دانلود کتاب The Mathematics of Medical Imaging: A Beginner’s Guide (Springer Undergraduate Texts in Mathematics and Technology)

فهرست مطالب :

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
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
AppendixA Integrability
A.1 Improper integrals
A.2 Iterated improper integrals
A.3 L1 and L2
A.4 Summability
AppendixB Topics for Further Study
References
Index