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2015 | Book

X-rays Read chapter Read first chapter

The Mathematics of Medical Imaging

A Beginner’s Guide

Author: Timothy G. Feeman

Publisher: Springer International Publishing

Book Series : Springer Undergraduate Texts in Mathematics and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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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 adeed 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, 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 to a real-world problem. …T

his 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

Table of Contents

Frontmatter
1. X-rays
Abstract
A computerized axial tomography (CAT or CT) scan is generated from a set of thousands of X-ray beams, consisting of 160 or more beams at each of 180 directions. To comprehend this large collection of X-rays, we must first understand just one beam.
Timothy G. Feeman
2. The Radon Transform
Abstract
For a given function f defined in the plane, which may represent, for instance, the attenuation-coefficient function in a cross section of a sample, the fundamental question of image reconstruction calls on us to consider the value of the integral of f along a typical line \(\,\ell_{t,\,\theta }\). For each pair of values of t and \(\,\theta\), we will integrate f along a different line.
Timothy G. Feeman
3. Back Projection
Abstract
Let us begin the process of trying to recover the values of an attenuation-coefficient function f(x, y) from the values of its Radon transform \(\,\mathcal{R}f\,\).
Timothy G. Feeman
4. Complex Numbers
Abstract
There is no real number a for which \(\,a^{2} + 1 = 0\,\). In order to develop an expanded number system that includes solutions to this simple quadratic equation, we define the “imaginary number” \(\,i = \sqrt{-1}\,\). That is, this new number i is defined by the condition that \(\,i^{2} + 1 = 0\,\).
Timothy G. Feeman
5. The Fourier Transform
Abstract
The idea behind this definition is that, for each value of ω, the value of \(\,\mathcal{F}f(\omega )\,\) captures the component of f that has the frequency ω∕(2π) (and period 2π∕ω ).
Timothy G. Feeman
6. Two Big Theorems
Abstract
The ideas discussed in this chapter involve interactions between three transforms — the Radon transform, the Fourier transform, and the back-projection transform. Each of these transforms is defined in terms of improper integrals on infinite intervals. This raises the somewhat technical matter of determining which functions may appropriately be considered, an issue that is addressed in Appendix A. For the time being, we assume that any function being considered here meets the requirements. For those functions that arise in the practical world of medical imaging this is certainly the case.
Timothy G. Feeman
7. Filters and Convolution
Abstract
Of constant concern in the analysis of signals is the presence of noise, a term which here means more or less any effect that corrupts a signal. This corruption may arise from background radiation, stray signals that interfere with the main signal, errors in the measurement of the actual signal, or what have you. In order to remove the effects of noise and form a clearer picture of the actual signal, a filter is applied.
Timothy G. Feeman
8. Discrete Image Reconstruction
Abstract
We have seen that, when complete continuous X-ray data are available, then an attenuation-coefficient function f(x, y) can be reconstructed exactly using the filtered back-projection formula, Theorem 6.​2.
Timothy G. Feeman
9. Algebraic Reconstruction Techniques
Abstract
To this point, we have studied how Fourier transform methods are used in image reconstruction. This is the approach taken in the seminal work of Cormack [14] and used in the algorithms of today’s CT scan machines. However, the first CT scanner, designed in the late 1960s, by Godfrey Hounsfield, used an approach grounded in linear algebra and matrix theory to generate an image from the machine readings. Algorithms that adopt this point of view are known as algebraic reconstruction techniques, or ART, for short. In this chapter, we look at a few basic mathematical elements of ART.
Timothy G. Feeman
10. MRI — an overview
Abstract
Magnetic resonance imaging, or MRI, is an imaging technique that has grown alongside CT and that, like CT, has produced Nobel laureates of its own. Where the physics of CT is fairly straightforward — X-rays are emitted and their changes in intensity measured — MRI is based on the generation of a complex of overlapping, fluctuating electromagnetic fields that must be precisely controlled. Mathematically, the effects of the electromagnetic fields on the atomic nuclei in the sample being studied are modeled with differential equations. The Fourier transform is the primary tool for analyzing the electrical signals generated by the motions of atomic nuclei under the influence of these fields.
Timothy G. Feeman
Backmatter
Metadata
Title
The Mathematics of Medical Imaging
Author
Timothy G. Feeman
Copyright Year
2015
Electronic ISBN
978-3-319-22665-1
Print ISBN
978-3-319-22664-4
DOI
https://doi.org/10.1007/978-3-319-22665-1

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