The Deceptively Asymmetric Unit Sphere(tangramvision.com) |
The Deceptively Asymmetric Unit Sphere(tangramvision.com) |
[1]: https://press.princeton.edu/books/paperback/9780691203706/vi...
https://www.youtube.com/watch?v=mKtctCyd0rs&list=PLWEiAJhCw-...
Math is like skiing or playing the guitar, you don't get better by watching others do it.
Personally I find videos useful to augment books, but rarely a substitute for them. But I am bad about pausing, ruminating and practicing, you may be more successful than I.
But practice is required IMHO.
Find an efficient (class of) algorithm(s) to select a large number N of uniformly distributed points on S^2, where "uniformly distributed" is given in a more flexible sense than the usual one. For instance, you may want to minimize the Weyl discrepancy between average and integral, or you may want to focus more on minimizing the number of ε-clusters of distances.
One of the most elegant approaches to this problem is the classic work of Lubotzky, Phillips and Sarnak: Hecke Operators and Distributing Points on the Sphere I and II. They translate the problem to one of generating good sequences of elements of SO(3), which they attack with a combination of harmonic analysis on the semisimple groups, homogeneous dynamics and number theory with Hecke operators as their central tool.
“In precise mathematical terms, this set of rays is called the oriented real projective plane and is commonly denoted by T^2. If you’ve seen this terminology before, you’ll notice that this is a torus. This is because in real-projective geometry, we also add the points and lines “at infinity”.”
The oriented real projective plane is a sphere, not a torus.
The projective points at infinity (one point for every 1-D angle (R mod 2pi)) form the equator of the sphere.
The T in T² is for "two-sided" , not Torus.
The torus explanation that Tangram gives doesn't make sense. In a pinhole projection, the horizontal and vertical infinites do not "wrap around" to meet. There is no meaningful "horizontal" and "vertical", the system is rotationally symmetric, which forms a hemisphere of curve it to make it compact. (Half sphere because you can only see one half of the space outside a pinhole camera)
https://en.m.wikipedia.org/wiki/Oriented_projective_geometry
Quaternions and SLERP are absolutely a fundamental part of 3D vision (and game development too). However, I wanted to focus this post mainly on the question "why is optimizing on the unit sphere difficult?" As the post stands, it's already quite verbose.
Maybe I'll find some time to do a deep dive on common Lie Groups used in computer vision e.g. SO(3), SE(3) and Sim(3) and also the common representations used for those groups.
+1
Also, great article!
I wish someone had told me about these computer vision applications of Lie groups when I was writing my dissertation on Lie groups, but better late than never!
> A manifold is a space that is locally Euclidean, but globally might be complicated, e.g. a torus or sphere, or etc.
Okay, so that's reasonably simple. As an intuition, if we imagine a 2D character living on the surface of the sphere, if they walk forward, from their perspective they just move forward in 2D, but from our outside perspective, they're moving in 3D on the curved surface of a sphere.
Once I have this, I try and read until I get lost again. I don't try to rigorously solve or follow through with each equation, but to rather approximately understand what the idea is.
For example in this article, I got completely stuck on how the Exp and Log functions he's talking about relate to the usual definitions of those functions, so now I'm going down that rabbit hole.
That being said, I began studying Differential Geometry and Lie Groups as part of my graduate degree in Electrical Engineering. Engineers think about problems very differently than mathematicians and I've benefited a lot from taking a more geometric-based and visual approach to learning in the years following my undergraduate.
So, my prescription would be to play around with math ideas when you see them. Create a script to draw what you are trying to visualize. This was my first time using the `manim` library and I gained a deeper appreciation and intuition for the ideas presented in the article even though I've studied them dozens of times!
Overall, learning math is a slow and deliberate exercise. Don't get down on yourself if you don't understand something at first glance. Feel free to pause, verify an idea (either visually or with a formal proof) and then continue on a more firm base of understanding.
For differential geometry the prerequisites are linear algebra and multi-variable calculus, and the prerequisite for multi-variable calculus is single variable calculus, and the prerequisites for each of those is basic algebra, trigonometry, and elementary geometry.
You don't need to know everything about each of these to get things at the next level, but a thorough grounding in the basics is essential in my experience. There's a reason every STEM field begins with calculus and linear algebra - they're used everywhere in anything at a higher level. Once you get through those you will find things open up for you.
I don't know your level so it's difficult to make any concrete recommendations, but in general I find Lang's books to be clear and efficient sources. His Short Calculus covers all the basics of single variable calculus in less than 200 pages, instead of ~500 pages like many intro to calc books. Similarly his Calculus of Several Variables is ~300 pages instead of 500-700. Alternatively a mathematical methods for physicists book like the one by Riley, Hobson, and Bence might suit you. It's huge (~1300 pages), but you can pick out the chapters you want to learn from and it builds up from the basics to some quite sophisticated mathematics and has references if you want more depth on some topic, and great problems.
I find I don't really learn anything from watching videos. They can be complementary, but most of the learning with maths comes from doing problems after reading through an introduction to the concepts
I tend to try and take on the whole thing and not really understand it then repeat the process (often from different sources) after a while I just seem to understand more and more
>> For convenience of notation, we’ll drop the explicit parametrization of the curves and denote this vector shift as [...]
FYI for clarity I wish the explicit parametrization was kept, even though it is more verbose.