Mathematicians and practitioners working in the area of tomography will enjoy this book. For engineers and developers, the author offers a gentle mathematical introduction delineating the importance of the Radon transform and filtered backprojection. Further, the author increases the complexity of the book in each chapter, aiding even the more seasoned mathematicians and researchers in the analytical aspects of tomography. The book presents an impressive collection of references tied together with an interesting historical background.

The book is very didactic in describing the problem of acquiring knowledge from the interior of objects, and in presenting the mathematical solution to this problem. It starts with a wonderful introduction motivating the reader with x-ray and computerized tomography equipment. Chapter 1 contains many graphs and images, helping the reader visualize the problem and have a glimpse at the solution. In this introductory chapter, the most basic derivations are presented without much mathematical rigor, explaining several abstract ideas visually in a straightforward way. The ideas range from tomography itself and the Radon transform to filtered backprojection and photon attenuation.

After a solid introduction, chapter 2 is dedicated to the foundation of tomography, the Radon transform. It recapitulates the basic concepts presented in the previous chapter, focusing on rigorously defining the Radon transform and its elementary properties. The beginning of the chapter defines important mathematical objects, such as the Euclidean n-space and hyperplanes, and then moves on to describe more advanced concepts, such as Riemann and Lebesgue integration. As theorems and proofs are laid out, the author illustrates the concepts with figures and examples. When the author is going beyond the prerequisites established for this or other chapters, he simply summarizes the mathematical rigor and points the reader to appropriate references. Another strategy the author uses to improve readability is to construct complex derivations from very simple ones, allowing readers with basic mathematical knowledge to understand the text easily, and readers with more knowledge to check the formulas with delight. The second chapter ends with examples, interesting historical remarks, and considerations of algorithmic and computational performance, highlighting the commitment of the book to pure and applied mathematics.

The remaining chapters concentrate on generalizing the basic theory of analytic tomography, requiring at least some graduate-level mathematics. More specifically, chapter 3 generalizes the Radon transform to the k-plane transform, chapter 4 characterizes the range of this general transform, and chapter 5 studies other generalizations and variants of the Radon transform. These generalizations offer readers the necessary background for researching in this area. In the last two chapters, the author reduces the mathematical rigor by not providing full proofs, returning to the style of the beginning of the book.

The result is a book that is at the same time deep in its subject and broad in its intended audience. In several chapters, the author presents corrections of important references, placing the book as a strong and comprehensive reference on the subject. Also, throughout the book, the author refers to very interesting and alternate applications of tomography, from economy to astronomy. The limitations of the book are clearly stated: geometric and impedance tomography are not treated and there are no exercises. These limitations, however, do not compromise the quality and usefulness of this great book.