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Keenan Crane

@keenanisalive • 38,630 subscribers

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

Shorts

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]

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]

275,399 Aufrufe

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!…🫣)

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!…🫣)

34,020 Aufrufe

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.

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.

185,296 Aufrufe

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 🤩

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 🤩

71,412 Aufrufe

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.

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.

181,753 Aufrufe

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.

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.

269,127 Aufrufe

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.”

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.”

131,168 Aufrufe

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

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

63,591 Aufrufe

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:

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:

15,355 Aufrufe

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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.

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.

52,802 Aufrufe • vor 4 Tagen

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]

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]

456,475 Aufrufe • vor 1 Jahr

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.

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.

493,866 Aufrufe • vor 1 Jahr

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.

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.

100,587 Aufrufe • vor 1 Jahr

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)

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)

111,869 Aufrufe • vor 2 Jahren

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 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]

91,919 Aufrufe • vor 2 Jahren

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

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

33,699 Aufrufe • vor 11 Monaten

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.

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.

29,470 Aufrufe • vor 1 Jahr

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!

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!

66,403 Aufrufe • vor 3 Jahren

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.

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.

30,536 Aufrufe • vor 1 Jahr

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.

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.

17,785 Aufrufe • vor 11 Monaten

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.

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.

23,368 Aufrufe • vor 2 Jahren

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:

16,352 Aufrufe • vor 2 Jahren

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