Analytic Tomography

Look, I'm not an applied mathematician, and the analysis I know is fairly limited. So I wasn't going to review this book. Then I read the first paragraph of the introduction:

This book is about tomography, which is a way to see what is inside an object without opening it up. If you are intrigued with this idea, then, no matter what your background, you will find that at least some portion of this book will provide interesting reading. If this idea is not intriguing, then I would recommend some other publication for your reading pleasure.

That was too tempting to resist! Reading on, I discovered that Markoe had attempted to do something remarkable: the first chapter of his book is a no-prerequisites, just-pictures attempt to explain the Radon transform. The second chapter brings in some calculus, and goes into more detail at a level that a (well-motivated) undergraduate should be able to follow. After that, things get much more technical.

So what is the book about. Here's a two-dimensional version (heavily based on the discussion in Markoe's first two chapters). Suppose you have a two-dimensional object of varying density given by a function f(x,y). Let θ be a unit vector and s be a real number, and consider the line onsisting of the vectors x such that θ∙x = s; define a new function Rf by letting Rf(θ,s) denote the average density of the object along that line (which is of course the integral of f(x,y) along the line in question). Rf is called the Radon Transform of f, and the crucial question studied by this book is how to get f if we know Rf. (Of course, after the first chapter, the book works in a far more general setting, with an n-dimensional object sliced by k-planes.)

The canonical example of this procedure is, of course, the CT scan. By sending an X-ray beam through an object and measuring its attenuation, we can get an estimate of the average density along the beam, and so we can measure Rf. Finding f is what allows the beautiful and clear diagnostic images we have all seen. (Note, however, that Radon proved his theorem in 1917, and the first CT scans were done in the 1960s.) The book does discuss this aspect of things, but its focus is really on the mathematical problem rather than in the applications. In particular, there is no discussion of discretization and/or algorithms, though Markoe does provide ample references for anyone who is interested.

So what we have here is a beautiful bit of mathematics that turns out to be incredibly useful. The first chapter is very easy to read. The second chapter would make a very good independent study assignment for an interested undergraduate. The book overall is very well done. I'm not an expert (very far from an expert), so I can't assess Markoe's choice of content and the details of his arguments. But he writes well and has a sense of humor, and that counts for a lot.

The author writes of his approach that "this will probably annoy most of my colleagues, and for that I hastily offer an apology. However, I think other readers will be thankful for the amount of detail I have provided." He's right. If you or your students are curious about how one can see inside something without opening it, you'll want a copy of this book.

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Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME. As the editor of FOCUS and of MAA Reviews, he has learned to value clear and intelligent writing.