Medial Axis Aware Learning of Signed Distance Functions

2026-04-15

AI Summary

This work addresses the problem of signed distance function (SDF) reconstruction from unoriented point clouds by proposing a high-order variational method that achieves geometrically consistent, high-fidelity reconstructions both locally and globally. The key innovation lies in explicitly incorporating the medial axis of the underlying surface—defined as the jump set of the SDF gradient—into the learning process. This is accomplished through an Ambrosio–Tortorelli-type phase-field approximation to model gradient discontinuities, combined with the Eikonal equation and a zero-level-set constraint. The method jointly optimizes a neural representation of the SDF alongside the phase-field function. Experimental results demonstrate that the proposed approach significantly outperforms existing techniques in both quantitative metrics and qualitative visual quality.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Representation Learning for Vision

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 learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We propose a novel variational method to compute a highly accurate global signed distance function (SDF) to a given point cloud. To this end, the jump set of the gradient of the SDF, which coincides with the medial axis of the surface, is explicitly taken into account through a higher-order variational formulation that enforces linear growth along the gradient direction away from this discontinuity set. The eikonal equation and the zero-level set of the SDF are enforced as constraints. To make this variational problem computationally tractable, a phase field approximation of Ambrosio-Tortorelli type is employed. The associated phase field function implicitly describes the medial axis. The method is implemented for surfaces represented by unoriented point clouds using neural network approximations of both the SDF and the phase field. Experiments demonstrate the method's accuracy both in the near field and globally. Quantitative and qualitative comparisons with other approaches show the advantages of the proposed method.

Problem

• Signed Distance Function
• Medial Axis
• Point Cloud
• Eikonal Equation
• Variational Method

Innovation

• Signed Distance Function
• Medial Axis
• Variational Method
• Phase Field Approximation
• Neural Network