Wiggly lines help catching crime

by Paul Curzon, Queen Mary University of London

Computer scientists rely on maths a lot. As mathematicians devise new mathematical theories and tools, computer scientists turn them into useful programs. Mathematicians who are interested in computing and how to make practical use of their maths are incredibly valuable. Ingrid Daubechies is like that. Her work has transformed the way we store images and much besides. She works on the maths behind digital signal processing - how best to manipulate things like music and images in computers. It boils down to wiggly lines.

Pixel pictures

The digital age is founded on the idea that you can represent signals: whether sound or images, radio waves, or electrical signals, as sequences of numbers. We digitise things by breaking them into lots of small pieces, then represent each piece with a number. As I look out my window, I see a bare winter tree, with a robin singing. If I take a picture with a digital camera, the camera divides the scene into small squares (or pixels) and records the colour for each square as a number. The real world I'm looking at isn't broken into squares, of course. Reality is continuous and the switch to numbers means some of the detail of the real thing is lost. The more pieces you break it into the more detail you record, but when you blow up a digital image too much, eventually it goes blurry. Reality isn't fuzzy like that. Zoom in on the real thing and you see ever more detail. The advantage of going digital is that, as numbers, the images can be much more quickly and easily stored, transmitted and manipulated by Photoshop-like programs. Digital signal processing is all about how you store and manipulate real-world things, those signals, with numbers.

Curvy components

There are different ways to split signals up when digitising them. One of the bedrocks of digital signal processing is called Fourier Analysis. It's based on the idea that any signal can be built out of a set of basic building blocks added together. It's a bit like the way you can mix any colour of paint from the three primary colours: red, blue and yellow. By mixing them in the right proportions you can get any colour. That means you can record colours by just remembering the amounts of each component. For signals, the building blocks are the pure frequencies in the signal. The line showing a heartbeat as seen on a hospital monitor, say, or a piece of music in a sound editing program, can be broken down into a set of smooth curves that go up and down with a given frequency, and which when added together give you the original line - the original signal. Their negative parts of one wave can cancel out positive parts of another just as two ripples meeting on a pond combine to give a different pattern to the originals.

This means you can store signals by recording the collection and strength of frequencies needed to build them. For images the frequencies might be about how rapidly the colours change across the image. An image of say a hazy sunset, where the colours are all similar and change gradually, will then be made of low frequencies with rolling wave components. An image with lots of abrupt changes will need lots of high frequency, more spiky, waves to represent all those sudden changes.

Blurry bits

Now suppose you have taken a picture and it is all a bit blurry. In the set of frequencies that blurriness will be represented by the long rolling waves across the image: the low frequencies. By filtering out those low frequencies, making them less important and making the high frequency building blocks stronger, we can sharpen the image up.

By filtering in different ways we can have different effects on the image. Some of the most important help compress images. If a digital camera divides the image into fewer pixels it saves memory by storing less data, but you end up with blocky looking pictures. If you instead throw away information by losing some of the frequencies of a Fourier version, the change may be barely noticeable. In fact, drawing on our understanding of how our brains process the world to choose what frequencies to drop we might not see a change in the image at all.

The power of Fourier Analysis is that it allows you to manipulate the whole image in a consistent way, editing a signal by editing its frequency building blocks. However, that power is also a disadvantage. Sometimes you want to have effects that are more local - doing something that's more like keyhole surgery on a signal than butchering the whole thing.

Wiggly wavelets

That is where wavelets come in. They give a way of focussing on small areas of the signal. The building blocks used with wavelets are not the smooth, forever undulating curves of Fourier analysis, but specially designed functions, ie wiggly lines, that undulate just in a small area - a bit like a single heart beat signal. A 'mother' wavelet is combined with variations of it (child wavelets) to make the full set of building blocks: a wavelet family.

Wavelets were perhaps more a curiosity than of practical use to computer scientists, until Ingrid Daubechies came up with compact wavelets that needed only a fixed time to process. The result was a versatile and very practical tool that others have been able to use in all sorts of ways. For example, they give a way to compress images without losing information that matters. This has made a big difference with the FBI's fingerprint archive, for example. A family of wavelets allows each fingerprint to be represented by just a few wavelets, so a few numbers, rather than the many numbers needed if pixels were stored. The size of the collection takes up 20 times less storage space as wavelets without corrupting the images. That also means it can be sent to others who need it more easily. It matters when each fingerprint would otherwise involve storing or sending 10 Megabytes of data.

People have come up with many more practical uses of Wavelets, from cleaning up old music to classifying stars and detecting earthquakes. Not bad for a wiggly line.