Medical imaging — including, for instance, CAT scanning and MRI — is among those applications of mathematics that provide immense value with essentially no public recognition of the role of mathematics. How many people are aware that the now commonplace use of CAT scans and MRI would be impossible without application of sophisticated mathematics? Yet, as of a few years ago, more than 65 million CAT scans were performed every year in the United States, and MRIs are now a common daily diagnostic procedure.
Because most books that address the mathematics of medical imaging are aimed at a fairly sophisticated audience, it can be difficult for a newcomer to get a foothold. This new book by Timothy Feeman, truly intended to be a beginner’s guide, makes the subject accessible to undergraduates with a working knowledge of multivariable calculus and some experience with vectors and matrix methods. The basic issue — the need to solve an inverse problem — is easily stated: if we know the values of the integral of a two-or three-dimensional function along all possible cross-sections, how can we reconstruct the function itself? Johann Radon first studied the problem in the early twentieth century; he developed a sophisticated approach based on the theory of transforms and integral operators. Then Allan Cormack, who with Godfrey Hounsfield developed the idea of computerized axial tomography in the late 1960s, essentially rediscovered Radon’s basic idea, and did so at a time when the required technology — including the computational power – was just becoming available.
Of course the devil is in the details. Radon’s inversion methods assume that the behavior of the function is known on every cross-section, but in practice only a discrete number of cross-sections can be sampled. So it is possible to construct only an approximate solution. Then, even to get this approximate solution, it’s necessary to apply a lot of computational power to process a large number of discrete measurements. Computational capability like this has only been available for the last couple of decades.
The current book begins with a description of the imaging problem in the simplest possible situation, where the physics and geometry are clearest. The author introduces the two-dimensional Radon transform in Chapter 2, establishes its properties and provides two examples where he computes the Radon transform explicitly. The next chapter introduces back projection, an important component of the reconstruction process that produces a kind of smoothed version of the desired image. What’s needed next is a technique called filtered back projection, and that introduces the Fourier transform. The foundations of the reconstruction process are two main theorems presented in Chapter 6 that describe interactions between the Radon, Fourier and back-projection transforms that lead to an explicit formula for reconstruction.
In practice, of course, all the incoming data are discrete, so the author revisits the problem from that perspective in Chapter 8. That includes important discussions of sampling and interpolation as well as discrete versions of the Radon, Fourier and back-projection transforms. The remaining chapters introduce some related topics: an algebraic reconstruction technique, convolutions and low pass filters to deal with noisy data, and a short introductory chapter describing the physics of magnetic resonance imaging.
Although this is not an easy subject, the author handles the material with clarity and grace. He does not try to do too much — he omits a number of technical issues and relegates questions of integrability to an appendix. What makes the book work is its focus on the primary questions. The author wisely declines to include a discussion of computer implementation of the reconstruction algorithms. Doing that in a system like MATLAB or Maple would make for a very nice independent project.
Bill Satzer (email@example.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.