Main description:

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.

Feature:

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

Inherently interdisciplinary between students of mathematics, physics, computer science, and biomedical engineering

Covers current medical imaging development and improvement regarding CAT scans, ultrasounds, MRIs, and more

Back cover:

A Beginner's Guide to the Mathematics of Medical Imaging presents the basic mathematics of computerized tomography – the CT scan – 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.

The text is self-contained with a range of practical exercises, topics for further study, and an ample bibliography, making it ideal for use in an undergraduate course in applied or engineering mathematics, or by practitioners in radiology who want to know more about the mathematical foundations of their field.

Contents:

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