Radon Transform

In Chapter 12 of the 4th Edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce the Radon transformation. It consists of finding the projections F(θ, x') at different angles θ from a function f(x,y). But why is it called the “Radon” transformation, and does it have anything to do with the radioactive gas radon discussed in Chapter 16?

Well, it has nothing to do with the element radon. Instead, and predictably, the term honors Johann Radon, the Austrian mathematician who investigated this transformation. In “A Tribute to Johann Radon” in the IEEE Transactions on Medical Imaging (Volume 5, Page 169, 1986, reproduced long after his death to honor his memory) Hans Hornich wrote

With the death in Vienna on 25 May 1956 of Dr. Johann Radon, Professor of the University of Vienna, not only the mathematical world and Austrian science but also the German Mathematical Union has suffered a severe loss, as have also many other scientific bodies of which the deceased was a prominent member, and who spent most of his teaching life in German universities.

Radon was born in the small town of Tetschen in Bohemia near the border of Saxony on December 16, 1887. He studied at Vienna University where, alongside Mertens and Wirtinger, Escherisch above all was the great influence on Radon’s development: Escherisch had, as one of the first in Austria, imparted to his students the world of ideas of Weierstrass and his rigorous foundations of analysis. Through Escherich, Radon was led next to variational calculus….

A few years later appeared his “Habilitationsschrift” “Theory and application of absolute additive weighting functions” (S. Ber. math. naturw., Kl. K. Akad. Wiss. Wien II Abt., vol. 122, pp. 1295–1438, 1913), which played a leading role in the development of analysis; the Radon integral and the Radon theorem laid the foundations of functional analysis. As an application Radon somewhat later treated the first and second boundary value problem of the logarithmic potential in a very general way.

The Radon transformation has important applications in medical imaging, and plays a crucial role in computed tomography, positron emission tomography, and single photon emission tomography. I found a nice layman’s description of the Radon transform in an essay at the website http://www.ams.org/samplings/feature-column/fcarc-tomography, written by Bill Casselman.

The original example of this sort of technology [involving a collaboration between medicine and mathematics], and the ancestor of many of these technologies, is what is now called computed tomography, for which Allan Cormack, a physicist whose research became more and more mathematical as time went on, laid down the theoretical foundations around 1960. He shared the 1979 Nobel prize in medicine for his work in this field.

In fact the basic idea of tomography had been discovered for purely theoretical reasons in 1917 by the Austrian mathematician Johann Radon, and it had been rediscovered several times since by others, but Cormack was not to know this until much later than his own independent discovery. The problem he solved is this: Suppose we know all the line integrals through a body of varying density. Can we reconstruct the body itself? The answer, perhaps surprisingly, is that we can, and furthermore we can do so constructively. In practical terms, we know that a single X-ray picture can give only limited information because certain things are obscured by other, heavier things. We might take more X-ray pictures in the hope that we can somehow see behind the obscuring objects, but it is not at all obvious that by taking a lot—really, a lot—of X-ray pictures we can in effect even see into objects, which is what Radon tells us, at least in principle. Making Radon’s theorem into a practical tool was not a trivial matter.”

You can listen to a lecture on tomography and inverting the Radon transform here.