Main 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.

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