Principles of Magnetic Resonance Imaging : A Signal Processing Perspective

by Liang, Zhi-Pei; Lauterbur, Paul C.

• ISBN13: 9780819435163
• ISBN10: 0819435163
• Format: Hardcover
• Copyright: 1999-10-01
• Publisher: Society of Photo Optical

Table of Contents

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