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Tables of Contents for Principles of Magnetic Resonance Imaging
Chapter/Section Title
Page #
Page Count
Preface
xiii
Acknowledgments
xv
Introduction
1
12
What Is MRI?
2
3
A System Perspective
5
2
The Main Magnet
5
1
The Gradient System
6
1
The RF System
7
1
A Signal Processing Perspective
7
2
Organization of the Book
9
4
Exercises
11
2
Mathematical Fundamentals
13
44
Vectors
13
4
Basic Concepts of Matrix Algebra
17
2
Some Commonly Used Functions
19
7
Unit Step Function
19
1
Signum Function
19
1
Rectangular Window Function
19
1
Triangle Window Function
19
1
Hamming Window Function
19
1
Gaussian Function
20
1
Dirac Delta Function
20
2
Kronecker Delta Function
22
1
Comb Function
22
1
Sinc Function
23
1
Dirichlet Function
23
1
Bessel Functions
24
2
Convolution
26
2
The Fourier Transform
28
8
Definition
29
1
Properties
30
2
Examples
32
4
The Radon Transform
36
21
Two-Dimensional Radon Transforms
36
2
Higher-Dimensional Radon Transforms
38
2
Partial Radon Transforms
40
3
Basic Properties
43
1
Sinogram
43
2
The Projection-Slice Theorem
45
5
Convolution Theorem
50
2
Exercises
52
5
Signal Generation and Detection
57
50
Magnetized Nuclear Spin Systems
58
11
Nuclear Magnetic Moments
58
6
Bulk Magnetization
64
4
More on the Larmor Frequency
68
1
RF Excitations
69
22
Resonance Condition
69
1
Characteristics of an RF Pulse
70
2
Rotating Frame of Reference
72
4
The Bloch Equation
76
1
On-Resonance Excitations
77
10
Off-Resonance Excitations
87
1
Frequency Selectivity of an RF Pulse
88
3
Free Precession and Relaxation
91
3
Signal Detection
94
13
Basic Detection Principles
94
1
Signal Expressions
95
6
Exercises
101
6
Signal Characteristics
107
34
Basic Assumptions
107
2
Free Induction Decays
109
5
RF Echoes
114
17
Two-Pulse Echo
114
6
Three-Pulse Echoes
120
5
Extended Phase Graphs
125
5
The CPMG Echo Train
130
1
Gradient Echoes
131
10
Gradient Fields
131
2
Formation of Gradient Echoes
133
3
Exercises
136
5
Signal Localization
141
46
Slice Selection
142
11
Slice Equation
142
1
Slice-Selective Gradients
143
2
Slice-Selective RF Pulses
145
4
Some Practical Considerations
149
4
Spatial Information Encoding
153
12
Frequency Encoding
153
2
Phase Encoding
155
2
A k-Space Interpretation
157
8
Basic Imaging Methods
165
8
One-Dimensional Imaging
165
2
Two-Dimensional Imaging
167
4
Three-Dimensional Imaging
171
2
Sampling of k-Space-Space
173
14
The Sampling Theorem
173
3
Sampling Requirements of k-Space Signals
176
4
Exercises
180
7
Image Reconstruction
187
30
General Issues of Image Reconstruction
188
2
Reconstruction from Fourier Transform Samples
190
9
Problem Formulation
190
1
Basic Theory
190
5
Computational Algorithms
195
4
Reconstruction from Radon Transform Samples
199
14
Problem Formulation
199
1
The Inverse Radon Transform
200
2
Backprojection
202
2
Practical Reconstruction Algorithms
204
9
Appendix
213
4
Exercises
214
3
Image Contrast
217
16
Introduction
217
1
Saturation-Recovery Sequence
218
3
Inversion-Recovery Sequence
221
2
Basic Spin-Echo Imaging
223
2
Basic Gradient-Echo Imaging
225
2
Discussion
227
6
Exercises
230
3
Image Resolution, Noise, and Artifacts
233
58
Resolution Limitations
233
6
Point Spread Function
233
2
PSF of Fourier Reconstructions
235
2
PSF of Backprojection Reconstructions
237
2
Image Noise
239
12
Basic Concepts of Random Signals
239
6
Noise Characteristics in the Data Domain
245
1
Noise in Direct FFT Reconstruction
246
2
Noise in Zero-Padded FFT Reconstruction
248
2
Noise in Filtered Backprojection Reconstruction
250
1
Image Artifacts
251
40
Gibbs Ringing Artifact
251
4
Aliasing Artifacts
255
3
Chemical Shift Artifact
258
2
Motion Artifacts
260
21
Artifacts Due to Corrupted Data
281
4
Exercises
285
6
Fast-Scan Imaging
291
30
Fast Spin-Echo Imaging
291
6
Basic Concept
292
3
Practical Issues
295
2
Fast Gradient-Echo Imaging
297
6
Spoiled Steady-State Imaging
297
3
Steady-State Imaging
300
3
Echo-Planar Imaging
303
8
Zigzag Trajectory
304
3
Rectilinear Trajectory
307
1
Spiral Trajectory
308
3
Discussion
311
1
Burst Imaging
311
10
Exercises
315
6
Constrained Reconstruction
321
46
Half-Fourier Reconstruction
322
9
Phase Estimation
323
1
Phase-Constrained Reconstruction
323
4
Discussion
327
4
Extrapolation-Based Reconstruction
331
8
Bandlimited Extrapolation
332
2
Maximum Entropy Reconstruction
334
3
Discussion
337
2
Parametric Reconstruction Methods
339
16
The Autoregressive Moving Average Model
340
7
The Generalized Series Model
347
8
Appendix
355
12
The Direct Least-Squares Method
356
1
SVD-Based Methods
357
6
Exercises
363
4
A Mathematical Formulas
367
4
A.1 Sums
367
1
A.2 Power Series
367
1
A.3 Complex Numbers
368
1
A.4 Trigonometric Identities
368
1
A.5 Short Tables of Convolutions
369
1
A.6 A Short Table of Fourier Transforms
370
1
B Glossary
371
12
C Abbreviations
383
2
D Mathematical Symbols
385
4
E Physical Constants
389
2
Bibliography
391
18
Index
409
6
About the Authors
415