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

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 $\theta$, for which we derive a bounded phase indicator $P=\tanh(\beta\theta)$ 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 $\beta$ 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 [MPFs-GitHub](https://github.com/JIAYI-Scarlett/ICML2026-MPF) for source code and test models.

Existing neural 3D reconstruction methods often struggle with thin structures such as shells, plates, and layered surfaces. Methods based on signed distance fields usually produce smooth closed surfaces but have difficulty handling thin or open geometries, while unsigned distance field methods are more flexible but can become unstable during training and surface extraction. We introduce Metric—Phase Fields (MPFs), a new representation that separates geometric distance from structural phase information when reconstructing shapes from unoriented point clouds. By learning these two components independently and combining them in a smooth and adaptive way, MPFs produce more stable surface representations with reliable near-surface behavior. Experiments on both synthetic and real scanned data show that our method reconstructs thin and layered structures more faithfully than recent approaches, while also improving training stability and surface extraction robustness.