💻 computer science

Points as Tori: Fast Pointwise Signed Distance for Point Clouds

This paper introduces a fast, feed-forward method that reconstructs point clouds into analytical torus-based parameterizations using a pre-trained network, enabling efficient, parallelizable pointwise signed distance queries and direct geometric operations without requiring global optimization or spatial discretization.

Original authors: Nicole Feng, Ioannis Gkioulekas, Keenan Crane

Published 2026-07-21

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CC BY 4.0

Original authors: Nicole Feng, Ioannis Gkioulekas, Keenan Crane

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are holding a handful of glittering dust, and you want to know the shape of the invisible cloud that dust is floating in. In the world of computer graphics and 3D modeling, this "dust" is called a point cloud—a collection of millions of tiny dots that represent the surface of an object, like a statue or a car, captured by a scanner. But here's the tricky part: just looking at the dots doesn't tell you which side is "inside" and which is "outside." To make the computer understand the object's true shape, we need a special map called a Signed Distance Function (SDF). Think of an SDF as a magical ruler that, for any point in space, tells you exactly how far you are from the object's surface and whether you are standing inside the object or outside it. This map is the secret sauce for everything from video game physics to robot navigation. However, creating this map for a messy cloud of dots has traditionally been a slow, heavy, and complicated math problem, often requiring the computer to solve a giant puzzle for the entire object at once.

This paper, titled "Points as Tori," introduces a clever new way to draw that map instantly, point by point, without needing to solve the giant puzzle. The authors, Nicole Feng, Ioannis Gkioulekas, and Keenan Crane, propose a method that treats every single dot in the point cloud as if it were the center of a tiny, invisible torus (a donut shape). Instead of trying to guess the whole shape at once, their method uses a pre-trained neural network to look at a small neighborhood around each dot and figure out exactly what kind of "donut" fits best there. Because the math for the distance to a donut is already known and very fast, the computer can instantly calculate the distance for any point in space by blending the distances from all these tiny donuts together.

The magic happens because the authors realized that while older methods tried to force the dots into flat planes or complex curves that were hard to measure, fitting them to donuts (tori) is a sweet spot. A donut can look like a flat sheet, a curved hill, or a saddle shape depending on how you stretch it, and it has a simple, closed-form formula for distance. By using a neural network to learn the perfect "stretch" for each dot's local neighborhood, the method bypasses the need for slow, global calculations. The result is a system that can take a point cloud with millions of dots and answer "how far is this point from the surface?" in a fraction of a second (specifically, about $10^{-4}$ seconds for a single query on a cloud with 4,096 points).

The paper explicitly argues against older, "naive" approaches that try to simply average distances or use flat planes, showing that these methods often fail or produce jagged, inaccurate results when the data is sparse or noisy. They also demonstrate that while other methods might try to learn the entire shape from scratch using massive neural networks, their approach is smarter: it only uses learning to figure out the local shape around each dot, then uses simple math to do the rest. This means the method is not only incredibly fast but also robust; it handles messy data from real-world scans, 3D Gaussians, and even neural implicit models without breaking.

In their tests, the authors found that their "Points as Tori" method could reconstruct surfaces from point clouds with 29 million points in about 12.5 minutes, and then evaluate the distance for any point in the scene in just a few milliseconds. They showed that this allows for cool applications like instantly creating "offset" surfaces (shells around the object), performing Boolean operations (cutting and joining shapes), and even visualizing the object directly in a video game shader using a technique called sphere tracing. While the pre-computation step (training the local donuts) takes some time, the actual querying is so fast that it opens the door for using raw point clouds directly in applications that previously required a full, slow surface reconstruction first. The authors suggest that while their method is a significant leap forward, there is still room to improve how it handles extremely sparse data or to make the pre-computation even faster, but the core idea of using donuts to map the world of points is a solid, proven step forward.

Technical Summary: Points as Tori: Fast Pointwise Signed Distance for Point Clouds

Problem Statement

Computing a Signed Distance Function (SDF) for point clouds is a fundamental challenge in geometric modeling, simulation, and rendering. While SDFs are well-defined for watertight geometry in $\mathbb{R}^n$ , point clouds lack an explicit definition of "inside" and "outside," making the sign of the distance ill-defined. Existing methods for reconstructing surfaces or computing distances from point clouds typically rely on global optimization, spatial discretization (e.g., grids or octrees), or expensive iterative solvers. These approaches often struggle with scalability, real-time interaction, and large-scale data, as they require costly global solves that preclude efficient pointwise evaluation at arbitrary resolutions. Furthermore, many learning-based approaches attempt to learn SDFs directly from entire shapes, requiring per-shape training and failing to generalize to imperfect or noisy data without explicit surface reconstruction.

Methodology

The paper proposes Points as Tori (PAT), a method that computes signed distance to point clouds by locally fitting tori to the data and blending their analytical SDFs. The core insight is that tori possess closed-form SDF expressions and can locally approximate surfaces up to the second order (capturing spherical, ellipsoidal, saddle, cylindrical, and planar geometries as limiting cases).

1. Theoretical Foundation: Unification of Distance and Occupancy

The authors establish a theoretical unification between signed distance and classic reconstruction methods (winding numbers and Poisson surface reconstruction) using the signed Hopf-Cole transformation.

They derive that signed distance can be viewed as the solution to a screened Laplace equation with jump boundary conditions.
As the screening parameter $\lambda \to \infty$ , the solution converges to the signed distance function.
The paper identifies two convolutional distance approximation formulas. They demonstrate that the standard "LogSumExp" style formula (Equation 16) is brittle for point clouds because it attempts to minimize distance directly, often snapping to the nearest point.
Instead, they utilize a self-normalized kernel density estimator (Equation 17):
$\phi(x) = \frac{\sum_{i=1}^{|P|} g_i(x) \exp(-\lambda_x \|x - p_i\|)}{\sum_{i=1}^{|P|} \exp(-\lambda_x \|x - p_i\|)}$
Here, $g_i(x)$ represents the signed distance function of a local geometric proxy fitted at point $p_i$ . This formulation allows for pointwise queries without global solves.

2. Local Geometric Proxy: Torus Fitting

To define $g_i(x)$ , the method fits a torus to the neighborhood of each point $p_i$ .

Parameterization: A torus is defined by a major radius $R$ , minor radius $r$ , center $c$ , and axis $u$ . These are derived from the local surface geometry (principal curvatures $\kappa_{\max}, \kappa_{\min}$ and directions) and a shift coefficient.
Learning-Based Fitting: Instead of using brittle heuristics or expensive robust statistics to estimate local curvature and shifts, the authors train a neural network to predict the six coefficients of a local polynomial surface ( $a_{0,0}, \dots, a_{2,0}$ ) for each point's $k$ -nearest neighbor neighborhood.
Network Architecture: The network uses a transformer-based architecture with self-attention to weigh the importance of neighboring points. It takes a size- $k$ neighborhood as input and outputs the polynomial coefficients, which are then analytically converted into torus parameters (radii, center, axis, and sign).
Sign Determination: The sign of the torus SDF is determined by the relative signs of the principal curvatures, ensuring the torus osculates the surface from the correct side (interior or exterior).

3. Inference and Evaluation

Once the tori are fitted (a precomputation step), the global SDF $\phi(x)$ is evaluated at any query point $x$ using the weighted average formula (Equation 25).

Efficiency: The evaluation is purely analytical and parallelizable. It does not require solving linear systems or iterating.
Adaptive $\lambda$ : The screening parameter $\lambda_x$ is chosen dynamically based on the local density of points to ensure numerical stability and precision, using a shifted exponential formula to avoid underflow.
Acceleration: For large point clouds, the summation is restricted to points within a fixed radius $R_{eval}$ , falling back to the $k$ -nearest neighbors if necessary.

Key Contributions

Fast Pointwise Evaluation: The method allows for the evaluation of signed distance at arbitrary spatial resolution in $O(1)$ time per query (after precomputation), scaling linearly with point cloud size for precomputation. It avoids global optimization and spatial discretization.
Analytical Parameterization: By using tori, the method provides a closed-form SDF that is differentiable and output-sensitive, enabling direct use in applications like sphere tracing, Boolean operations, and morphological operations without explicit mesh reconstruction.
Theoretical Unification: The paper provides a new theoretical framework unifying signed distance, winding numbers, and Poisson reconstruction via the signed Hopf-Cole transformation, clarifying why certain convolutional formulas succeed while others fail for point clouds.
Learning-Driven Local Fitting: The approach replaces brittle heuristic curvature estimation with a pre-trained neural network that learns to fit local surface parameters, achieving robustness to noise, outliers, and inconsistent sampling without per-shape training.

Results and Performance

Accuracy: On datasets including ABC (CAD models), Thingi10K, and procedurally generated shapes, PAT achieves lower mean absolute error in signed distance compared to baseline methods like Signed Hopf-Cole distance, Smoothed Signed Planar Distance, and the Signed Heat Method (SHM). It performs particularly well on sparse point clouds where other methods suffer from catastrophic failures or over-smoothing.
Reconstruction Quality: The zero-level set of the computed SDF produces surface reconstructions competitive with mature methods like Screened Poisson Surface Reconstruction (SPSR) and NN-VIPSS, especially for sparse or noisy data.
Speed:
- Query Time: Evaluating the SDF for a single point on a point cloud with millions of points takes approximately $10^{-4}$ to $10^{-3}$ seconds.
- Precomputation: Fitting tori for a point cloud with 1 million points takes roughly 40 seconds on a standard CPU/GPU setup.
- Scalability: The method scales linearly with point cloud size, handling datasets with up to 29 million points with precomputation times of ~12.5 minutes and query times remaining in the millisecond range.
Applications: The authors demonstrate direct applications including offset surface generation, Boolean operations on point clouds, morphological operations (erosion/dilation), and direct visualization via sphere tracing. It also successfully computes signed distance for 3D Gaussians and neural implicit functions.

Significance and Claims

The paper claims that Points as Tori offers a paradigm shift in handling point cloud data by treating them as direct inputs for signed distance queries, bypassing the need for explicit surface reconstruction in many workflows.

Robustness: The method is tolerant to noise, outliers, imperfect sampling, and inconsistent orientations, outperforming classical point-set methods that rely on hand-tuned parameters.
Differentiability: The representation is fully differentiable, making it suitable for inverse rendering, shape optimization, and generative modeling tasks.
Efficiency vs. Quality Trade-off: While the authors acknowledge that specialized reconstruction methods might outperform PAT on perfect, dense geometry, PAT provides a unique balance of speed, robustness, and the ability to perform geometric operations directly on point clouds.
Limitations: The authors modestly note that the method is not specialized for reconstruction and may struggle with point clouds having sampling characteristics significantly different from the training data (e.g., large holes or extreme sparsity). They also note that the current precomputation step is non-negligible, though future work could improve acceleration and compression.

In summary, the paper presents a method that unifies classical geometric theory with modern learning techniques to provide a fast, robust, and analytically tractable signed distance function for point clouds, enabling a wide range of geometric processing tasks without the overhead of global surface reconstruction.