- The video discusses the use of geometric algebra in a project involving drones and computer vision.
- Geometric algebra is a mathematical framework that extends the concept of outer product and includes inner and outer products.
- It was developed by William Clifford in the 19th century but gained practical application in the 1960s by David Hestanus.
- Geometric algebra incorporates scalars, vectors, bivectors, and trivectors to represent points, lines, planes, and volumes.
- It simplifies space-time physics and relativity by providing a unified language for mathematics and physics.
- Geometric algebra has been proven and has practical applications, but it is not widely used, possibly due to historical reasons and the availability of other mathematical systems.

- The speaker discusses the limitations of vector calculus and linear algebra in dimensions other than 3D.
- The speaker's husband became obsessed with geometric algebra and its applications in cosmology and quantum physics.
- Geometric algebra is found to be useful in engineering, particularly in dealing with rotations.
- Quaternions are mentioned as a method for rotations, but geometric algebra provides a more general approach.
- The speaker applied geometric algebra to computer vision in a postdoc position.
- Computer vision at the time was focused on geometry and Bayesian statistics.
- The speaker received funding from the Royal Society to pursue applications of geometric algebra in engineering.
- Early projects involved translating classic projective geometry in computer vision into geometric algebra.

- Geometric algebra provides an origin and eliminates the need for coordinate systems, making it easier to work with rotations and translations.
- Geometric algebra allows for differentiation and calculus on geometric objects, making it useful for computer vision algorithms.
- Conformal geometric algebra extends to a five-dimensional space, allowing for the representation of points, lines, planes, circles, and spheres as objects.
- Geometric algebra simplifies the implementation of algorithms and intersections of objects.
- Geometric algebra can be used in different underlying geometries, such as hyperbolic or spherical geometry.
- Geometric algebra is applicable to physics and can simplify complex transformations.
- Modern computer vision is moving towards machine learning techniques.

- The use of machine learning is becoming more prevalent in various fields, but there are still geometric problems that need to be addressed.
- Geometric algebra can provide a way to solve these problems and enable new insights into conventional techniques.
- In day-to-day research, machine learning is applied to tasks such as medical time series data analysis and image segmentation, but geometric algebra is especially useful for tasks involving moving cameras and triangulation.
- Geometric algebra can be used to parameterize and learn geometric objects, as well as analyze motion in images.
- In the field of self-driving cars, conventional techniques like Bayesian methods are primarily used for tasks like segmentation and recognition.
- Geometric algebra serves as a unifying language and can simplify complex problems in various fields, including thin shell elasticity and electromagnetism.
- Currently, there is a lack of teaching and interest in geometric algebra, but students who learn it find it to be a valuable tool.
- Porting traditional equations to geometric algebra has not been widely explored, but there are tools available for performing numerical computations efficiently.

- The speaker discusses the difficulty of translating equations from Pauli spinors to geometric algebra.
- They mention a paper they wrote on geometric algebra that gained attention on Hacker News.
- The speaker mentions that geometric algebra has gained popularity and there are now groups interested in it.
- They discuss the range of comments on the paper, from positive to skeptical.
- There are no naysayers of geometric algebra, just people who question its practicality.
- Geometric algebra is seen as a tool that requires a learning curve but can provide advantages in thinking and problem-solving.
- It is predicted to have an impact in physics and engineering fields.
- The speaker also mentions their interest in promoting physical activity and mobility as they are a runner.
- Various books, articles, and code resources are available for those interested in learning more about geometric algebra.

The speaker is discussing a web version of a project that is being released. They mention that there is a lot of content available and provide the web address for people to access and test it out. They encourage users to provide feedback on any issues they encounter.

The video discusses the use of geometric algebra in a project involving drones and computer vision. Geometric algebra is a mathematical framework that extends the concept of outer product and includes inner and outer products. It simplifies space-time physics and relativity and has practical applications in engineering, particularly in dealing with rotations. Geometric algebra has gained popularity and is seen as a valuable tool, although it is not widely used. It can be used in various fields, including computer vision and physics, and can simplify complex problems. There are resources available for those interested in learning more about geometric algebra. The speaker also mentions the release of a web version of a project and encourages users to provide feedback.

Joan Lasenby - https://www-sigproc.eng.cam.ac.uk/Main/JL - is a University Reader in the Signal Processing and Communications Group of the Cambridge University Engineering Department, and is a College Lecturer and Director of Studies in Engineering at Trinity College. Here's a list of her published work - http://publications.eng.cam.ac.uk/view/people/jl.html

In this episode we talk about Joan’s research into 3D reconstruction from multiple cameras and her interest in geometric algebra.

The YC podcast is hosted by Craig Cannon - https://twitter.com/craigcannon

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Topics

00:00 - What's a tangible example of geometric algebra?

1:20 - What is geometric algebra?

6:15 - What resparked interest in geometric algebra?

7:10 - Why is it important?

11:00 - When did Joan start working on it?

12:55 - Rotations

16:45 - Computer vision in the early 90s

19:00 - Joan's fellowship at the Royal Society

23:00 - What's changed in computer vision since the 90s to allow for Joan's drone research?

29:35 - Machine learning in computer vision

31:20 - How Joan and her students are applying machine learning

34:30 - Unifying qualities of geometric algebra

40:30 - Joan's paper ending up on Hacker News

45:00 - Where could geometric algebra take hold?

47:00 - Running and mobility

48:00 - Where to learn more about geometric algebra

So Joan, as we walk through geometric algebra, I think the best place to start might be through a more tangible example.

So you're doing a project with drones here at Cambridge.

Yes, so we're doing a project with drones.

This is joint with the architecture department.

And what we'd like to do is use drones to look at the built environment generally.

And the built environment is made up of lots of lines.

So what we would like to do is to do a lot of our processing, our vision processing with lines.

And lines are much more difficult classically in computer vision than points.

A lot of reconstruction is done with points.

You get a point cloud, you can get structure from motion.

So motion capture, for instance, when you see someone in the suit with all the ping pong balls, they're connecting points.

But then even, not with motion capture, just with cameras that are moving, you can get points, you can match points.

Lines are difficult and our mathematical framework that we will use for this is geometric algebra.

So Grassman was a mathematician, and Grassman had something called an outer product.

So for example, I can take 2 vectors, I can put a wedge between them, a wedge product or an outer product, and I get A wedge B.

So this quantity is now a different thing, and I'll explain what that is.

So Grassman had this outer product, And Clifford, William Clifford, who was actually at Trinity before he moved to London.

So he came along and he extended this outer product.

He effectively had an inner product plus an outer product.

So for example, if I have 2 vectors, A dot B, where the dot is an inner product giving me a scalar, and A wedge B, which gives me this other thing, is a Clifford product.

And he had an algebra for this Clifford product, which is called Clifford algebra.

And it's been in the mathematics literature and research program forever, since Clifford died.

He died at the age of 34, I think, of TB.

So he did a lot, but clearly could have done much more.

So, but as a kind of applied tool, it wasn't really used.

And it was David Hestanus who came in the 1960s and said, gosh, look at this.

I'm gonna call it geometric algebra and I'm gonna do all these wonderful things with it.

So effectively, so that's the background.

scalars, so just numbers, vectors, so things with a magnitude and a direction.

So things that have 2 vectors, that makes a plane.

So a plane would have a position and a magnitude and a handedness.

A handedness is, so suppose I have 3 points that make up a plane.

I can sort of go from A to B to C, or from A to C to B.

So if I take A wedge B to form my plane in vectors, then B wedge A will give me minus the plane.

So it's, you have to then start to think of these planes as geometric objects which have a sign as well.

So if I take vectors A, wedge B, wedge C, it gives me a volume.

If I live in 4 dimensions, A, which B, which C, which D, it'll give me a 4 volume.

So at some point, I get to the highest element in space.

And that has a special place in my algebra.

So imagine I have in 3D, scalars plus vectors plus bivectors plus trivectors.

I have an algebra which takes these things as objects and I can add them, I can multiply them, I can differentiate with respect to them.

So it's a kind of abstract concept, but it is amazingly powerful.

So that's the kind of rough idea of this.

Right, how Clifford algebra moved to geometric algebra in the modern.

And this is all predating computers, So this isn't being rendered anywhere yet.

So we have a community, not a massive community, But there are lots of people who are very interested in potentially getting instructions for chipsets, etc.

of course, we have quite a lot of programs that people have been building so that the community can test these things out.

And so what rekindled it in the 60s to make it happen now?

So David Hestener was doing his PhD and he was a physicist and looking at space-time.

So basically his PhD turned into a book called Space-Time Algebra from which basically in the 80s, people got hold of and started to get interested in.

Even though David had been working on it throughout since the 60s.

What started David on the Clifford algebra?

But he very quickly realized that this algebra simplified a lot of space-time physics.

Right, so that's kind of what I wanted to get at.

What was the realization that made it like, why is this important now?

So, in effect, space-time physics, quantum physics, relativity is extremely complicated.

And it's the area where you do have to have a lot of background knowledge and a lot of background mathematical knowledge.

You do need to be proficient in, for example, for general relativity, you need differential geometry.

You need a lot of these mathematical systems.

So, I mean, David could see that with this algebra, he could work entirely within an algebra of geometric objects, transformations between these objects.

Transformations, linear transformations functions were geometrically intuitive.

There were mappings of objects to objects.

We didn't have to go to another space, a sort of dual space as you do in differential geometry.

So things became easy and he started to see that you could interpret a lot of things like Dirac matrices.

Well, actually, they're not matrices at all.

They were elements of the algebra and immediately, they became easy to deal with.

So his big motivation was that here was a unifying language for mathematics and physics, basically.

So if you know this language, you can not only do rigid body dynamics and engineering, classical mechanics.

You can do linear algebra without matrices and without tensors.

And you can do complex things in quantum space time physics with the same mathematical system.

And this is not theorized, this is proven at this point.

It's not like this idea that maybe this might work someday.

No, but Part of the problem is that we all, I mean, your next question might be, well, if it's so wonderful, well, why doesn't everyone use it?

So A, Clifford died when he was 34, and it was at the point where Gibbs, Heaviside came in and produced vector algebra and the cross product.

And so have you ever thought about cross product?

So I have 2 vectors and I take A cross B.

It gives me the vector, which is perpendicular to the plane, effectively.

Now, that's all very nice, but it only works in 3D.

Because in a plane, I've got no perpendicular.

In 4 dimensions, there's no concept of a perpendicular to a plane.

So, but of course, we all grow up with vector calculus, linear algebra, and matrices, and then we have our research areas, and it's very difficult to actually, to listen to somebody if they come along and say, I've got this super duper new mathematical system that you ought to take notice of.

It would say, if someone comes up to me and says, look at this, I will say, well, you know, I've got my own research and it takes me all my time to do what I'm doing.

Okay, and so what planted the seed in you that you wanted to follow this path?

So the truth is, My husband met David Heston, Anthony Lazenby.

He became fairly obsessed with geometric algebra.

He could see that There was this thing which told him what the Cauchy-Riemann equations are, for people who know it, that the Pauli matrices and the Dirac matrices in space-time quantum physics were just really interpretable.

He became obsessed with it, totally obsessed.

So Anthony had a PhD student called Chris Doran.

Chris and Anthony, They've written a book on geometric algebra physicists.

And it was very difficult for me to actually talk to my husband at that point without actually finding out something about it.

Because he thought it was 1 of the most exciting things he'd ever seen.

And you were at the time just pursuing math, a PhD?

So I did have some time actually, because I was, you know, not that much time, of course, when you've got a small child.

But yeah, I was, I'd gone back as an engineer, I was a postdoc in engineering, I was doing imaging, I was imaging flames.

But I actually then began to realize that this algebra would be really useful in parts of engineering, particularly things which involve rotations.

And maybe I should talk about rotations because in engineering and physics, rotation, the way geometric algebra deals with rotations is totally key.

So quaternions are, well, if we go back, people rotate with rotation matrices.

So you have, say in 3 dimensions, you have a 3 by 3 matrix, you act on a vector and it rotates it.

Now, a rotation has just 3 degrees of freedom, which have 9 components in a 3 by 3 matrix.

So rotation matrices are not numerically nice to deal with because you have to keep them on the manifold.

You have to make sure if you change, if you update rotation matrices, you have to make sure it obeys the constraints.

Now in graphics, et cetera, and satellite motion, it's long been known that the best things to use is quaternions.

Other methods are Euler angles, like a rotation about the x-axis, the y-axis, the z-axis.

But quaternions have been particularly nice because they are minimally parameterized.

They are smooth, they don't suffer from singularity problems.

And Hamilton created quaternions as an extension of complex numbers.

So complex numbers, Everybody, people know that complex number I, I effectively rotates in the plane and multiplication by I.

So Hamilton spent many, many years in his later life trying to extend complex numbers to 3 dimensions.

So quaternions have these elements, IJK, which all square to minus 1.

But everybody knew they were great libraries.

People have been using quaternions since the early days of satellites.

So if you actually look at code, et cetera, people will use this for rotations.

Now, very early on, it's in David's book that you see that if I square 1 of these bivectors in 3D, let's stick in 3D for now.

If I square it, if it's a unit bivector, I get minus 1.

So I have a real object that squares to minus 1.

Which is kind of telling me that I, the unit imaginary, and this jk are probably unit planes.

It turns out that the quaternions are just rotations.

The ijk give you rotations about the unit planes in 3 dimensions, the X, Y, the Y, Z, and the X, Z plane.

So immediately you see that you can have complete generalizations of things that do rotations in any dimension.

And so at that point you're doing this post-doc position and you realize that you could apply it.

Not to flames but to mainly computer vision.

And what was the stage, like the state of computer vision at this time?

Yeah, but a lot of the, you know, it was it was really starting to move forward.

There was no machine learning in computer vision, but it was all geometry,

So people had used projectable, you know, people had been using the ideas of projective geometry in computer vision for a long time, which is a 4 dimensional space.

Still with matrices, so I rotate and translate things.

So I have lots of cameras, I want to find from my images, the rotations and translations between my cameras.

Once I've done that, I can triangulate and I can do 3D reconstruction.

There was also at that stage, a lot of Bayesian statistics were coming into computer vision, so tracking things in images and finding the most probable tracks

So Computer vision was really starting to take off.

Okay, and so you're, you finished up your 2 year, you're creating these flames, these graphics, essentially.

What do you jump into to actually give it a go?

So I was extremely lucky, because by that point I had 2 small children.

I don't know if you heard the Royal Society.

In, well, the Royal Society, the body in London.

Yeah, so you probably have, because it has journals and lots of historical people with big names in history were fellows of the Royal Society.

So the Royal Society has something called University Research Fellows.

And I applied to do, it was probably quite a step for them because I applied to do applications of geometric algebra in engineering.

So I then had a 5 year effectively postdoc and I could choose what teaching I did And the engineering department were very good.

I didn't need to do a lot of teaching or admin, but I basically had kind of 5 years to try and get this off the ground.

And What was an example of an early project?

So an early project was actually with, I think at the time, the internet wasn't really like it, quite like it was today.

But there was some mailing lists and there was somebody who called Eduardo Bayro Correchano, who's now in Mexico, but he put this email out on some list saying, anybody working with computer vision and geometric algebra.

And those early days, we did quite a lot of translating all the classic projective geometry computer vision,

was quite mathematical at the time, into geometric algebra.

And I mean, this may sound kind of basic, but 1 of the really nice things about putting your problem, et cetera, into geometric algebra is you have an origin.

So some inertial frame, everything is with respect to that.

don't have matrices, so I don't have to worry about coordinate systems.

I don't stack up coordinate system upon coordinate system and then worry about what on earth that translation vector, what coordinate system is that in.

And with vision, where you're measuring things in an image, so what coordinate frame is that in?

It is quite confusing, and I know this for a fact because students are very confused about it all.

When you've got a rotation, where is it with respect to in these complicated systems, et cetera?

And was the goal of this 5 year period to then apply it to some product use or just do the basic research and see how it goes?

At that time, it was just really seeing what we could get out of

So I did some work with a company called PhaseSpace, who's their motion capture company in the States, not far from Berkeley.

And, you know, looking at algorithms to calibrate cameras.

Because 1 of the things I haven't mentioned is that there is another aspect to this system, which is extremely useful.

So I've said that I have these geometric objects, my rotations are objects, so I can write down coordinate-free expressions.

But not only can I write them down, I can differentiate with respect to them easily?

So because I have this algebra of objects, I can do calculus on them.

And that's quite hard to do conventionally because you've got people can do it, but you're differentiating with respect to a matrix or a tensor or a vector, all this.

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Joan Lasenby on Applications of Geometric Algebra in Engineering