© 2015

The Mathematics of Medical Imaging

A Beginner’s Guide

  • Offers concise treatment of mathematics for undergraduates solely within the context of medical imaging

  • Covers current medical imaging development and improvements regarding CT scans, ultrasounds, MRIs, and more

  • Offers short computer-based assignments and more than 20 examples using R

  • SpringerLink features R code used throughout the text


Table of contents

  1. Front Matter
    Pages i-xiv
  2. Timothy G. Feeman
    Pages 1-11
  3. Timothy G. Feeman
    Pages 13-37
  4. Timothy G. Feeman
    Pages 39-46
  5. Timothy G. Feeman
    Pages 47-53
  6. Timothy G. Feeman
    Pages 55-70
  7. Timothy G. Feeman
    Pages 71-76
  8. Timothy G. Feeman
    Pages 77-95
  9. Timothy G. Feeman
    Pages 97-136
  10. Timothy G. Feeman
    Pages 137-161
  11. Timothy G. Feeman
    Pages 163-175
  12. Back Matter
    Pages 177-197

About this book


The basic mathematics of computerized tomography, the CT scan, are aptly presented for an audience of undergraduates in mathematics and engineering. Assuming no prior background in advanced mathematical analysis, topics such as the Fourier transform, sampling, and discrete approximation algorithms are introduced from scratch and are developed within the context of medical imaging. A chapter on magnetic resonance imaging focuses on manipulation of the Bloch equation, the system of differential equations that is the foundation of this important technology.

Extending the ideas of the acclaimed first edition, new material has been added to render an even more accessible textbook for course usage. This edition includes new discussions of the Radon transform, the Dirac delta function and its role in X-ray imaging, Kacmarz’s method and least squares approximation, spectral filtering,  and more.  Copious examples and exercises, several new computer-based exercises, and additional graphics have been added to further delineate concepts.  The use of technology has been revamped throughout with the incorporation of the open source programming environment R to illustrate examples and composition of graphics. All R code is available as extra source material on SpringerLink.

From the reviews of the first edition:

“This book is valuable, for it addresses with care and rigor the relevance of a variety of mathematical topics t

o a real-world problem. …This book is well written. It serves its purpose of focusing a variety of mathematical topics onto a real-world application that is in its essence mathematics.”
The Journal of Nuclear Medicine, Vol. 51 (12), December, 2010


“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. …author handles the material with clarity and grace…”
The Mathematical Association of America, February, 2010


“All theoretical material is illustrated with carefully selected examples which are easy to follow. …I highly recommend this interesting, accessible to wide audience and well-written book dealing with mathematical techniques that support recent ground-breaking discoveries in biomedical technology both to students…and to specialists.”
Zentralblatt MATH, Vol. 1191, 2010


Bloch equation Fourier transform Kaczmarz's method MRI R code medical imaging RF field algebraic reconstruction techniques back projection complex numbers convolution discrete Fourier transform eigenvalue decomposition fast Fourier transform (FFT) integrability mathematics medical imaging mathematics medical imaging textbook adoption radon transform singular value decomposition ultrasound

Authors and affiliations

  1. 1.Villanova University Dept. Mathematical SciencesVillanovaUSA

About the authors

Timothy G. Feeman is professor of mathematics, Villanova University, in Lancaster, Pennsylvania. His original area of research is the theory of operators on Hilbert spaces once described as "the field of mathematics that has the strongest interaction with the scientific and technological developments which are characteristic of the twentieth century." Since the mid- to late-1990s, his scholarly efforts have become more diversified.

Bibliographic information

Industry Sectors
Finance, Business & Banking


“The text is concise, lucidly written and coherently structured. It is also self-contained and easily accessible to any undergraduate student having a solid command of mathematics at the level of some introductory courses in algebra and analysis. The code written in R is a genuine asset of high practical and educational value. Overall, this textbook provides good, highly informative, and useful material to students and all of those with interest in medical imaging.” (Witold Pedrycz, zbMATH 1351.92002, 2017)

“I believe that the book is a useful starting point for undergraduate students from mathematics, computer science, and related fields who want to learn how CT works; it also provides interesting reading for people from medical areas who want to find out the technical and mathematical background of the tools that they use.” (Kai Diethelm, Computing Reviews,, May, 2016)