Medial Axis Aware Learning of Signed Distance Functions

arXiv cs.CV / 4/21/2026

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Key Points

The paper introduces a variational framework to compute an accurate global signed distance function (SDF) from an input point cloud.
It explicitly models the medial axis by incorporating the jump set of the SDF’s gradient into a higher-order variational formulation that enforces linear growth away from that discontinuity set.
The method constrains the solution by enforcing both the eikonal equation and the SDF’s zero level set.
To keep the optimization tractable, it uses an Ambrosio–Tortorelli-type phase-field approximation, where the phase field implicitly represents the medial axis.
Neural networks are used to approximate both the SDF and the phase field for unoriented point clouds, and experiments show improved accuracy versus existing methods in both local (near-field) and global settings.

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.