Shadow and light
Where we discover how Chinese shadow plays and an obscure mathematical result led to enlightenment in medical imaging.
I was still a kid when I saw a Chinese shadow play for the first time. And already back then I wondered how a pair of hands could form the image of a rabbit. But what interested me even more was, if I saw a shadow, or maybe a few different shadows, could I deduce the position of the hands from them?
This is an example of something that we call inverse problems in science, and that still interests me today. Namely, can you deduce the object when you cannot observe the object of interest directly (in this case the hands), but only an indirect image of the object (the shadow)? In general, this seems quite impossible, but if I know something about the original object (that it is a hand, for example), maybe I can then deduce its constellation?
The described setup actually applies to much more important problems nowadays, such as in tomography images in medicine. Furthermore, these are not just black and white images, because if one shines an X-ray beam through a human body, the beam is attenuated differently by the individual body parts. In other words, we get an X-ray image with fuzzy and weak elements inside due to the type and amount of “stuff” that the beam traverses (for example, bones provide more attenuation than soft tissue). Also, the result depends on the direction from which the beam travels through the body. If you shine an X-ray from another direction, you get a different image created by a beam that traverses the body. But if you collected many such images from many directions, would you then get the information necessary to reconstruct the attenuation inside the body, that is the skeleton and other body parts?
The answer is yes, and how to do that was discovered by the Austrian mathematician Johann Radon long before the invention of tomography. In 1917 he proved that if you had projections from all directions, you could recover the attenuation. Interestingly, Radon himself did not expect that his discovery would be highly useful one day (as with many fundamental discoveries in mathematics). In fact, without this mathematical formula called the Radon transform, the creation of tomography images would be impossible today. Furthermore, thanks to Moore’s Law (see my previous column on that topic), what used to require a huge computer is now done by a regular computer on a daily basis, making tomography an affordable technology available in most hospitals around the world.
But as mentioned in the beginning, tomography is just one example of inverse problems. In my opinion, inverse problems represent some of the most interesting mathematical questions in science. And their putative application is very wide, too. You can find them in technologies around acoustics, radars, astronomy and, of course, behind many internet technologies. I was thus not completely wrong when as a child I felt that there was something deeply magic and profound behind these Chinese shadow plays…
