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Thanks, this means a lot!

SGI is an amazing example of how one can bring excellent new graduate students from a variety of backgrounds into a research field. It's one of the few examples that I've seen work in practice. It's going to disappear in 2026 if new funding is not secured. If you can help, contact Justin!

"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!

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]

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]

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]

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]

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]

(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]