#94 - Joan Lasenby

**Craig Cannon** (0:00)
Hey, how's it going? This is Craig Cannon, and you're listening to Y Combinator's podcast.
Today's episode is with Joan Lasenby. Joan is a university reader in the Signal Processing and Communications Group of the Cambridge University Engineering Department. She's also a college lecturer and director of studies and engineering at Trinity College. In this episode, we talk about Joan's research into 3D reconstruction for multiple cameras and her interest in geometric algebra. If you'd like to learn more about Joan's work, I'll link it up in the description. All right, here we go.
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. Can you explain that first?

**Joan Lasenby** (0:44)
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 or 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.

**Craig Cannon** (1:20)
So motion capture, for instance, when you see someone in the suit with all the ping pong balls, they're connecting points.

**Joan Lasenby** (1:26)
They are all points, exactly. But then even not with motion capture, just with cameras that are moving, you can get points. You can match points.
So we would like to do this with lines. Lines are difficult and our mathematical framework that we will use for this is geometric algebra.

**Craig Cannon** (1:47)
So let's define it. Let's start there.

**Joan Lasenby** (1:50)
Me, shall I define it? Okay, and we'll see if it ties in with what you've read about it. So do you want some history?
So Grassman was a mathematician and Grassman had something called an outer product. So for example, I can take two 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 two vectors, A dot B, where the dot is an inner product, giving me a scalar, and A, which B, which gives me this other thing, is a Clifford product. So seems like a strange thing to do. 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, which was in the 1870s. He died at the age of 34, I think, of TB. So he did a lot, but clearly could have done much more.
But as a kind of applied tool, it wasn't really used. And it was David Hestonus who came in the 1960s and said, gosh, look at this, Clifford called it geometric algebra. I'm going to call it geometric algebra, and I'm going to do all these wonderful things with it. So effectively, so that's the background.
The basis is, so imagine I have scalars, so just numbers, vectors, so things with a magnitude and a direction.
In 3D, we can think of this. Five vectors, they're planes, right? So things that have two vectors, that makes a plane. So a plane would have a position and a magnitude. And a handedness.

**Craig Cannon** (4:25)
Yeah, what is that?

**Joan Lasenby** (4:26)
A handedness is, so suppose I have three points that make up a plane. I can sort of go from A to B to C or from A to C to B.

**Craig Cannon** (4:37)
Got it, okay.

**Joan Lasenby** (4:38)
So if I take A wedge B to form my plane in vectors, then B wedge A will give me minus the plane. So 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. Again, it will be an oriented volume. If I live in four dimensions, A wedge B wedge C wedge D will give me a four volume.
So at some point, I get to the highest element in the space. In 3D, that will be a volume. I can't go any bigger, and that has a special place in my algebra.