Nicole Feng
Back to home
Points as Tori: Fast Pointwise Signed Distance for Point Clouds
ACM Transactions On Graphics (SIGGRAPH 2026)
Nicole Feng<br>Carnegie Mellon University
Ioannis Gkioulekas\(^\dagger\)<br>Carnegie Mellon University
Keenan Crane\(^\dagger\)<br>Carnegie Mellon University
\(^\dagger\) indicates equal contribution
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.
Paper (28.3mb)
poster PDF (11.9mb)
poster PNG (6.2mb)
For SIGGRAPH poster session.
Social media
X (Twitter) thread
-->
GitHub repository
Python API, demo visualization, training scripts and data.
Acknowledgements
This work was funded by NSF awards 2047341, 2212290 and 2504890, and a gift from nTopology. The authors thank Alexander Belyaev and Pierre Fayolle for inspiring conversations about distance computation; Chris Wojtan, Christian Hafner, Samara Ren, Aleksei Kalinov, and Mikaël Ly for helpful discussion; Rohan Sawhney and Josua Sassen for discussion of early ideas; and Hanyu Chen and Benran Hu for providing point cloud data.
@article{Feng:2026:PAT,<br>author = {Feng, Nicole and Gkioulekas, Ioannis and Crane, Keenan},<br>title = {Points as Tori: Fast Pointwise Signed Distance for Point Clouds},<br>year = {2026},<br>issue_date = {July 2026},<br>publisher = {Association for Computing Machinery},<br>address = {New York, NY, USA},<br>volume = {45},<br>number = {4},<br>issn = {0730-0301},<br>url = {https://doi.org/10.1145/3811385},<br>doi = {10.1145/3811385},<br>journal = {ACM Trans. Graph.},<br>month = {jul},<br>articleno = {53},<br>numpages = {24}
Fast Forward
Talk at SIGGRAPH 2026
Slides for 10-minute SIGGRAPH talk (17.8mb)
Slides for 50-minute talk (coming soon!)
-->
Extended 20-min talk (director's cut)
Video coming soon!
-->
Errata
In the original version of the paper, I unfortunately misspelled Anand Rangarajan's name in Figures 5 and 6. This has been corrected in the version on my website.
What is this paper about?
This paper introduces a method called points as tori (PAT) for computing signed distance to point clouds, that allows fast pointwise evaluation of signed distance at arbitrary spatial resolution, without requiring discretization, global optimization, or explicit reconstruction.
In a nutshell, signed distance computation has been either fast or robust (i.e. able to generalize from imperfect shape observations like point clouds) — but not both. We explain why, and develop a simple algorithm that directly estimates signed distance from point clouds.
In the end, given a point cloud with normals \(P = \{(\mathbf{p}_i, \mathbf{n}_i)\}_{i=1}^{|P|}\), our method evaluates signed distance \(\phi(\mathbf{x})\) at a given point \(\mathbf{x}\in\mathbb{R}^3\) simply using the kernel density estimator \[\phi(\mathbf{x}) = \frac{\sum_{i=1}^{|P|} g_i(\mathbf{x}) \exp\left(-\lambda_{\mathbf{x}}\|\mathbf{x}-\mathbf{p}_i\|\right)}{\sum_{i=1}^{|P|} \exp\left(-\lambda_{\mathbf{x}}\|\mathbf{x}-\mathbf{p}_i\|\right)}, \hspace{1cm} (1)\]<br>giving an exponentially-weighted average of per-point functions \(g_i\) associated with each point. We define \(g_i\) as signed distance functions (SDFs) of tori , because torus SDFs have efficient closed-form expressions, and tori can locally approximate surfaces up to second order. The parameter \(\lambda_{\mathbf{x}}\) is automatically chosen with a simple expression. Variants of Equation 1 have been used as heuristics in other point set methods, but were underexplored for signed distance.
The main challenge in our method is fitting tori to a given point cloud \(P\), and we determine these tori by pre-training a neural network to learn surface parameters that give good signed distance to point clouds. This network takes as input a size-\(k\)...