Metric--Phase Fields: Decoupling Distance and Sign for Thin-Structure Reconstruction from Unoriented Point Clouds

2026-05-25

AI Summary

Existing neural signed distance functions struggle to reconstruct thin structures and open boundaries, while unsigned distance fields (UDFs) suffer from gradient singularities at the zero-level set, hindering optimization and surface extraction. This work proposes a decoupled implicit representation that separates geometric metric from topological phase by jointly learning a UDF and a smooth phase field. A learnable phase coefficient adaptively controls the sharpness of phase transitions, yielding a signed implicit function with stable near-surface gradients. By integrating a tanh-based soft sign indicator and a gated metric fusion strategy, the method significantly outperforms existing SDF- and UDF-based approaches on both synthetic and real-world thin-shell and thin-plate data, achieving high-fidelity recovery of fine, slender geometries alongside more robust training dynamics and reliable surface reconstruction.

Abstract

Neural Signed Distance Functions (SDFs) excel at reconstructing watertight manifolds but fail on thin structures and open boundaries due to strict inside--outside constraints. Conversely, Unsigned Distance Fields (UDFs) accommodate general geometries but suffer from gradient singularities at the zero-level set, hindering optimization and extraction. We introduce Metric--Phase Fields (MPFs), a decoupled implicit representation that separates metric proximity from topological phase. Given an unoriented point cloud, MPFs learn (i) an unsigned metric field $r$ and (ii) a smooth phase field $θ$, for which we derive a bounded phase indicator $P=\tanh(βθ)$ that provides soft inside--outside cues where they are meaningful. We couple the two fields via a gated-metric formulation with a residual phase injection to obtain a signed implicit function with stable near-surface gradients. The phase coefficient $β$ is learnable, allowing MPFs to adaptively control the sharpness of the phase transition and the degree of saturation of the soft sign indicator. Experiments on both synthetic and scanned thin-shell and thin-plate shapes demonstrate that MPFs preserve thin and layered structures more faithfully than recent SDF-based methods, while also enabling more robust training and more reliable surface extraction than UDF-based approaches. Check out \href{https://github.com/JIAYI-Scarlett/ICML2026-MPF}{MPFs-GitHub} for source code and test models.

Problem

• thin-structure reconstruction
• unoriented point clouds
• signed distance functions
• unsigned distance fields
• implicit surface representation

Innovation

• Metric–Phase Fields
• Signed Distance Function
• Unsigned Distance Field
• Thin-Structure Reconstruction
• Implicit Representation