NumGrad-Pull: Numerical Gradient Guided Tri-plane Representation for Surface Reconstruction from Point Clouds

High-fidelity surface reconstruction from multi-view images is a core problem in 3D computer vision. While neural implicit surfaces like SDFs offer smooth geometry, they are often bottlenecked by the computational intensity of volume rendering. Conversely, 3D Gaussian Splatting (3DGS) provides rapid training but lacks geometry continuity, often leading to fragmented surfaces. This paper presents a novel framework that integrates Signed Distance Fields directly into the splatting pipeline. By leveraging the continuous nature of SDFs to regularize Gaussian primitives, our method effectively fills geometric holes and suppresses noise inherent in sparse point clouds. Unlike hybrid approaches that rely on heavy volumetric sampling, our approach utilizes the efficiency of splatting to achieve faster convergence. Extensive evaluations demonstrate that our method produces high-quality surfaces with significantly fewer primitives, offering a more compact and efficient representation for both indoor and outdoor environments.

We describe a method for computing signed distance to point clouds that allows fast pointwise evaluation at arbitrary spatial resolution. As input, our method takes a point cloud with normals; as output, it provides an analytical parameterization that allows queries of signed distance to the approximate underlying surface at arbitrary points - simultaneously providing reconstruction and distance. Our key idea is to reconstruct shapes by locally fitting point clouds with tori, which have closed-form signed distance functions. Tori are fitted in a feed-forward manner, using a pre-trained network to output per-point curvature and shift parameters. Importantly, our method does not require costly global optimization or spatial discretization, and is easily parallelizable. Underlying our method is a new theory that unifies signed distance with the classic reconstruction methods of winding numbers and Poisson surface reconstruction. We use our method to compute signed distance to point clouds arising from photogrammetry, meshes, 3D Gaussians, and neural implicits. Our method allows point clouds to be used directly in applications, without explicit surface reconstruction: as examples, we take offsets of point clouds, apply morphological and Boolean operations, and directly visualize offset surfaces using sphere tracing.

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.