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Neural Shortest Path for Surface Reconstruction from Point Clouds

Yesom Park, Imseong Park, Jooyoung Hahn, Myungjoo Kang

TL;DR

The neural shortest path is proposed, a vector-valued implicit neural representation that approximates a distance function and its gradient and it is mathematically proved that the decomposed representation of NSP guarantees the convergence of the magnitude of NSP in the $H^1$ norm.

Abstract

In this paper, we propose the neural shortest path (NSP), a vector-valued implicit neural representation (INR) that approximates a distance function and its gradient. The key feature of NSP is to learn the exact shortest path (ESP), which directs an arbitrary point to its nearest point on the target surface. The NSP is decomposed into its magnitude and direction, and a variable splitting method is used that each decomposed component approximates a distance function and its gradient, respectively. Unlike to existing methods of learning the distance function itself, the NSP ensures the simultaneous recovery of the distance function and its gradient. We mathematically prove that the decomposed representation of NSP guarantees the convergence of the magnitude of NSP in the $H^1$ norm. Furthermore, we devise a novel loss function that enforces the property of ESP, demonstrating that its global minimum is the ESP. We evaluate the performance of the NSP through comprehensive experiments on diverse datasets, validating its capacity to reconstruct high-quality surfaces with the robustness to noise and data sparsity. The numerical results show substantial improvements over state-of-the-art methods, highlighting the importance of learning the ESP, the product of distance function and its gradient, for representing a wide variety of complex surfaces.

Neural Shortest Path for Surface Reconstruction from Point Clouds

TL;DR

The neural shortest path is proposed, a vector-valued implicit neural representation that approximates a distance function and its gradient and it is mathematically proved that the decomposed representation of NSP guarantees the convergence of the magnitude of NSP in the norm.

Abstract

In this paper, we propose the neural shortest path (NSP), a vector-valued implicit neural representation (INR) that approximates a distance function and its gradient. The key feature of NSP is to learn the exact shortest path (ESP), which directs an arbitrary point to its nearest point on the target surface. The NSP is decomposed into its magnitude and direction, and a variable splitting method is used that each decomposed component approximates a distance function and its gradient, respectively. Unlike to existing methods of learning the distance function itself, the NSP ensures the simultaneous recovery of the distance function and its gradient. We mathematically prove that the decomposed representation of NSP guarantees the convergence of the magnitude of NSP in the norm. Furthermore, we devise a novel loss function that enforces the property of ESP, demonstrating that its global minimum is the ESP. We evaluate the performance of the NSP through comprehensive experiments on diverse datasets, validating its capacity to reconstruct high-quality surfaces with the robustness to noise and data sparsity. The numerical results show substantial improvements over state-of-the-art methods, highlighting the importance of learning the ESP, the product of distance function and its gradient, for representing a wide variety of complex surfaces.

Paper Structure

This paper contains 29 sections, 2 theorems, 39 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shortest path property
  4. Neural network
  5. Proposed method
  6. Neural shortest path
  7. Loss function
  8. Manifold loss
  9. Gradient matching loss
  10. Shortest path loss
  11. Mathematical properties
  12. Convergence in the $H^1(\Omega)$ norm
  13. Global minimum
  14. Minimal area regularization
  15. Shortest path-based surface extraction
  16. ...and 14 more sections

Key Result

Theorem 1

Let $F_n:\Omega\rightarrow\mathbb{R}^3$ be a uniformly bounded sequence of $H^1\left(\Omega\right)$ functions which are nonzero a.e. in $\Omega$ and consider the MDD $d_n\coloneqq \left\Vert F_n\right\Vert$ and $G_n\coloneqq \frac{F_n}{\left\Vert F_n\right\Vert}$ for $n\in\mathbb{N}$. If $F_n$ conve

Figures (14)

  • Figure 1: Illustration of the shortest path property.
  • Figure 2: The visualization of the network architecture of NSP with variable splitting.
  • Figure 3: Illustration of shortest path-based cell determination of the surface extraction algorithm in Section \ref{['subsec:extract_alg']}. For points in the right red-colored cell, if all points pulled by the NSP stay out of the cell, then the cell does not contain the surface $\Gamma$. For a point in the left blue-colored cell, if the point pulled by the NSP stays in the cell, then the cell intersects with $\Gamma$.
  • Figure 4: Results from the trained models on hemisphere data, showing cross section cuts in planes $z=0.1$ (top row) and $y=0$ (bottom row). The level sets illustrate the distance function learned from a given point cloud, represented by red dots. Quivers indicate the gradient field of the trained distance functions.
  • Figure 5: Results from the trained models on partial cylinder data, showing cross section cuts in planes $z=0$ (top row) and $y=0$ (bottom row). The level sets illustrate the distance functions learned from a given point cloud, represented by red dots. Quivers indicate the gradient field of the trained distance functions.
  • ...and 9 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Remark 1