Neural Shortest Path for Surface Reconstruction from Point Clouds

2025-02-09

AI Summary

This paper addresses high-fidelity surface reconstruction from noisy and highly sparse point clouds. We propose Neural Shortest Path (NSP), a vector-valued implicit neural representation that jointly models the signed distance function (SDF) and its gradient. NSP robustly recovers geometry by learning the exact shortest path from any spatial point to its nearest surface point. Key contributions include: (1) the first decomposition-based NSP representation, provably convergent under the H¹ norm; and (2) a novel loss function incorporating variable splitting and gradient consistency constraints, ensuring the global optimum corresponds to the true shortest path. Extensive experiments on multiple benchmark datasets demonstrate that NSP significantly outperforms state-of-the-art methods, particularly under severe noise and extreme sparsity—achieving superior reconstruction accuracy and stability.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Representation Learning for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Approximates distance function and gradient
• Ensures simultaneous recovery of both components
• Improves surface reconstruction from point clouds

Innovation

• Neural shortest path learning
• Distance function gradient recovery
• Novel loss function design