iiinova — where your digital self keeps evolving

"filthy engineer", haha. A big reason why I document my work is so people who are better than me at engineering, especially, can make these methods better. Thanks for reading!

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Finally, like for all my past projects, I’ve also put an informal writeup of the project on my webpage. Full project page: nzfeng.github.io/research/PointsAsTori/ </🧵>

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This work was done jointly with Ioannis Gkioulekas and @keenanisalive, and I’d like to thank many people for helpful discussions over the last few years. [34/35]

There's lots still to explore. Future directions I'm looking forward to: using PAT as a differentiable shape representation for inverse rendering/optimization, improving robustness (very noisy data with flipped normals, etc.), even faster precomputation. [33/35]

On the other hand, precomputation (torus fitting) can take minutes because it requires neural network evaluations, but I’m optimistic the network can be better engineered — or perhaps there’s a nice non-neural solution after all. [32/35]

One query of our SDF takes between 10^−4 — 10^−3 seconds for a single query to point clouds with millions of points. The kernel formula is simple, trivially parallelizable, and analytically differentiable. [31/35]

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The payoff is faster inference time, and faster and easier training (pretraining only — no per-shape optimization!) We also sidestep the difficulties of fitting neural fields to the highly nonlinear eikonal equation. [30/35]

We intentionally do *not* attempt end-to-end SDF learning. We rely on classical theory for convergence to true signed distance, and use the network only for the one component with no easy analytical solution (local surface fitting). [29/35]

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We use a small transformer network that predicts the 6 polynomial coefficients encoding the local surface underlying a neighborhood. It trains in ~45 hours on one NVIDIA RTX 3090, and can then be applied to any point cloud without extra training. [28/35]

So instead of tweaking one particular surface-fitting model, we simply use fixed-size neighborhoods with a *data-driven* approach that can fit to an extremely large set of examples, with a far more sophisticated data-fitting model than we could craft purely by hand. [27/35]

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But estimating the true underlying surface, even locally, is extremely difficult. Decades of literature have been dedicated to designing point-based methods for (local) reconstruction. Often, even just estimating neighborhoods is really difficult. [26/35]

For any point on a given surface, a best osculating torus is determined easily and in closed-form. [25/35]

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So instead, we fit *tori* to each point, because tori (1) have efficient closed-form SDFs, (2) can locally approximate surfaces up to second order (spherical, ellipsoidal, saddle). In 2D, we'd use circles; in 3D, tori generalize circles to two curvature directions. [24/35]

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But many past methods simply use linear functions g_i, leading to ugly shape completions and brittle behavior. [23/35]

Due to these fundamental challenges, we instead consider a 2nd class of distance formulas. Using a similar asymptotic argument as before, this formula instead implements closest point interpolation of a function g – letting us control the shape of the estimated surface. [22/35]

Our signed heat method from 2024 sidestepped these challenges by essentially decoupling distance from reconstruction, instead diffusing gradient vectors – at the cost of requiring a global solve. (Thread here: x.com/nicolefeng_/...) [21/35]

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(As an aside, our insights about distance extend to manifold learning in general, including flow-based generative models — the latter face exactly the same “memorization-generalization” tradeoff suffered by naive convolutional distance formulas.) [20/35]

Naive convolutional distance is essentially a single-point approximation; the exponential even dominates any regularization we might try to use, and unfortunately λ is coupled to both regression quality and distance accuracy, in opposite directions. [19/35]

In particular, they all give a distance function to the *sampled* geometry, rather than to the *underlying surface* from which the points were sampled. [18/35]

So amazingly, signed distance, (regularized) winding numbers, and Poisson surface reconstruction can be unified by a single class of PDEs and a simple exponential transform! But unfortunately, *none of the resulting distance formulas work for point clouds*. [17/35]

In turn, generalized winding number is a special case of Poisson surface reconstruction (PSR) — our jump screened Laplace equation can also be written as a screened Poisson equation, yielding PSR and regularized winding numbers. [16/35]

Through careful derivation of boundary conditions, we establish a close relationship between classic occupancy methods and signed distance. As λ -> 0, the jump screened Laplace equation yields generalized winding number, an inside/outside function. [15/35]

In more detail, a distance function is described by an eikonal equation; boundary conditions determine whether distance is signed. We show how a variant of the classic Hopf-Cole transformation turns a *signed eikonal equation* into a *jump screened Laplace equation*. [14/35]

In our paper, we show how one of these perspectives in fact unifies signed distance with inside/outside computation! [13/35]

So letting 𝜑(x) = |x-z| yields the first general convolutional distance formula. There are many special cases of this formula throughout math, science, and engineering (Varadhan, Hopf-Cole, LogSumExp, Kreisselmeier-Steinhauser, Schrödinger distance…) [12/35]

In fact, an integral of the following form, asymptotically speaking, becomes dominated by the contribution at the minimizer as λ -> ∞. So taking a log makes this whole process act like a minimum operator. [11/35]

The key to distance is the exponential: applying exp to (the negation of) a function φ supercharges extreme values: the difference between the largest (resp. smallest) values of φ vs. the rest becomes exponentially exaggerated, with λ controlling the severity. [10/35]

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On one level, these formulas simply blend exponentially-weighted functions in the style of countless other meshless interpolation methods (radial basis functions, moving least squares, partition-of-unity, smoothed particle hydrodynamics, etc.) [9/35]

This formula is not just a heuristic: we show that lots of literature boils down to just two “convolutional formulas” that approximate distance through a summation of kernel functions concentrated on the shape’s boundary, and that only one is viable for point clouds. [8/35]

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(Note that we’re taking a distance-weighted average of SDFs, *not*, for example, a union of shapes — taking Booleans of SDFs doesn’t result in a true SDF.) [7/35]

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The algorithm for PAT is deceptively simple: (1) Around each point, we fit a torus using a small pre-trained network; (2) Signed distance is evaluated using a kernel density estimator, taking a weighted average of the *SDFs of the fitted tori*. [6/35]

Our method also gives better reconstructions than other fast point-based methods like winding numbers, especially for sparse data, and provides signed distance to boot. [5/35]

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In contrast to past distance methods for point clouds, our method is designed to properly estimate the underlying surface. [4/35]

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The full paper, code, poster, and recorded 10-minute SIGGRAPH talk are all on the following project page: nzfeng.github.io/research/PointsAsTori [3/35]

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Our goal is signed distance to the true underlying surface. This requires both surface reconstruction & distance, so past methods use expensive global solves or neural field fitting. PAT allows evaluation without discretization, global solves, or per-shape training. [2/35]

The motivation is that point clouds make up a huge portion of geometric data — but computing a signed distance function (SDF) from a point cloud is surprisingly hard. (This was my final PhD project and ended up jam-packed, so a warning this thread is long) [1/35]

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Nicole Feng @nicole--f.bsky.social · 21/07/2026

At #SIGGRAPH2026, I presented "points as tori" (PAT) for fast pointwise evaluation of signed distance for point clouds. Some code (Python w/ C++ extensions) and demo up: github.com/nzfeng/point... Contributions from people/LLMs welcome. Explainer thread below for the project: 🧵

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The lighting and camera effects (and your voiceover at the live session) are too good 😂

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In the same session, I'll be presenting my own paper, "Points as Tori: Fast Pointwise Signed Distance for Point Clouds", which shows how to *compute* signed distance in the especially challenging case of point clouds (nzfeng.github.io/research/PointsAsTori). Come & give us your best Qs! </🧵>

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The paper also develops both primal and dual versions of the algorithm, and include results/comparisons for both. So I think it's a good idea, good execution, and I think this algorithm will be useful for a lot of people! (basically, anyone who works with implicits) [10/n]

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Like classic marching algorithms, SMT is still deterministic, parallelizable over all cells, guarantees compatibility of surface patches with neighbors, and guarantees manifold, intersection-free output. It also works on surfaces that are open or inconsistently oriented 😀 [9/n]

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SMT avoids the aliasing problem by intelligently making use of *all* grid crossings per cell, by generalizing normal coordinates from geometric topology. The theory is elegant, and for any given level of refinement, the result is better than naive adaptive methods. [8/n]

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Hossein's algorithm, Subgrid Marching Tetrahedra (SMT), extracts much more accurate isosurfaces than past methods, both geometrically and topologically speaking. SMT follows the first paradigm, yet has no lower bound on recoverable feature size for a fixed grid. [7/n]

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