5 ms·

Quaternions in Signal and Image Processing

93 points by teleforce 2y ago

teleforce 2y ago
Abstract: Quaternions are still largely misunderstood and often considered an “exotic” signal representation without much practical utility despite the fact that they have been around the signal and image processing community for more than 30 years now. The main aim of this article is to counter this misconception and to demystify the use of quaternion algebra for solving problems in signal and image processing. To this end, we propose a comprehensive and objective overview of the key aspects of quaternion representations, models, and methods and illustrate our journey through the literature with flagship applications. We conclude this work by an outlook on the remaining challenges and open problems in quaternion signal and image processing.
- the__alchemist 2y ago
  I can't comment on the DSP applications in the article, but for representing rotations and orientations in 3D (Computer rendersing, robotics, aerospace etc), they can be effectively treated as black boxes. There are a handful of operations you can perform between quaternions and other quaternions, vectors, and scalers, to do anything you would want in this domain. Practially, you may just be calling functions like "Quaternion::from_unit_vecs()` etc. Call it a Rotor if you'd like; it doesn't matter when viewing it this way! Both implementation and application will be the same. The conceptual aspect people prefer about rotors is N/A here. You could also call it a `Orientation` or `Rotation` (eg the struct/class name); maybe that's better than either.
  - teleforce 2y ago
    Check this application of quaternion for robust UAV flight control: Quaternion vs Euler Angles for UAV position control: https://www.youtube.com/watch?v=0VAc_G79POE https://www.youtube.com/watch?v=0VAc_G79POE
frozenport 2y ago
“exotic” signal representation without much practical utility despite the fact that they have been around the signal and image processing community for more than 30 years now. Maybe they aren't that good? Maxwell's equations got a lot better when they dumped them, same thing with the few uses in video game physics/camera tracing.
- rowanG077 2y ago
  I didn't think quaternions are even exotic and have used the extensively to represent rotations. I'm not familiar with video games but it seems a pretty natural fit to the problem. What is the alternative that is so much better?
  - okaleniuk 2y ago
    Rotors. https://marctenbosch.com/quaternions/ https://marctenbosch.com/quaternions/
    - MyFirstSass 2y ago
      Thank you for this link. I just dabbled in webgl/threejs and tried creating a small movement engine and was confused from beginning to end. Quaternions as a black box is pretty accurate.
    - empiricus 2y ago
      From your link: "We can notice that 3D Rotors look a lot like Quaternions". "In fact the code/math is basically the same!"
      - dandanua 2y ago
        Yeah, why remove quaternions if the math is the same as with rotors? They also have their own value, without applications to geometry. The article is good, though. BTW, it's easy to understand the relations between i,j,k if you're familiar with Pauli matrices.
    - 2y ago
roger_ 2y ago
Preprint: http://w3.cran.univ-lorraine.fr/perso/sebastian.miron/documents/SPM_Quaternion23.pdf http://w3.cran.univ-lorraine.fr/perso/sebastian.miron/docume...
theanonymousone 2y ago
My friend worked on this for his Msc thesis (2010-11). I went to his defence presentation but still couldn't figure out what these "things" are :-| These and Monads will bite me forever.
- BenoitP 2y ago
  You should jump straight to geometric algebra. It's actually simple there. Quaternions and Complex Numbers are about just some not so special multivectors. Here are 2 45mn videos you'll not regret: https://youtu.be/htYh-Tq7ZBI?si=qi9HfU40qKcc_HNc https://youtu.be/htYh-Tq7ZBI?si=qi9HfU40qKcc_HNc https://youtu.be/60z_hpEAtD8?si=1M3dDAh7vPjx2Q8Y https://youtu.be/60z_hpEAtD8?si=1M3dDAh7vPjx2Q8Y
adelpozo 2y ago
For the visually inclined I enjoyed Visualizing Quaternions by Andrew Hanson. Haven’t yet read the article in the post but I am curious on the applications mentioned.
James_K 2y ago
Quaternions cannot be understood without first understanding geometric algebra, at which point the quaternions themselves cease to be of consequence.
- chpatrick 2y ago
  They're not that magical. Multiplication with complex numbers is a neat way to do 2D rotation and scaling. It's the same idea with multiple dimensions.
  - James_K 2y ago
    An idea which is encapsulated much better by geometric algebra, in which quaternions and complex numbers follow the same rules.
    - chpatrick 2y ago
      That's true but you definitely don't need to understand it first.
      - James_K 2y ago
        I think it would be self-damaging not to.
- catgary 2y ago
  I think this is pretty on par with telling someone they need to understand category theory if they’re playing around with Haskell/FP. I have lots of experience using quaternions in robotics/animation, but I never found geometric algebra particularly compelling - and I’ve given it several tries to sit down and read through various textbooks (and I have a PhD in Category theory/differential geometry, I don’t think it can be attributed to a skill issue with algebra or geometry).
  - James_K 2y ago
    Notice my use of the term "understand". If you are just using from some library quaternions as a black-box, you really needn't understand anything about them, but if you want understand them it's probably easier to go through geometric algebra first. Otherwise you're basically just learning geometric algebra with half the rules missing and it makes no sense.

93 points by teleforce 2y ago

Abstract: Quaternions are still largely misunderstood and often considered an “exotic” signal representation without much practical utility despite the fact that they have been around the signal and image processing community for more than 30 years now. The main aim of this article is to counter this misconception and to demystify the use of quaternion algebra for solving problems in signal and image processing. To this end, we propose a comprehensive and objective overview of the key aspects of quaternion representations, models, and methods and illustrate our journey through the literature with flagship applications. We conclude this work by an outlook on the remaining challenges and open problems in quaternion signal and image processing.

the__alchemist 2y ago
I can't comment on the DSP applications in the article, but for representing rotations and orientations in 3D (Computer rendersing, robotics, aerospace etc), they can be effectively treated as black boxes. There are a handful of operations you can perform between quaternions and other quaternions, vectors, and scalers, to do anything you would want in this domain. Practially, you may just be calling functions like "Quaternion::from_unit_vecs()` etc. Call it a Rotor if you'd like; it doesn't matter when viewing it this way! Both implementation and application will be the same. The conceptual aspect people prefer about rotors is N/A here. You could also call it a `Orientation` or `Rotation` (eg the struct/class name); maybe that's better than either.
- teleforce 2y ago
  Check this application of quaternion for robust UAV flight control: Quaternion vs Euler Angles for UAV position control: https://www.youtube.com/watch?v=0VAc_G79POE https://www.youtube.com/watch?v=0VAc_G79POE

frozenport 2y ago

“exotic” signal representation without much practical utility despite the fact that they have been around the signal and image processing community for more than 30 years now. Maybe they aren't that good? Maxwell's equations got a lot better when they dumped them, same thing with the few uses in video game physics/camera tracing.

rowanG077 2y ago
I didn't think quaternions are even exotic and have used the extensively to represent rotations. I'm not familiar with video games but it seems a pretty natural fit to the problem. What is the alternative that is so much better?
- okaleniuk 2y ago
  Rotors. https://marctenbosch.com/quaternions/ https://marctenbosch.com/quaternions/
  - MyFirstSass 2y ago
    Thank you for this link. I just dabbled in webgl/threejs and tried creating a small movement engine and was confused from beginning to end. Quaternions as a black box is pretty accurate.
  - empiricus 2y ago
    From your link: "We can notice that 3D Rotors look a lot like Quaternions". "In fact the code/math is basically the same!"
    - dandanua 2y ago
      Yeah, why remove quaternions if the math is the same as with rotors? They also have their own value, without applications to geometry. The article is good, though. BTW, it's easy to understand the relations between i,j,k if you're familiar with Pauli matrices.
  - 2y ago

roger_ 2y ago

Preprint: http://w3.cran.univ-lorraine.fr/perso/sebastian.miron/documents/SPM_Quaternion23.pdf http://w3.cran.univ-lorraine.fr/perso/sebastian.miron/docume...

theanonymousone 2y ago

My friend worked on this for his Msc thesis (2010-11). I went to his defence presentation but still couldn't figure out what these "things" are :-| These and Monads will bite me forever.

BenoitP 2y ago
You should jump straight to geometric algebra. It's actually simple there. Quaternions and Complex Numbers are about just some not so special multivectors. Here are 2 45mn videos you'll not regret: https://youtu.be/htYh-Tq7ZBI?si=qi9HfU40qKcc_HNc https://youtu.be/htYh-Tq7ZBI?si=qi9HfU40qKcc_HNc https://youtu.be/60z_hpEAtD8?si=1M3dDAh7vPjx2Q8Y https://youtu.be/60z_hpEAtD8?si=1M3dDAh7vPjx2Q8Y

adelpozo 2y ago

For the visually inclined I enjoyed Visualizing Quaternions by Andrew Hanson. Haven’t yet read the article in the post but I am curious on the applications mentioned.

James_K 2y ago

Quaternions cannot be understood without first understanding geometric algebra, at which point the quaternions themselves cease to be of consequence.

chpatrick 2y ago
They're not that magical. Multiplication with complex numbers is a neat way to do 2D rotation and scaling. It's the same idea with multiple dimensions.
- James_K 2y ago
  An idea which is encapsulated much better by geometric algebra, in which quaternions and complex numbers follow the same rules.
  - chpatrick 2y ago
    That's true but you definitely don't need to understand it first.
    - James_K 2y ago
      I think it would be self-damaging not to.
catgary 2y ago
I think this is pretty on par with telling someone they need to understand category theory if they’re playing around with Haskell/FP. I have lots of experience using quaternions in robotics/animation, but I never found geometric algebra particularly compelling - and I’ve given it several tries to sit down and read through various textbooks (and I have a PhD in Category theory/differential geometry, I don’t think it can be attributed to a skill issue with algebra or geometry).
- James_K 2y ago
  Notice my use of the term "understand". If you are just using from some library quaternions as a black-box, you really needn't understand anything about them, but if you want understand them it's probably easier to go through geometric algebra first. Otherwise you're basically just learning geometric algebra with half the rules missing and it makes no sense.