This concise and nicely written book grew out of a course offered by the author in 2008 to undergraduate mathematics majors and minors at Villanova University. It has everything one needs for understanding of fundamental mathematical ideas behind computerized axial tomography (CAT), known also as a CT scan. In 1979, the English electrical engineer Allan McLeod Cormack and the South African-born American physicist Godfrey Newbold Hounsefield received the Nobel Prize for Medicine and Physiology for pioneering work on developing the diagnostic technique of X-ray CAT. As a part of this project, mathematical algorithms used to create an image from X-ray data have been suggested. The Radon transform, named after the Austrian mathematician Johann Radon, plays an important role in these algorithms. It is used, among other things, to reconstruct a two- or three-dimensional function using the values of its integrals along all possible cross-sections.
The material in the book is divided into ten conveniently sized chapters, each providing sufficient information for a couple of one-hour lectures. Introducing the reader to X-rays in the very first chapter, the author proceeds with the Radon transform, back projection, two fundamental theorems, complex numbers, the Fourier transform, filters and convolution, discrete image reconstruction, algebraic reconstruction techniques, and concludes the presentation with an overview of magnetic resonance imaging. The book is well structured; the exposition is neat and transparent. All theoretical material is illustrated with carefully selected examples which are easy to follow. Each chapter has a section with exercises, some barely computational and some theory-oriented. There are two appendices devoted, respectively, to integrability issues and topics for further study. The bibliography contains forty one items, and there is also a useful index.
I highly recommend this interesting, accessible to a wide audience and well-written book dealing with mathematical techniques that support recent ground-breaking discoveries in biomedical technology both to students in mathematics, computer science, physics, biomedical sciences, engineering and to specialists.