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1-Lipschitz Neural Distance Fields

Guillaume Coiffier, Louis Bethune

TL;DR

This work addresses robust neural distance field learning by enforcing a $1$-Lipschitz constraint to guarantee non-overshooting distance estimates and enable reliable geometric queries. It introduces the hinge-Kantorovitch-Rubinstein ($hKR$) loss, which semi-supervisedly optimizes an occupancy-based objective, pushing the gradient toward unit norm and enabling signed or unsigned distance fields without exact ground-truth distances. A $1$-Lipschitz architecture (SLL) guarantees the Lipschitz bound, while inside/outside labeling via the generalized winding number enables robust learning from noisy or incomplete data. The approach supports large-scale geometric queries, constructive solid geometry, and resilience to noise, with clear tradeoffs controlled by the margin $m$ and improvements possible through future architectural and sampling enhancements.

Abstract

Neural implicit surfaces are a promising tool for geometry processing that represent a solid object as the zero level set of a neural network. Usually trained to approximate a signed distance function of the considered object, these methods exhibit great visual fidelity and quality near the surface, yet their properties tend to degrade with distance, making geometrical queries hard to perform without the help of complex range analysis techniques. Based on recent advancements in Lipschitz neural networks, we introduce a new method for approximating the signed distance function of a given object. As our neural function is made 1- Lipschitz by construction, it cannot overestimate the distance, which guarantees robustness even far from the surface. Moreover, the 1-Lipschitz constraint allows us to use a different loss function, called the hinge-Kantorovitch-Rubinstein loss, which pushes the gradient as close to unit-norm as possible, thus reducing computation costs in iterative queries. As this loss function only needs a rough estimate of occupancy to be optimized, this means that the true distance function need not to be known. We are therefore able to compute neural implicit representations of even bad quality geometry such as noisy point clouds or triangle soups. We demonstrate that our methods is able to approximate the distance function of any closed or open surfaces or curves in the plane or in space, while still allowing sphere tracing or closest point projections to be performed robustly.

1-Lipschitz Neural Distance Fields

TL;DR

This work addresses robust neural distance field learning by enforcing a -Lipschitz constraint to guarantee non-overshooting distance estimates and enable reliable geometric queries. It introduces the hinge-Kantorovitch-Rubinstein () loss, which semi-supervisedly optimizes an occupancy-based objective, pushing the gradient toward unit norm and enabling signed or unsigned distance fields without exact ground-truth distances. A -Lipschitz architecture (SLL) guarantees the Lipschitz bound, while inside/outside labeling via the generalized winding number enables robust learning from noisy or incomplete data. The approach supports large-scale geometric queries, constructive solid geometry, and resilience to noise, with clear tradeoffs controlled by the margin and improvements possible through future architectural and sampling enhancements.

Abstract

Neural implicit surfaces are a promising tool for geometry processing that represent a solid object as the zero level set of a neural network. Usually trained to approximate a signed distance function of the considered object, these methods exhibit great visual fidelity and quality near the surface, yet their properties tend to degrade with distance, making geometrical queries hard to perform without the help of complex range analysis techniques. Based on recent advancements in Lipschitz neural networks, we introduce a new method for approximating the signed distance function of a given object. As our neural function is made 1- Lipschitz by construction, it cannot overestimate the distance, which guarantees robustness even far from the surface. Moreover, the 1-Lipschitz constraint allows us to use a different loss function, called the hinge-Kantorovitch-Rubinstein loss, which pushes the gradient as close to unit-norm as possible, thus reducing computation costs in iterative queries. As this loss function only needs a rough estimate of occupancy to be optimized, this means that the true distance function need not to be known. We are therefore able to compute neural implicit representations of even bad quality geometry such as noisy point clouds or triangle soups. We demonstrate that our methods is able to approximate the distance function of any closed or open surfaces or curves in the plane or in space, while still allowing sphere tracing or closest point projections to be performed robustly.
Paper Structure (21 sections, 1 theorem, 20 equations, 11 figures)

This paper contains 21 sections, 1 theorem, 20 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Signed Distance Function
  4. Lipschitz Implicit Representations
  5. Neural Implicit Surfaces
  6. Lipschitz Neural Networks
  7. Robust Learning of a Signed Distance Function
  8. $1$-Lipschitz Neural Architecture
  9. The hinge-Kantorovitch-Rubinstein Loss Function
  10. Inside/Outside Partitionning
  11. Distance field of shapes without interior
  12. Results and Applications
  13. Implementation details
  14. Gradient Robustness
  15. Signed vs Unsigned Distance Field
  16. ...and 6 more sections

Key Result

theorem 1

Let $D,\Omega$ be compact subsets of $\real^n$ such that $\Omega \subset D$. Let $y$ be binary labels defined over $D$ as: Let $m>0$ and assume that $\rho(x)=0$ whenever $|S_\Omega(x)|\leqslant m$ and $\rho(x)>0$ otherwise. Let $f^*$ be a minimizer of $\loss_{KR}(f,y)$ under constraint that $\loss_{hinge}^m(f,y) = 0$, where the minimum is taken over all possible $1$-Lipschitz functions. Then:

Figures (11)

  • Figure 1: The value of the SDF $S_\Omega$ at point $x$ is the radius of the larger sphere centered at $x$ that does not intersect $\Omega$ (left). For an approximated SDF $f$, if $f$ overestimates the distance (middle), then the query is wrong and no guarantees can be drawn. If $f$ always underestimates the distance (right), iterating the query still converges to the correct result.
  • Figure 2: Plot of a the gradient norm of neural distance fields on a simple 2D dolphin silhouette. While minimizing the eikonal loss stabilizes the gradient norm, only a Lipschitz network guarantees a unit bound.
  • Figure 3: Overview of our method on a corrupted Botijo dataset. Given an input geometry in the form of an oriented point cloud or a triangle soup, we uniformly sample points in a domain containing the desired geometry. Defining negative samples as points of the geometry and positive samples everywhere else yields an unsigned distance field when minimizing the hKR loss. On the other hand, partitioning samples as inside or outside the shape leads to an approximation of the signed distance function of the object.
  • Figure 4: Generalized winding number field computed for the Gargoyle model on the original manifold mesh (top) and a point cloud of 50K points(bottom). Thresholding this field allows to partition a dataset of points into inside and outside of the shape, from which a neural signed distance field can be optimized by minimizing the hKR loss (right column).
  • Figure 5: Neural distance field of an open curve in 2D. Minimizing $\loss_{hKR}$ on this dataset yields a distance field to the curve up to the margin parameter $m$. Large $m$ create large but consistent underestimation of the true distance while smaller $m$ lead to instabilities in training and final result. A value of $m$ around $10^{-2}$ is a good trade-off in practice.
  • ...and 6 more figures

Theorems & Definitions (1)

  • theorem 1