Professor in Mathematics, Duke University, Former President of the International Mathematical Union (IMU), USA.
Breaking the Wall of Computer Stupidity. How Wavelet Analysis Improves Geophysics, Biology and Art History
Ok, computer stupidity: are computers stupid? Well, maybe. These are things that have happened to me. One thing: you know that regularly you have to log in places. So, you type your name, your password. You have to take something that you will remember, but it has all these rules to obey; so I found this wonderful password. Then it tells you: this is not the password for your records, try again. Ok, you try again, and then it says: a temporary password has been mailed to your account, because you were no good. Ok, you go to your e-mail account, you get it; it says: wrmsodmm. Ok, you try it, you type it, and so on, and then it says: you must change your password NOW. Your new password must be at least 8 characters, have both lower and upper case letters, at least one number, at least one non-alphanumeric, and cannot contain a word in the dictionary. Fine. Ok, well the previous one had all that, and it didn’t like it; it didn’t know it. Ok, lets try again. You confirm, and it says: this password is unacceptable. That was your previous password. This happened to me – not for this conference – but then you say: “Oh my God.” This didn’t happen, but it made me think of it; you remember Clippy? It has been a long day, and I have provided some light relief.
But computers have also really enabled many things: nice gadgets. We all know them, but not only that; these gadgets typically are now made on the other side of the world. The shipping to get them to us, and that is quite cheap these days, is something that needs a lot of computing, because they arrive in these big containers that then have to be dealt with, and so on. This is what a container port looks like. You see how many there are? To keep track of all that without very powerful computer programs, of which actually Berlin is one of the world experts, would not be possible.
Airfoil design. So, this is how the Wright brothers did it. Earlier this year I was at the Ben Franklin Museum, and they actually showed me the things that the Wright brothers did – I actually had no idea that they had done experiments. They had these tiny little airfoils of different shapes on which they did little experiments with a fan and so on and measuring how much it pulled or not. That is how they decided on the shape of the airfoil. Now wind-tunnel experiments these days are much, much larger; but these winglets, these very particular shapes of wings where the tips go up, these were not designed with wind-tunnel experiments. I am not sure that with just wind-tunnel experiments we ever would have gotten there. They were designed by a scientific computation.
Ok, lots of computing – also very nifty mathematics in all of that. I would like to show you a couple of case studies in which I am involved these days, which involve using computers and actually bringing it to scientists, in some cases who are quite used to computers; but as we go on to people who are less and less connected to the computer but for whom we make things possible.
Global tomography. You can find on the web a record of all the large earthquakes of – in this case – the last four or five years, where the colour gives you the age, with respect to the day on which you look it up, and the dimension of the circle – the magnitude of the earthquake, and of course the centre is where the earthquake took place. All these earthquakes are measured by many places, many seismic stations around the world, and you can look up all those seismograms. In fact, with these very large earthquakes, you can see that they can originate far away, and you can feel them on the other side of the Earth. Because the Earth is layered with many, more or less, spherically symmetric layers, these rays are actually bent through the Earth. So, you get information on places quite deep in the mantle by looking at these seismograms. Typically what has happened is that we can compute very well when the time of arrival would be, if the Earth was completely spherically symmetric, but it is not quite. Because it is not quite, the time of arrival is a little changed, and from that information we can actually compute properties in the mantle that are very localised: hot plumes, bigger hot up-wellings, like under southern Africa.
How do you analyse? You analyse typically when you talk about functions on the sphere; you use spherical harmonics. Mathematicians, physicists and geophysicists, know them very well, and love them a lot. They are very good for things that live on the whole sphere and have a certain pattern. So, a function that looked like that – where blue is negative and reddish is positive – on the Earth, on an Earth map and Mercator projection would be very easy to express in spherical harmonics. But we are not talking about things that are very localised. So, what we are doing is we are building special wavelet type things that make that computation, that are much more tailored to very localised functions, and that help us analyse and do global tomography like this. It is on-going – to be continued.
Another case study: functional magnetic resonance imaging (fMRI). We use similar localisation techniques to improve our reading of functional magnetic resonance imaging that are used by neuroscientists to understand brain function. You have all seen this in scientific papers where they show you a brain that lights up in some spaces when people were asked to perform a certain task.
First of all, brains. These are two human brains, and all brains are folded. If you took the surface of a human cortex, you would get an area that is about half a metre by half a metre – so a quarter metre squared. All that is folded up into just that one little skull. The folds are different for everybody. Look, I followed here on these two brains one particular folding, and you see that it is very different. Typically what is done if people do experiments for certain tasks and seeing how people register them, is they take the data and they already kind of – imagine a balloon all folded like that and they blow it up – inflate it in order to get all these wrinkles out, because the wrinkles are different for everybody.
But, of course, by doing that, you distort the data. So we have developed – and this is also on-going – special wavelets that use a view of the cortex for each individual subject and build wavelets that manage to localise on there in order to get to a much more accurate reading of the fMRI experiments. Here are, in fact, on the left you have what would be done with the conventional analysis, and on the right, the same data, but because we have these much better localised data, we are able to capture much more precisely what the connection is – what the activation is. Again, it is: to be continued.
Next what we will do is we look at the function for different subjects in order to do registration, rather than using the geometry, which is not perfect since brains do look so different for different people; in order to understand them better – to be continued.
Anatomical surfaces. So, many surfaces you have seen in applications; you can get something scanned. You can even get your body scanned and then send it to a catalogue so that they will get you clothes exactly to your measure. That is not really science. So, objects can be scanned; they can be scanned in different ways. What you typically acquire is a cloud of points. From that cloud of points we can go to a mesh, a triangulated mesh for the surface. Fine. So, now we have a table of numbers representing the surface. But if you do that two times with the same thing, you get two different lists of numbers. How do you recognise it is the same surface? Or, if it is not the same surface, but they are very similar, how do you recognise that they are so similar? Or, if they are two things that are inspired by the same thing? Here you have a real beetle and a dinky toy beetle. A dinky toy beetle actually is different in proportion from the real beetle, because it didn’t look right. It is shorter. It is more squashed. Otherwise, a real beetle is really longer than the dinky toy proportions.
But how would you quantify that – that kind of difference from these point clouds? So, we develop mathematics that uses both local and global properties in order to capture that similarity, to capture distance between surfaces. Let me give you an example of where we apply this to biology – people who work on the phenotypes, not the genotype, but the phenotypes of animals. In this case we are looking at different bones in mammals. We have molars, a piece of radius, and we have a metatarsal. I think the radius and the metatarsal have been switched actually. So, they study these, and they learn very carefully how they develop. They learn how to put landmarks on them, which are visualised here with these little coloured dots. Then they look at these landmarks and how the landmarks change from one sample to another. It takes a lot of training to do this, and then it takes a lot of drudgery in order to really put those points on and then make that comparison.
The problem is that it is really not only the local landmarks, but the whole global organisation of them that guides the biologist in putting in those landmarks. So, you need a lot of training to do this, and it is not very reproducible. So, “Janet, don’t trust this”, they say. The phenotype people are very frustrated by this.
What we have done is we have developed methods that make it possible for us to compare when we have two samples. The whole global geometry that make it possible for us to define mappings from one to another, which we can then, if we want to test it, use it, and this shows how given such a mapping we would take the landmarks put on by the biologist on one, transport them to the other, and you see we end up with something very similar to what they do. Of course, it is not our goal to transport landmarks. We want to get to a method where the landmarks wouldn’t even be needed. This is a paper that was just published, and some biologists are very excited about it – to be continued.
Finally, here, I am moving the furthest away from people who are used to working with computers and programming computers: art historians. This is a case study in which we were given a lot of paintings by Goosen van der Weyden, who was grandson of the much more famous Rogier van der Weyden, who is an early 16th century painter and who had a large workplace in which he had an usual number of apprentices. As was usual in that time, there were underdrawings under the paintings. He used different styles of underdrawings, and people thought: “well, maybe this was because of an evolution through his career.” Then they found that paintings in the same period might have those different styles. So, let me show you underdrawings.
Here are four samples. On the left, Class 1, you see little parallel lines indicating shading, like woodcut. You see them mostly where they were indicating something that wasn’t present, like shading – in exactly that form in the painting – or when the painter changed his opinion in making it – so when a mistake is corrected. So, on the second, Class 2, you see that nose and the mouth were placed in different places; you see the underdrawing; so you also see through these little parallel lines that indicate volume. In Class 3, you see that there are a whole lot of lines that indicate a sketch of what the face should be. Class 4 is not even a sketch. I would say that this is just a very course indication of: there is nose here and some eyes there. So, these are the different classes of things that we were looking at.
The question was: if I give you the painted surface, can you distinguish from the painted surface what the style of underdrawing was? The idea being that maybe the style of underdrawing was telling you whether it was the master himself that was going to finish it or one of the apprentices. So, we were given blind data sets, and indeed, we learned – and I am sweeping a whole lot of mathematics under the rug here – to distinguish them, and that we found that out of the seven images in our blind data set that were by Goosen van Weyden, we got it right for six. The seventh we were completely wrong with – completely. We said: “how is this possible?”
We looked at it, and we said it was Class 3, and they had told us Class 4. Then we looked at it and said: “but look, this is not Class 3; this is not parallel lines; this is a sketch.” We had it right. They had mislabelled it. So, we are very enthusiastic, and this is: to be continued.
Here I am stopping. Thank you.