Points as Tori: Fast Pointwise Signed Distance for Point Clouds
Authors: Nicole Feng, Ioannis Gkioulekas, Keenan Crane
Organizations: Carnegie Mellon University, USA · Roblox, USA
Abstract
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.
Reconstructing signed distance functions (SDFs) from point cloud data benefits many robot autonomy capabilities, including localization, mapping, motion planning, and control. Methods that support online and large-scale SDF reconstruction often rely on discrete volumetric data structures, which affects the continuity and differentiability of the SDF estimates. Neural network methods have demonstrated high-fidelity differentiable SDF reconstruction but they tend to be less efficient, experience catastrophic forgetting and memory limitations in large environments, and are often restricted to truncated SDF. This work proposes OREN, a hybrid method that combines an explicit prior from octree interpolation with an implicit residual from neural network regression. Our method achieves non-truncated (Euclidean) SDF reconstruction with computational and memory efficiency comparable to volumetric methods and differentiability and accuracy comparable to neural network methods. Extensive experiments demonstrate that OREN outperforms the state of the art in terms of accuracy and efficiency, providing a scalable solution for downstream tasks in robotics and computer vision.
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.
Reconstructing continuous surfaces from unoriented and unordered 3D points is a fundamental challenge in computer vision and graphics. Recent advancements address this problem by training neural signed distance functions to pull 3D location queries to their closest points on a surface, following the predicted signed distances and the analytical gradients computed by the network. In this paper, we introduce NumGrad-Pull, leveraging the representation capability of tri-plane structures to accelerate the learning of signed distance functions and enhance the fidelity of local details in surface reconstruction. To further improve the training stability of grid-based tri-planes, we propose to exploit numerical gradients, replacing conventional analytical computations. Additionally, we present a progressive plane expansion strategy to facilitate faster signed distance function convergence and design a data sampling strategy to mitigate reconstruction artifacts. These components are synergistically integrated into a unified tri-plane-based pulling framework, in which numerical gradients, progressive expansion, and complementary sampling jointly address the locality and sparsity challenges of learning SDFs from unoriented point clouds. Our extensive experiments across a variety of benchmarks demonstrate the effectiveness and robustness of our approach. Codes are available at: https://github.com/cuiruikai/numgrad-pull.