The Mathematics of Medical Imaging (eBook)

A Beginner’s Guide

Timothy G. Feeman (Autor)

eBook Download: PDF

2010 | 2010
XII, 141 Seiten
Springer New York (Verlag)
978-0-387-92712-1 (ISBN)

Infos

Medical imaging is a major part of twenty-first century health care. This introduction explores the mathematical aspects of imaging in medicine to explain approximation methods in addition to computer implementation of inversion algorithms.

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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.

Autor

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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 10.3.2010
• Reihe/Serie: Springer Undergraduate Texts in Mathematics and Technology
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Informatik ► Theorie / Studium ► Künstliche Intelligenz / Robotik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Medizin / Pharmazie ► Gesundheitsfachberufe
• Themenwelt: Medizinische Fachgebiete ► Radiologie / Bildgebende Verfahren ► Radiologie
• Themenwelt: Medizin / Pharmazie ► Physiotherapie / Ergotherapie ► Orthopädie
• Themenwelt: Technik ► Medizintechnik
• Schlagworte: CAT scan • convolution • discrete Fourier transform • discrete image reconstruction • Fast Fourier transform • fast Fourier transform (FFT) • Fourier Tranform • Fourier transform • Functional Analysis • MRI • Radon Transform • Ultrasound • X-Ray
• ISBN-10: 0-387-92712-3 / 0387927123
• ISBN-13: 978-0-387-92712-1 / 9780387927121