# Points as Tori: Fast Pointwise Signed Distance for Point Clouds

arXiv: https://arxiv.org/abs/2607.16946 · HTML: https://mlsanity.com/paper/2607.16946/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 45 of its 161 references that are in the map.
- First posted on arXiv 18 Jul 2026.

## Builds on (most cited first, 30 of 45)

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- [Grounding Image Matching in 3D with MASt3R](https://mlsanity.com/paper/2406.09756/) (2024)
- [Extracting Training Data from Diffusion Models](https://mlsanity.com/paper/2301.13188/) (2023)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
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- [BlendedMVS: A Large-scale Dataset for Generalized Multi-view Stereo Networks](https://mlsanity.com/paper/1911.10127/) (2019)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [Neural Unsigned Distance Fields for Implicit Function Learning](https://mlsanity.com/paper/2010.13938/) (2020)
- [Thingi10K: A Dataset of 10,000 3D-Printing Models](https://mlsanity.com/paper/1605.04797/) (2016)
- [Neural-Pull: Learning Signed Distance Functions from Point Clouds by Learning to Pull Space onto Surfaces](https://mlsanity.com/paper/2011.13495/) (2020)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [POCO: Point Convolution for Surface Reconstruction](https://mlsanity.com/paper/2201.01831/) (2022)
- [DiGS : Divergence guided shape implicit neural representation for unoriented point clouds](https://mlsanity.com/paper/2106.10811/) (2021)
- [Deep Implicit Moving Least-Squares Functions for 3D Reconstruction](https://mlsanity.com/paper/2103.12266/) (2021)
- [Neural Fields as Learnable Kernels for 3D Reconstruction](https://mlsanity.com/paper/2111.13674/) (2021)
- [Critical Regularizations for Neural Surface Reconstruction in the Wild](https://mlsanity.com/paper/2206.03087/) (2022)
- [Mosaic-SDF for 3D Generative Models](https://mlsanity.com/paper/2312.09222/) (2023)
- [DeepFit: 3D Surface Fitting via Neural Network Weighted Least Squares](https://mlsanity.com/paper/2003.10826/) (2020)
- [Phase Transitions, Distance Functions, and Implicit Neural Representations](https://mlsanity.com/paper/2106.07689/) (2021)
- [Neural-Singular-Hessian: Implicit Neural Representation of Unoriented Point Clouds by Enforcing Singular Hessian](https://mlsanity.com/paper/2309.01793/) (2023)
- [The Vector Heat Method](https://mlsanity.com/paper/1805.09170/) (2018)
- [StEik: Stabilizing the Optimization of Neural Signed Distance Functions and Finer Shape Representation](https://mlsanity.com/paper/2305.18414/) (2023)

## Abstract (arXiv)

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.