The Mathematics of Medical Imaging
A Beginner's Guide
Published on 1. November 2009
Book
Hardback
XII, 141 pages
978-0-387-92711-4 (ISBN)
Description
This text explores medical imaging, one of the most significant areas of recent mathematical applications, in a concise manner accessible to undergraduate students. The author emphasizes the mathematical aspects of medical imaging, including not only the theoretical background, but also the role of approximation methods and the computer implementation of the inversion algorithms.
In twenty-first century health care, CAT scans, ultrasounds, and MRIs are commonplace. Significant computational advances, along with the development, design, and improvement of the machines themselves, can only occur in conjunction with a proper understanding of the mathematics. This book is inherently interdisciplinary in nature, and therefore is appropriate for students of engineering, physics, and computer science, in addition to mathematics.
Reviews / Votes
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)More details
Series
Edition
2010
Language
English
Place of publication
NY
United States
Target group
College/higher education
Research
Illustrations
1 s/w Tabelle
1 black & white tables, biography
Dimensions
Height: 23.5 cm
Width: 17.8 cm
Weight
463 gr
ISBN-13
978-0-387-92711-4 (9780387927114)
DOI
10.1007/978-0-387-92712-1
Schweitzer Classification
Other editions
New editions
Book
11/2015
2nd Edition
Springer
€64.19
Shipment within 10-15 days
Additional editions
Book
11/2014
Springer
€42.75
Article exhausted; check for reprint
E-Book
03/2010
Springer
€39.99
Available for download
Person
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.
Content
Preface.- 1 X-rays.- 1.1 Introduction.- 1.2 X-ray behavior and Beer's law.- 1.3 Lines in the plane.- 1.4 Exercises.- 2 The Radon Transform.- 2.1 Definition.- 2.2 Examples.- 2.3 Linearity.- 2.4 Phantoms.- 2.5 The domain of R.- 2.6 Exercises.- 3 Back Projection.- 3.1 Definition and properties.- 3.2 Examples.- 3.3 Exercises.- 4 Complex Numbers.- 4.1 The complex number system.- 4.2 The complex exponential function.- 4.3 Wave functions.- 4.4 Exercises.- 5 The Fourier Transform.- 5.1 Definition and examples.- 5.2 Properties and applications.- 5.3 Heaviside and Dirac d.- 5.4 Inversion of the Fourier transform.- 5.5 Multivariable forms.- 5.6 Exercises.- 6 Two Big Theorems.- 6.1 The central slice theorem.- 6.2 Filtered back projection.- 6.3 The Hilbert transform.- 6.4 Exercises.- 7 Filters and Convolution.- 7.1 Introduction.- 7.2 Convolution.- 7.3 Filter resolution.- 7.4 Convolution and the Fourier transform.- 7.5 The Rayleigh-Plancherel theorem.- 7.6 Convolution in 2-dimensional space.- 7.7 Convolution, B, and R.- 7.8 Low-pass filters.- 7.9 Exercises.- 8 Discrete Image Reconstruction.- 8.1 Introduction.- 8.2 Sampling.- 8.3 Discrete low-pass filters.- 8.4 Discrete Radon transform.- 8.5 Discrete functions and convolution.- 8.6 Discrete Fourier transform.- 8.7 Discrete back projection.- 8.8 Interpolation.- 8.9 Discrete image reconstruction.- 8.10 Matrix forms.- 8.11 FFT-the fast Fourier transform.- 8.12 Fan beam geometry.- 8.13 Exercises.- 9 Algebraic Reconstruction Techniques.- 9.1 Introduction.- 9.2 Least squares approximation.- 9.3 Kaczmarz's method.- 9.4 ART in medical imaging.- 9.5 Variations of Kaczmarz's method.- 9.6 ART or the Fourier transform?.- 9.7 Exercises.- 10 MRI-An Overview.- 10.1 Introduction.- 10.2 Basics.- 10.3 The Bloch equation.- 10.4 The RF field.- 10.5 RF pulse sequences; T1 and T2 .- 10.6 Gradients and slice selection.- 10.7 The imaging equation.- 10.8 Exercises.- Appendix A Integrability.- A.1 Improper integrals.- A.2 Iterated improperintegrals.- A.3 L1 and L2 .- A.4 Summability.- Appendix B Topics for Further Study.- References.- Index