The Mathematics of Medical Imaging

1. Front Matter

   Pages i-x

   PDF

2. X-rays

   • Timothy G. Feeman

   Pages 1-10

3. The Radon Transform

   • Timothy G. Feeman

   Pages 11-18

4. Back Projection

   • Timothy G. Feeman

   Pages 19-24

5. Complex Numbers

   • Timothy G. Feeman

   Pages 25-31

6. The Fourier Transform

   • Timothy G. Feeman

   Pages 33-46

7. Two Big Theorems

   • Timothy G. Feeman

   Pages 47-52

8. Filters and Convolution

   • Timothy G. Feeman

   Pages 53-66

9. Discrete Image Reconstruction

   • Timothy G. Feeman

   Pages 67-100

10. Algebraic Reconstruction Techniques

    • Timothy G. Feeman

    Pages 101-114

11. MRI—An Overview

    • Timothy G. Feeman

    Pages 115-128

12. Back Matter

    Pages 1-12

    PDF

In 1979, the Nobel Prize for Medicine and Physiology was awarded jointly to Allan McLeod Cormack and Godfrey Newbold Houns eld, the two pioneering scienti- engineers primarily responsible for the development, in the 1960s and early 1970s, of computerized axial tomography, popularly known as the CAT or CT scan. In his papers [13], Cormack, then a Professor at Tufts University, in Massachusetts, dev- oped certain mathematical algorithms that, he envisioned, could be used to create an image from X-ray data. Working completely independently of Cormack and at about the same time, Houns eld, a research scientist at EMI Central Research Laboratories in the United Kingdom, designed the rst operational CT scanner as well as the rst commercially available model. (See [22] and [23]. ) Since 1980, the number of CT scans performed each year in the United States has risen from about 3 million to over 67 million. What few people who have had CT scans probably realize is that the fundamental problem behind this procedure is essentially mathematical: If we know the values of the integral of a two- or three-dimensional fu- tion along all possible cross-sections, then how can we reconstruct the function itself? This particular example of what is known as an inverse problem was studied by Johann Radon, an Austrian mathematician, in the early part of the twentieth century.

• Fourier transform
• convolution
• discrete Fourier transform
• fast Fourier transform
• fast Fourier transform (FFT)
• ultrasound

From the reviews:

“It is a textbook that presents a compact, rigorous treatment of basic tomographic image reconstruction at a level suitable for an undergraduate who is strong in math. … This book is valuable, for it addresses with care and rigor the relevance of a variety of mathematical topics to 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.” (Richard Wendt III, 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. … The current book begins with a description of the imaging problem in the simplest possible situation, where the physics and geometry are clearest. … author handles the material with clarity and grace. … Doing that in a system like MATLAB or Maple would make for a very nice independent project.” (William J. Satzer, The Mathematical Association of America, February, 2010)

“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. … 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. … 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.” (Svitlana P. Rogovchenko, Zentralblatt MATH, Vol. 1191, 2010)

• Dept. Mathematical Sciences, Villanova University, Villanova, U.S.A.

  Timothy G. Feeman

Dr. Timothy G. Feeman, veteran mathematics professor at Villanova University, has been published in all of the leading mathematics journals and has received an award for expository writing from the Mathematical Association of America. In 2002, the American Mathematical Society published his first book, Portraits of the Earth: A Mathematician Looks at Maps.

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