. 1, Figure 1 · p. 2
  • With positive anchoring weights, PNF estimation combines tangent-plane fitting, soft-PCA anchoring, and graph-based overlap regularization in a strongly convex problem with a unique global minimizer.The framework uses the decoded normal axes and their spectral confidence to select and weight directional sources for heat-based UDF construction.Source: §1 · p. 2

2 Related Work

Related work includes methods that learn scalar UDFs directly and alternatives that make geometric, topological, or directional information explicit. PNF is closest to VAD but differs in how it estimates the local normal field.

2.1 Direct UDF Learning

  • Direct UDF learning includes neural predictors, query-projection methods, and approaches that modify the field or its regularization to address near-surface behavior.Hybrid methods add sign, separation, or phase cues, while these approaches retain scalar distance fields or transformed distance fields as prediction targets.Source: §2.1 · p. 2

2.2 Geometry-First and Alternative Implicit Representations

  • Geometry-first alternatives include local or variational surface reconstruction and representations that explicitly predict closest points, directions, or displacement vectors.These alternatives encode correspondence or directional information rather than geometry solely through scalar distances.Source: §2.2 · p. 2
  • VAD estimates bidirectional normals by matching local projection-distance fields and gradients across Voronoi bisectors, whereas PNF jointly optimizes soft normal-axis tensors on a fixed graph without requiring a Voronoi construction.With positive anchoring weights, PNF's objective is strongly convex and has a unique global minimizer; the distinction is in local normal-field representation and optimization, not downstream diffusion.Source: §2.2 · pp. 2–3

3 Overview

The method constructs a smooth UDF approximation from point positions in two stages. It first estimates an orientation-free local normal field, then uses its decoded axes and confidence to build a global scalar field.

3 Overview

  • Given point positions for a possibly open, non-orientable, or non-manifold surface, the method constructs an unsigned distance field approximation in two stages.Stage I estimates local normal axes without imposing a global orientation.Source: §3 · p. 3
  • The method represents each unoriented normal axis with a rank-one projector, then relaxes hard projectors to soft tensors for joint optimization.The fixed-graph objective combines local tangent-plane evidence with neighboring-model compatibility and has a unique global minimizer under positive anchoring weights.Source: §3 · p. 3
  • In Stage II, decoded axes and confidence values guide directional-source selection and weighting for heat diffusion, followed by Poisson integration into a scalar UDF field.The overview's Figure 2 illustrates the process on a 2D toy model.Source: §3 · p. 3

4 Convex Optimization of Bidirectional Normals

Projective Normal Fields encode unoriented normal axes with sign-invariant projectors, then relax them to soft tensors for joint estimation. Their fixed-graph objective combines tangent-plane fitting, soft-PCA anchoring, and overlap regularization; positive anchoring weights guarantee a unique global minimizer.

4.1 Hard Normal-Axis Projectors

  • A hard normal-axis projector, P = nn⊤, represents n and −n identically, encoding their shared unoriented axis.It also represents the normal component of a displacement and defines the associated tangent plane.Source: §4.1 · p. 3

4.2 Soft Normal-Axis Tensors

  • Soft tensors relax the rank-one projector constraint to a convex set; their eigenvectors and eigenvalues retain candidate axes and their relative weights.The principal eigengap indicates how distinctly the optimized tensor favors its leading axis, not whether that axis is correct.Source: §4.2 · pp. 3–4

4.3 A Convex PNF Objective

  • Each point–tensor pair defines a local quadratic squared-distance model, with a soft tensor blending distances to planes associated with its candidate axes.For a locally planar neighborhood, the direction of least point spread approximates the surface normal; the soft-PCA prior favors that direction while retaining competing directions when geometry is ambiguous.Source: §4.3 · p. 4
  • The objective combines tangent-plane fitting to neighboring samples with soft-PCA anchoring, which balances local fidelity against graph-based evidence.Larger anchoring weights place greater emphasis on the local soft-PCA estimate.Source: §4.3 · p. 4
  • Overlap regularization couples neighboring local models by penalizing Hessian, midpoint-gradient, and midpoint-value discrepancies, with residuals normalized by edge length and its square.A fixed undirected neighborhood graph is constructed before optimization and remains unchanged during the solve.Source: §4.3 · pp. 4–5
  • With every anchoring weight positive, the fixed-data objective is strongly convex and has a unique global minimizer, but this does not guarantee geometrically compatible graph edges.The optimized tensors are hardened into normal projectors, and their eigengaps provide confidence values used to modulate directional diffusion.Source: §4.3 · p. 5

5 Experimental Results

PNF jointly optimizes normal axes to address ambiguous local geometry, and the experiments report competitive reconstruction across corrupted inputs and strong accuracy near evaluated non-manifold structures.

PCA versus PNF: ambiguous local neighborhoods

  • Joint optimization combines tangent-plane evidence, soft-PCA anchoring, and neighboring-model compatibility to address noisy, outlier-contaminated, and thin-structure neighborhoods where PCA can be unreliable.The reported sensitivity to neighborhood size is lower for PNF than for local PCA.Source: §5 · p. 5

Confidence as a geometric cue

  • The eigengap confidence modulates directional-source contributions and suppresses sources below a fixed threshold without automatically discarding their input samples.Low confidence can indicate competing axes or inconsistent support, but it does not identify sample type and high confidence does not guarantee a correct normal.Source: §5 · p. 5

Benchmark and evaluation

  • PNF achieves lower CD and HD than VAD under all five tested input conditions, with the lowest CD in four and lowest HD under 0.8% noise.GeoUDF has the lowest HD in four conditions; PNF has the second-lowest CD under 0.8% noise.Source: §5 · p. 5

PCA versus PNF: ambiguous local neighborhoods

  • On illustrated thin structures and 0.8% noise, PNF yields higher normal-axis accuracy and improved surface reconstructions than local PCA.Figure 3 colors per-point normal error and reports mean absolute cosine similarity; higher similarity is better.Source: §5 · p. 5, Figure 3 · p. 6

Non-manifold reconstruction

  • PNF achieves the lowest global CD, junction-region CD, and HD95 on all three synthetic non-manifold models, with 100% junction recall in each case.The models are a cross-junction, a multi-junction, and the self-intersecting Henneberg surface.Source: §5 · p. 6

6 Conclusion

The conclusion presents PNF as an orientation-free way to separate normal-axis estimation from UDF construction, with experiments showing competitive reconstruction under corruption and improved accuracy near evaluated non-manifold junctions.

6 Conclusion

  • PNF relaxes rank-one projectors to positive-semidefinite unit-trace tensors and combines tangent-plane evidence, soft-PCA anchoring, and overlap consistency in strongly convex optimization on a fixed graph.With positive anchoring weights, the optimization has a unique global minimizer; the resulting axes and spectral confidence guide directional-source selection, heat diffusion, and UDF construction.Source: §6 · p. 6
  • Experiments show reduced sensitivity to neighborhood size versus local PCA, competitive reconstruction under noise and outliers, and improved accuracy near evaluated non-manifold junctions.Figure 4 qualitatively shows PNF reconstructions that follow junction geometry more smoothly and coherently than several baselines.Source: §6 · p. 6, Figure 4 · p. 7
  • Figure 4 compares reconstructions under spatially varying noise and outliers and shows cross-sections for a cross-junction, multi-junction, and self-intersecting Henneberg surface.Several baselines show holes, irregularities, or floating fragments; DEUDF failed on the three synthetic non-manifold models.Source: Figure 4 · p. 7

A Proof of Proposition 4.1

The proof establishes the structure of the convex feasible set of soft tensors, including its compactness and the characterization of its extreme points.

Convexity

  • Convexity follows because convex combinations of feasible symmetric positive-semidefinite unit-trace matrices retain those properties.The proof checks the quadratic form, symmetry, and trace under a convex combination.Source: A Proof of Proposition 4.1 · p. 9

Compactness

  • The feasible set is compact because positive semidefiniteness and unit trace define a closed set, while nonnegative eigenvalues summing to one make it bounded.Closedness and boundedness imply compactness in the finite-dimensional space Sym(3).Source: A Proof of Proposition 4.1 · p. 9

Convex-hull identity

  • The spectral theorem expresses each feasible matrix as a convex combination of hard projectors, establishing the convex-hull identity.The proof identifies the hard projectors as the extreme points of the feasible set.Source: A Proof of Proposition 4.1 · p. 9

Extreme points

  • A hard projector cannot be a nontrivial convex combination of distinct feasible matrices, so every hard projector is an extreme point.The proof shows the component matrices vanish on the orthogonal complement of the projector's axis and therefore equal that projector.Source: A Proof of Proposition 4.1 · p. 9

B Proof of Proposition 4.2

The proof reduces finding a nearest hard projector to maximizing a quadratic form on unit vectors, then relates the optimizer and its uniqueness to the leading eigenspace.

Nearest hard projector

  • Finding the nearest hard projector is equivalent to maximizing n⊤Mn over unit vectors, whose maximizers lie in the leading eigenspace.The minimum squared distance is 1 + tr(M2) − 2λ1.Source: B Proof of Proposition 4.2 · p. 9

Uniqueness

Eigenvalue bound

  • The squared-eigenvalue sum is at least 1/3, with equality when all three eigenvalues equal 1/3 and M = I/3.The eigenvalues are nonnegative and sum to one.Source: B Proof of Proposition 4.2 · p. 9

C Proof of Theorem 4.3

The proof establishes that the feasible tensor fields form a nonempty compact convex set and that the objective has a unique global minimizer. Strong convexity follows from the anchoring terms, while the other objective terms preserve convexity.

C Proof of Theorem 4.3

  • The feasible set is a nonempty, compact, convex product of tensor domains.It is nonempty because each domain contains I/3.Source: C Proof of Theorem 4.3 · p. 9
  • With fixed inputs and graph data, the objective is continuous because it is a finite sum of linear and quadratic terms.Source: C Proof of Theorem 4.3 · p. 9
  • The fitting term is linear, while anchoring supplies strong convexity and nonnegative overlap weights preserve convexity.Source: C Proof of Theorem 4.3 · p. 10
  • Continuity on the feasible set ensures a global minimizer, and strong convexity rules out two distinct minimizers.Source: C Proof of Theorem 4.3 · p. 10

D Confidence-Guided Heat Diffusion for UDF Construction

Stage II turns the optimized normal axes and their confidence into bidirectional directional sources, propagates them through the domain, and integrates them into a scalar UDF approximation. Confidence truncation reduces weak directional contributions, but does not remove samples or guarantee that the propagated field is an exact UDF gradient.

D Confidence-Guided Heat Diffusion for UDF Construction

  • Each decoded normal axis yields two offset sources with opposite directions, providing two-sided guidance without globally orienting input normals.Reversing the chosen normal representative only swaps the pair; placing them at distinct positions avoids cancellation at one point.Source: D Confidence-Guided Heat Diffusion for… · p. 10
  • Heat-kernel propagation weights paired sources by eigengap confidence, with an empirically selected 0.1 threshold suppressing weaker contributions.Both sources in a pair receive the same weight, preserving reversal invariance.Source: D Confidence-Guided Heat Diffusion for… · p. 10
  • Truncation suppresses weak directional evidence, including potentially valid junction axes, but does not remove samples or alter their scalar-value constraints.Source: D Confidence-Guided Heat Diffusion for… · p. 10
  • The propagated field can be queried continuously, but cancellation and ambiguity near surfaces, junctions, or medial structures mean it need not equal an exact UDF gradient.Eigengap confidence measures preference for an axis, not agreement with an unknown ground-truth normal.Source: D Confidence-Guided Heat Diffusion for… · p. 10
  • Directional guidance drives scalar integration on a discretized domain, followed by DCUDF extraction and vertex refinement toward the zero level set.For non-orientable or non-manifold targets, extraction may retain a double-layered geometric approximation.Source: D Confidence-Guided Heat Diffusion for… · p. 10

E Benchmark

The benchmark covers varied geometry and sampling conditions across 60 point-cloud models. The following discussion examines graph construction, normal estimation, confidence guidance, comparison, computational cost, and limitations.

E Benchmark

  • The 60-model benchmark spans medial geometry, garments, complex-topology shapes, open or non-manifold objects, and indoor scenes, with roughly 50,000–150,000 samples per cloud.The categories include thin sheets, open boundaries, closely spaced surface layers, junctions, and complex scenes.Source: E Benchmark · pp. 10–11
  • All 60 models are tested with clean samples, two noise levels, and two outlier levels under matched conditions across methods.Noise perturbs local geometry; outliers add points that do not belong to the surface.Source: E Benchmark · p. 11

F Additional Results and Discussion

F.1 Ablation of Normal Estimation and Confidence Guidance

Across the reported comparisons, PNF outperforms VAD on the stated corruption and non-manifold metrics, while ablations show benefits from confidence truncation and graph-based joint estimation. The section also reports sparse-input robustness, pipeline runtime, and limits on what its optimization guarantee establishes.

F.1 Ablation of Normal Estimation and Confidence Guidance

  • Raw confidence weighting improves CD from 6.98 to 6.44 and F-score from 95.65% to 96.06%, while HD95 rises from 26.07 to 26.53.With uniform weights, PNF improves CD and F-score over PCA and VAD, although VAD has slightly lower HD95.Source: §F.1 · p. 11
  • Confidence truncation achieves the best reported results across all three metrics: CD 3.73, HD95 6.22, and F-score 99.20%.Source: §F.1 · p. 11

F.2 Robustness to Input Density

  • As point counts fall from 30K to 3K, PNF produces stable, coherent reconstructions while competing methods show distortions or reconstruction failures.The reported examples retain fine structures and complex topological configurations at lower densities.Source: §F.2 · p. 11, Figure 6 · p. 14

F.3 Comparison with VAD

  • Unlike VAD’s Voronoi-based iterative formulation, PNF optimizes soft tensors over a fixed neighborhood graph and has a unique-global-minimizer guarantee with positive anchoring weights.The reported guarantee applies to the fixed-graph soft-tensor optimization.Source: §F.3 · p. 11
  • PNF reports lower directed CD and HD than VAD under every corruption condition, and lower global CD, junction-region CD, and HD95 on all three non-manifold models.Table 5 reports global and junction-region metrics for three synthetic shapes; ties are included among best results.Source: §F.3 · p. 11, Table 5 · p. 15

F.1 Ablation of Normal Estimation and Confidence Guidance

  • On the evaluated thin structures and non-manifold models, PNF maintains comparable normal-estimation accuracy across three graph types and outperforms local PCA on both reported metrics.The tested graphs are symmetrized k-NN, radius-based, and Voronoi adjacency; the comparisons use fixed settings.Source: §F.1 · p. 11, Table 3 · p. 12

F.4 Computational Cost and Runtime

  • On evaluated non-manifold point clouds, PNF optimization averages 8.299 s excluding graph construction; field computation and surface extraction average 329.269 s and 35.036 s.The pipeline uses 500 optimization iterations, with O(N + |E|) work per iteration; field computation accounts for most reported time.Source: §F.4 · p. 13

F.5 Limitations

  • The global-optimality guarantee covers Stage I optimization, not exact surface recovery; extraction can retain a double-layered approximation, and single-layer output may require post-processing.Reconstruction also depends on graph connectivity, local evidence, and numerical discretization.Source: §F.5 · p. 13
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