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Introduction to Radon Transforms

Boris Rubin
Cambridge University Press
Publication Date: 
Number of Pages: 
Encyclopaedia of Mathematics and Its Applications 160
[Reviewed by
Jeff Ibbotson
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What is the Radon transform? This should seem like a question with a simple and direct answer but it isn’t. Depending on one’s point of reference there is a broad variety of answers. The original application of the Radon transform was to the field of Tomography — given a set of averages of “slices” of an object, how does one assemble these into a three-dimensional image of the object under exploration? This question was answered (at least in the two dimensional case) in 1917 by Johann Radon in a now-famous paper that remained unread by the scientific public for many years (probably until Fritz John’s 1934 paper on the topic of plane and spherical means for differential operators). Given the use of X-rays in medical tomography it seems like a natural question.

The Radon transform has come to mean much more — in the general setting of \(\mathbb{R}^n\), it refers to an integral transform defined over sets of \((n-1)\)-dimensional hyperplanes. Or, even further beyond, it is a tool for constructing a definition of values at a point for a function defined only on \(k\)-dimensional submanifolds which span a full \(n\)-dimensional manifold.

Or perhaps you, like Sigurdur Helgason, would prefer to define the Radon Transform as a Haar-like measure on \(k\)-dimensional subgroups of homogeneous topological groups? And did I mention that, at some point, the Radon Transform was recognized for its relevance to harmonic analysis on non-Euclidean spaces? So one can see that one needs to be a careful consumer in buying a book about Radon transforms. It might not be the one you imagine it to be. When the book contains the word “Introduction” the stakes are raised even further.

The book under discussion is emphatically not an introduction at the conceptual level. It is a compilation of hard analysis results often presented as “Show that operator \(T: U\to V\) is continuous when restricted to \(L^p\cap U\) etc.” More than this, it is an “introduction” only in that it seeks to build up the machinery of the Radon Transform using an approach via fractional derivatives. At the first reading this struck me as “the unknown explained through the even more unknown”.

There are, of course, almost as many varieties of fractional derivatives as there are transforms. The traditional version is that of Riemann and Liouville: define an integral operator \(I(f)=\int_a^x f\) and then interate to get \(I^{\alpha+1}(f)=\int_a^x \frac{(t-a)^\alpha}{\alpha!} f(t)\,dt\). Replace the factorial with the gamma function, show that these operators satisfy the semigroup law \(I^rI^s=I^{r+s}\) and then it’s off to the races: define the fractional derivative \(D^\alpha=I^{-\alpha}\) and there you have it.

This, however, is the only the beginning: the author spends at least 120 pages proving detailed results about Riesz Potentials before the Radon Transform as such is finally introduced in Chapter 4. In that chapter we encounter many of the members of the Radon transform family — Fourier transforms, the dual Radon Transform, Semyanistyi’s fractional integrals, Radon transforms of finite measures, spherical harmonics and finally the Heisenberg Group.

Succeeding chapters explore the Funk transform. The hemispherical transform, and applications of the Radon transform to hyperbolic and convex geometries. This approach assumes a facility with integration theory from advanced real analysis and often uses some detailed “machinery”. It also constitutes a fairly encyclopedic rendering of the field. As such it is well-suited to anyone looking for specific results to be used in analysis. Beginners should likely look elsewhere. 

Jeff Ibbotson holds the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.

1. Preliminaries
2. Fractional integration: functions of one variable
3. Riesz potentials
4. The Radon transform on Rn
5. Operators of integral geometry on the unit sphere
6. Operators of integral geometry in the hyperbolic space
7. Spherical mean Radon transforms.