Using Mathematics To Detect Skin Cancer

Image Analysis is the science of processing digitally stored images, often with a view to stressing important features or enhancing overall quality of the picture. Related methods of image analysis can be used to ...

• improve the quality of television pictures -- this was one of its earliest applications;
• focus your camera -- most auto-focus devices in single-lens reflex cameras operate by taking the first two derivatives of the picture in the viewfinder;
• enhance photographs taken by satellite;
• identify and analyse skin lesions

... identifying just a small number, from many such applications.

I'm going to say something about the last of these, which is part of the extensive involvement of CSIRO's Division of Mathematics and Statistics in `Image Analysis'.

The first stage in analysing an image of a skin lesion is to determine its boundary. Then, expert knowledge built into computer software identifies special features, like particular colours, colour variation and texture. All the methods for these analyses are based on mathematical theory. For example, the boundary detection algorithm involves taking a mathematical transformation of the data, known as the ``top-hat transformation'' because parts of the simplified signal that it produces are shaped rather like a top hat. To detect boundaries, a sequence of mathematical questions is `asked' of the simplified signal, at each point where it exhibits high amplitude:

• In which direction is it changing fastest?
• How sharp is the curve along which the signal is changing most rapidly?
• How strong is the signal along that curve?

...and others of a similar nature. The answers to these questions are found by doing some high-speed computing. These are then combined using mathematical theory. This information is used to trace the boundary.