Keenan Crane

@keenanisalive • 39,060 subscribers

Digital Geometer, Assoc. Prof. of Computer Science & Robotics @CarnegieMellon @SCSatCMU and member of the @GeomCollective. There are four lights.

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Keenan Crane

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We often use discretization to approximate continuous laws of physics, but it also goes the other way: You can use continuous equations to approximate the behavior of discrete systems! Here we'll see how electrical circuits can be modeled using the Laplace equation Δφ=0. [1/n]

Jensen's inequality gives the difference between the average value of a convex function φ, and its value at the center, where both “average” and “center” are defined in terms of some distribution p_X. When the function φ is flat, or the distribution is narrow, they agree.

A discrete Markov chain is basically a random walk on a graph, where each outgoing edge has a fixed probability. Cool fact: any initial distribution on a sufficiently nice* Markov chain converges to its stationary distribution after many steps—because it's a contraction mapping.

Revisiting this game a few months later: (i) It's pretty impressive that general-purpose image generators can estimate meaningful depth from a single photo. (ii) Monocular depth remains pretty awful for 3D reconstruction! Here Gemini &gt; GPT &gt; Qwen. (At least I hope so!…🫣)

Cantor showed that the "integer grid" has the exact same number of points as the "integer line"—even though both have infinitely many points! The correspondence can be shown using two (&amp; only two!) bijective quadratic functions sending pairs (x,y)∈ℕ₀×ℕ₀ to integers n∈ℕ₀.

Fun &amp; likely unintended use case for #NanoBananaPro: image→3D point cloud Prompt: This is a rendering of a 3D model of a person. Make an accurate image of the corresponding normal map. Then I run Poisson-based normals→3D Pretty amazing for a general-purpose 2D image model 🤩

There is a selection bias in college-level mathematics education: We believe that the status quo works well because it works for the top students (who go on to do PhDs, etc.). Yet it often works poorly for students in the “middle of the class.”

A basic strategy for drawing from a random distribution is to first generate uniformly-distributed points, then "warp" these points so they're spread out according to the target distribution. For example, the Box-Muller transform takes uniform points to normally-distributed ones

If you missed Nicole Feng's brilliant paper on generalized signed distance at #SIGGRAPH2024, here's a 20 second summary. 📺🎧 In short: nice SDFs even from "garbage" input geometry. You can read all about it in her nice blog-style writeup here:

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Here's a nice "proof without words": The sum of the squares of several positive values can never be bigger than the square of their sum. This picture helps make sense of how ℓ₁ and ℓ₂ norms regularize and sparsify solutions (resp.). [1/n]

Entropy is one of those formulas that many of us learn, swallow whole, and even use regularly without really understanding. (E.g., where does that “log” come from? Are there other possible formulas?) Yet there's an intuitive &amp; almost inevitable way to arrive at this expression.

Giving a talk in the Stanford University SCIEN seminar this Wednesday (1/3) at 4:30pm: The topic is “normal coordinates”: a shape representation little-used outside of mathematics—but which turns out to have nice applications in geometry processing &amp; learning.

Mental map of Markov Chain Monte Carlo (MCMC) algorithms, and analogous machine learning (ML) algorithms [dashed = especially loose analogy]. Grey boxes are basic tools, and each arrow is annotated with the "delta" between algorithms.

This breathtaking imagery is neither generative AI nor computer graphics—but rather real physical liquids under a microscope, exhibiting "reaction-diffusion" behavior. These clips were created and filmed by artist Kamil Czapiga: (ig: cosmodernism)

Excited to announce our #SIGGRAPH2023 paper "Winding Numbers on Discrete Surfaces," with Nicole Feng and Mark Gillespie. In essence, we address the question: given a jumble of curves on a surface (possibly noisy &amp; broken), which points are "inside" vs. "outside"? [1/n]

Meshes with 90° angles are super useful, providing asymptotically faster convergence for finite element simulation, and optimal shape approximation (when aligned with curvature). Amazingly, no past quad meshing method could guarantee 90° angles under refinement—until now. #RSP

What's the best way to pack N circles into a square? Using less than 20 lines of code in Penrose, we can reproduce the best-known solutions ever found: Try it out for yourself!

Huge congratulations to my former PhD student Rohan Sawhney for winning the ACM SIGGRAPH Outstanding Doctoral Dissertation Award! 🥳 Rohan's thesis develops a whole new class of Monte Carlo methods for solving physical equations, without needing to break the world up into a mesh.

If you're at #SIGGRAPH2024, I will be giving a talk on "Repulsive Shells" tomorrow (Thursday) in the Geometry: Editing and Deformation session, which begins at 10:45am in Mile High 2A.

Quick “teaser” for a fun #SIGGRAPH2025 project, led by Hossein Baktash, on optimizing a shape to have the desired rolling statistics. Basically we can turn arbitrary objects into fair dice, or make dice which capture the statistics of other objects—like several coin flips.

Nice new implementation of "stripe patterns" in GeometryCentral, thanks to David Joudan: The stripes algorithm computes evenly-spaced stripes aligned with a given vector field. This version also carefully extracts isolines, even around singularities.

Excited to kick off a brand new course at Carnegie Mellon University: Monte Carlo Methods and Applications Monte Carlo uses random sampling to solve tough problems throughout science &amp; engineering. We cover everything from fundamentals to real-world implementation:

Revisiting this game a few months later: (i) It's pretty impressive that general-purpose image generators can estimate meaningful depth from a single photo. (ii) Monocular depth remains pretty awful for 3D reconstruction! Here Gemini > GPT > Qwen. (At least I hope so!…🫣)

Cantor showed that the "integer grid" has the exact same number of points as the "integer line"—even though both have infinitely many points! The correspondence can be shown using two (& only two!) bijective quadratic functions sending pairs (x,y)∈ℕ₀×ℕ₀ to integers n∈ℕ₀.

Fun & likely unintended use case for #NanoBananaPro: image→3D point cloud Prompt: This is a rendering of a 3D model of a person. Make an accurate image of the corresponding normal map. Then I run Poisson-based normals→3D Pretty amazing for a general-purpose 2D image model 🤩

Entropy is one of those formulas that many of us learn, swallow whole, and even use regularly without really understanding. (E.g., where does that “log” come from? Are there other possible formulas?) Yet there's an intuitive & almost inevitable way to arrive at this expression.

Giving a talk in the Stanford University SCIEN seminar this Wednesday (1/3) at 4:30pm: The topic is “normal coordinates”: a shape representation little-used outside of mathematics—but which turns out to have nice applications in geometry processing & learning.

Excited to announce our #SIGGRAPH2023 paper "Winding Numbers on Discrete Surfaces," with Nicole Feng and Mark Gillespie. In essence, we address the question: given a jumble of curves on a surface (possibly noisy & broken), which points are "inside" vs. "outside"? [1/n]

Excited to kick off a brand new course at Carnegie Mellon University: Monte Carlo Methods and Applications Monte Carlo uses random sampling to solve tough problems throughout science & engineering. We cover everything from fundamentals to real-world implementation: