Ray Tracing Harmonic Functions

Ray Tracing Harmonic Functions by Mark Gillespie, Denise Yang, Mario Botch, and Keenan Crane. SIGGRAPH 2024.
markjgillespie...
ShaderToy examples:
* Riemann Surfaces (www.shadertoy....)
* Nonplanar Polygons (www.shadertoy....)
* Point Cloud Reconstruction (www.shadertoy....)
* Gyroid (www.shadertoy....)
* Hyperspherical Harmonics (www.shadertoy....)
* Full playlist (www.shadertoy....)
The sphere tracing algorithm provides a fast and high-quality strategy for visualizing surfaces encoded by signed distance functions (SDFs), which have become a centerpiece in a wide range of visual computing algorithms. In this paper we introduce a sphere tracing algorithm for a completely different class of functions, harmonic functions, opening up a whole new set of possibilities. Harmonic functions are found throughout geometric and visual computing, where they arise naturally as the solution to interpolation problems, and in the physical sciences, where they appear as solutions to the Laplace equation. Our algorithm for harmonic functions is similar in spirit to the sphere tracing algorithm for SDFs: by using a conservative lower bound on the distance to the level set, we can take much larger steps than with naïve ray marching. Our key observation is that for harmonic functions such a bound is given by Harnack's inequality. Unlike Lipschitz bounds used in traditional sphere tracing, this Harnack bound requires only the value of the function at a point-we use this bound to develop a sphere tracing algorithm that can also handle jump discontinuities arising in angle-based harmonic functions. We show how this algorithm can be used to directly visualize smooth surfaces reconstructed from point clouds (via Poisson surface reconstruction) or polygon soup (via generalized winding numbers) without performing linear solves or mesh extraction. We also show how it can be used to render nonplanar polygons (including those with holes), and to visualize key objects from mathematics, including knots, links, spherical harmonics, and Riemann surfaces.

[@fibbooo1123]

Ah, the classic "go up a dimension and things work out better". Beautiful work!

[@Sloimay]

This video is so good! I'm not a big math guy so I barely understood what was being shown but the "You don't need to know how it works, just need to know it does *this*" sections were very helpful. It's abstraction we rarely see in nerd KZbin that greatly help my viewing experience so these were really nice. + The set up of motivations was really well written I think, the script in general is really clean and straight to the point while keeping it interesting the entire way through. And the actual tech being displayed is so elegant and cool!

[@MDNQ-ud1ty]

So it's so good but you have no real knowledge about the subject to actually judge if it was good? You should study some basic logic first.

[@drdca8263]

@@MDNQ-ud1tyI believe the point being made was that it was good at being as understandable as it could be even to people not familiar with the field. While I can imagine a situation in which someone who doesn’t know the background of some field could, after watching a video, incorrectly reach the conclusion that the video was good at that, with the inaccurate judgement being a consequence of not being familiar with the field, I really don’t think not-having-familiarity-with-the-field in general makes one totally unable to evaluate how well some video does at the task. So, I don’t think your criticism of the original comment makes the most sense.

[@sergehog]

PLEASE UPDATE VIDEO DESCRIPTION: you have same URL for different ShaderToy examples!!

[@markgillespie4572]

Thanks for the heads up, I've fixed the links

[@makerhq376]

A very high quality report on worthwhile research. Thank you for posting this online!!

[@dot32]

So awesome that you're both the genius behind this who invented this technique, and a brilliant presenter!

[@larswanderart]

very cool! what happens if you have a negative singularity within the ball's radius? how do you pick a constant to ensure your harmonic function is positive?

[@markgillespie4572]

Good question! When using our algorithm, you always have to set the ball radius small enough so that it does not contain any singularities (because Harnack's inequality only applies to harmonic functions without singularities in the ball). This also means that we can always pick a constant to make the harmonic function positive. We talk a little more about the details in section 4.3 of the paper, since the harmonic function that we use for surface reconstruction contains negative singularities

[@oraz.]

Maybe this will show up everywhere

[@sentinelav]

What an awesome paper!! Brilliantly presented, and it's great to see it applied to previous methods. Can this render Mandelbulbs and other fractals?

[@dot32]

I believe fractals can already be raytraced, i've seen it done by youtubers such as CodeParade with marble marcher and Sebastian Lague with his Ray Marching video.

[@minma02262]

Amazing work!

[@mzg147]

amazing and beautiful stuff, thanks for the video!

[@suricrasia]

this is awesome work, well done!

[@SampleroftheMultiverse]

That’s 9:18 deep

[@debblez]

incredible

[@Kavukamari]

wait if it can trace the implicit surface of a 3d shape, then it should also be a general solution to rasterization of fonts, should it not?

[@GU-jt5fe]

Good idea, I hope this gets a response.

[@MikeLeed]

You sound like Sean Carroll

[@MooImABunny]

damn that's really interesting, I wish I had the time to study this more, but I don't have much of a background in the world of ray tracing

[@MagicGonads]

any challenges of this in higher dimensions?

[@markgillespie4572]

Everything works out pretty much the same in higher dimensions, the only difference is that the formula for determining your safe step size gets more complicated. In 2D, the safe step size is the solution to a linear equation. In 3D, the safe step size is the solution to a quadratic equation, so you have to take some square roots. In 4D, the safe step size is the solution to a cubic equation, so the formula is nastier. And in n dimensions, the safe step size is the solution to a degree n-1 equation, which can get tricky to solve if your dimension is too high

[@MDNQ-ud1ty]

I'm starting a band and I need an 3 octave oscillator that runs about 0.5hz, interested?