demofoxDec 29, 2022
If you've heard of monte carlo geometry processing and wondered what it's all about, give this a read!
In short, it's a way of interpolating data using random walks. An implementation looks quite a bit like ray marching an SDF, including needing to know the distance to the closest feature, from within a loop!
https://blog.demofox.org/2020/07/11/interpolating-data-over-arbitrary-shapes-with-laplaces-equation-and-walk-on-spheres/
Interpolating Data Over Arbitrary Shapes With Laplace’s Equation and Walk on Spheres

A very cool paper was accepted to SIGGRAPH 2020 called “Monte Carlo Geometry Processing: A Grid-Free Approach to PDE-Based Methods on Volumetric Domains” The paper is here: You can find…

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Show threadj_bertolottiDec 29, 2022
@demofox Naïve question: is this faster and/or more efficient than solving the Laplace equation? I somehow had in my mind that there were pretty good Laplace solvers out there (notice that I might be badly wrong about it).
Show threadNikita LisitsaDec 29, 2022
@j_bertolotti @demofox IIRC the trick here is that it gives you the value of a solution at a specific point without requiring to solve for the whole domain.
Show threadj_bertolottiDec 29, 2022
@lisyarus @demofox What this is exploiting is that the solution of the Laplace equation is also the steady state of the diffusion equation. So while it is true you don't need to solve for parts of the domain far away from the boundaries (i.e. the data you want to interpolate/extrapolate), you still need to "diffuse" from all those points.
If you tell me that solving the Laplace equation is a lot more onerous than doing a random walk I am ready to believe you, but it doesn't feel obvious to me.
Show threaddemofox
@j_bertolotti @lisyarus one benefit is that you can do localized solves instead of solving it all globally. From a graphics pov, this means each pixel can solve for the data it needs on demand. You can zoom into part of the data and see that part, to the detail being viewed (even progressively) without having to calculate the whole solution in advance to full detail.
Dec 29, 2022 at 3:59pmWeb