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Keenan Crane Profile picture
@keenanisalive
Jun 14, 2021 • 12 tweets • 7 min read • Read on X
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Uniform edge lengths are a reasonable proxy for mesh quality, since equilateral triangles have good angles. But if you have more specific criteria, you can usually do better. "Hodge-Optimized Triangulations" provides a nice discussion of this perspective: geometry.caltech.edu/pubs/MMdGD11.p…
A popular approach in graphics is something like Botsch & Kobbelt, "Remeshing for Multiresolution Modeling" (Sec. 4): graphics.rwth-aachen.de/media/papers/r…

Basic idea: iteratively move vertices to equalize areas, split long edges, collapse short edges, & flip edges to improve vertex valence.
In this class of algorithms, I really like Chen & Holst, "Efficient mesh optimization schemes based on Optimal Delaunay Triangulations": math.uci.edu/~chenlong/Pape…

Here there's a clear notion of optimality, & an efficient preconditioner. For surfaces: restrict to tangential motions
Another classic is Alliez et al, "Isotropic Surface Remeshing": hal.inria.fr/inria-00071991…

Here the idea is to parameterize the mesh & nicely sample according to varying density in the plane. I don't sense this idea is used too much these days, due to the need to parameterize.
Though on second thought, there are methods that use sophisticated field-aligned parameterization to get very high-quality almost-regular meshes with few irregular vertices, such as Nieser et al, "Hexagonal Global Parameterization of Arbitrary Surfaces": mi.fu-berlin.de/en/math/groups…
More recently a popular tool is the award-winning "Instant Meshes" by @wenzeljakob et al: igl.ethz.ch/projects/insta…

Combines the strengths of field-aligned & local iterative methods to get extreme performance/scalability, albeit with more irregular vertices than, say, Nieser et al
Finally, a fun example where meshes with unit edge lengths show up is in Isenberg et al, "Connectivity Shapes": cs.unc.edu/~isenburg/rese…

The idea: if you have perfect unit edge lengths, you can throw away vertex positions and recover the geometry *purely* from the connectivity!
In simulation, the award-winning "El Topo" package of Brochu & Bridson uses near-unit edge lengths to provide high-quality free surface tracking: github.com/tysonbrochu/el…

(By the way, "award winning" in these posts refers to the SGP software award @GeometryProcess!)
@BrunoLevy01 has a great package called GEOGRAM that provides a lot of great regular meshing functionality: alice.loria.fr/software/geogr…

Bruno will be able to give better details about the algorithms used, and interesting applications like optimal transport that I didn't mention yet.
It's also important to mention that uniform edge lengths provide more detail than necessary in large regions with low curvature.

Other methods adapt to local curvature/feature size, like Dunyach et al, "Adaptive Remeshing for Real-Time Mesh Deformation": hal.archives-ouvertes.fr/hal-01295339/d…
Likewise, the Holst & Chen paper mentioned earlier allows spatial adaptivity, by providing a density function: math.uci.edu/~chenlong/Pape…
There are sure to be countless references I'm missing both to algorithms & applications—but that should provide a few useful pointers!

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More from @keenanisalive

Keenan Crane Profile picture
Sep 6
“Everyone knows” what an autoencoder is… but there's an important complementary picture missing from most introductory material.

In short: we emphasize how autoencoders are implemented—but not always what they represent (and some of the implications of that representation).🧵 Image
A similar thing happens when (many) people learn linear algebra:

They confuse the representation (matrices) with the objects represented by those matrices (linear maps… or is it a quadratic form?) Image
With autoencoders, the first (and last) picture we see often looks like this one: a network architecture diagram, where inputs get “compressed”, then decoded.

If we're lucky, someone bothers to draw arrows that illustrate the main point: we want the output to look like the input!Image
Read 14 tweets
Keenan Crane Profile picture
Aug 29
I can't* fathom why the top picture, and not the bottom picture, is the standard diagram for an autoencoder.

The whole idea of an autoencoder is that you complete a round trip and seek cycle consistency—why lay out the network linearly? Image
*Of course I do in reality know why people use this diagram: it fits into a common visual language used for neural networks.

But it misses some critical features (like cycle consistency). And often adds other nutty stuff—like drawing functions as complete bipartite graphs! Image
Like, yeah, I know that for a function from ℝⁿ to ℝᵐ each output can depend on any of the inputs. That's how a function works!

Maybe you can use some of the space in that diagram to help me understand what those particular functions mean, or aim to do?
Read 6 tweets
Keenan Crane Profile picture
May 21
Fun new paper at #SIGGRAPH2025:

What if instead of two 6-sided dice, you could roll a single "funky-shaped" die that gives the same statistics (e.g, 7 is twice as likely as 4 or 10).

Or make fair dice in any shape—e.g., dragons rather than cubes?

That's exactly what we do! 1/n Image
Here's the paper, which is an industry-funded collaboration between my PhD student Hossein Baktash at @SCSatCMU, @nmwsharp at @nvidia, and Qingnan Zhou & @_AlecJacobson at @AdobeResearch.



2/n cs.cmu.edu/~kmcrane/Proje…Image
In a nutshell, we show that the resting poses & statistics of a rigid body are easily computed from its geometry, without any dynamical simulation.

This simple geometric model enables us differentiate through dice designs, & optimize their shapes to match target statistics. 3/n Image
Read 5 tweets
Keenan Crane Profile picture
Apr 17
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]
These pictures are often batting around in my brain when I think about optimization/learning problems, but can take some time to communicate to students, etc. So, I thought I'd make some visualizations. [2/n]
Suppose we minimize the squared length of a vector x, equal to the sum of squares of its components.

To avoid the trivial solution x=0, we'll also require that the components sum to a nonzero value.

Equivalently: minimize the ℓ₂ norm ‖x‖₂, subject to ‖x‖₁=1. [3/n]
Read 13 tweets
Keenan Crane Profile picture
Apr 7
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]
The Laplacian Δ is central to numerous (continuous) physical equations like the heat equation, the wave equation, and so on.

I have a whole video about it here: [2/n]
The discrete or graph Laplacian L is typically viewed as a numerical approximation of Δ, giving the difference between the value ui at a node of a graph, and a weighted average of uj at all neighbors j:

(Lu)_i := Σ_j w_ij (ui - uj)

Here w_ij are edge weights. [3/n] Image
Read 23 tweets
Keenan Crane Profile picture
Dec 9, 2024
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.
When I first heard about entropy, there was a lot of stuff about "free states" and "disorder." Or about the number of bits needed to communicate a message.

These are ultimately important connections—but it's not clear it's the best starting point for the formula itself.
A better starting point is the idea of "surprise."

In particular, suppose an event occurs with probability p. E.g., if the bus shows up on time about 15% of the time, p = 0.15.

How *surprising*, then, is an event with probability p? Let's call this quantity S(p).
Read 17 tweets

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