Medical Image Reconstruction: A Conceptual Tutorial

by Zeng, Gengsheng Lawrence

Edition: 1st

ISBN13: 9783642053672

ISBN10: 364205367X

Format: Hardcover

Pub. Date: 2010-05-30

Publisher(s): Springer Verlag

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Summary

"Medical Image Reconstruction: A Conceptual Tutorial" introduces the classical and modern image reconstruction technologies, such as two-dimensional (2D) parallel-beam and fan-beam imaging, three-dimensional (3D) parallel ray, parallel plane, and cone-beam imaging. This book presents both analytical and iterative methods of these technologies and their applications in X-ray CT (computed tomography), SPECT (single photon emission computed tomography), PET (positron emission tomography), and MRI (magnetic resonance imaging). Contemporary research results in exact region-of-interest (ROI) reconstruction with truncated projections, Katsevich's cone-beam filtered backprojection algorithm, and reconstruction with highly undersampled data with l0-minimization are also included. This book is designed for engineers and researchers in the field of biomedical engineering with medical imaging specialty, and image processing with image reconstruction specialty.Gengsheng Lawrence Zeng is a leading expert in the algorithm development of medical image reconstruction, and is a professor at the Department of Radiology, University of Utah, Salt Lake City, Utah, USA.

Author Biography

Gengsheng Lawrence Zeng is an expert in the development of medical image reconstruction algorithms and is a professor at the Department of Radiology, University of Utah, Salt Lake City, Utah, USA.

Basic Principles of Tomography	p. 1
Tomography	p. 1
Projection	p. 3
Image Reconstruction	p. 6
Backprojection	p. 8
Mathematical Expressions	p. 10
Projection	p. 10
Backprojection	p. 11
The Dirac ¿-function	p. 12
Worked Examples	p. 14
Summary	p. 17
Problems	p. 18
References	p. 19
Parallel-Beam Image Reconstruction	p. 21
Fourier Transform	p. 21
Central Slice Theorem	p. 22
Reconstruction Algorithms	p. 25
Method 1	p. 25
Method 2	p. 26
Method 3	p. 27
Method 4	p. 28
Method 5	p. 28
A Computer Simulation	p. 30
ROI Reconstruction with Truncated Projections	p. 31
Mathematical Expressions	p. 36
The Fourier Tranform and Convolution	p. 36
The Hilbert Transform and the Finite Hilbert Transform	p. 36
Proof of the Central Slice Theorem	p. 39
Derivation of the Filtered Backprojection Algorithm	p. 40
Expression of the Convolution Backprojection Algorithm	p. 41
Expression of the Radon Inversion Formula	p. 41
Derivation of the Backprojection-then-Filtering Algorithm	p. 41
Worked Examples	p. 42
Summary	p. 45
Problems	p. 46
References	p. 46
Fan-Beam Image Reconstruction	p. 49
Fan-Beam Geometry and Point Spread Function	p. 49
Parallel-Beam to Fan-Beam Algorithm Conversion	p. 52
Short Scan	p. 54
Mathematical Expressions	p. 56
Derivation of a Filtered Backprojection Fan-Beam Algorithm	p. 57
A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform	p. 58
Worked Examples	p. 60
Summary	p. 63
Problems	p. 64
References	p. 65
Transmission and Emission Tomography	p. 67
X-Ray Computed Tomography	p. 67
Positron Emission Tomography and Single Photon Emission Computed Tomography	p. 71
Attenuation Correction for Emission Tomography	p. 75
Mathematical Expressions	p. 79
Worked Examples	p. 81
Summary	p. 83
Problems	p. 83
References	p. 84
3D Image Reconstruction	p. 87
Parallel Line-Integral Data	p. 87
Backprojection-then-Filtering	p. 90
Filtered Backprojection	p. 91
Parallel Plane-Integral Data	p. 92
Cone-Beam Data	p. 94
Feldkamp's Algorithm	p. 95
Grangeat's Algorithm	p. 96
Katsevich's Algorithm	p. 97
Mathematical Expressions	p. 101
Backprojection-then-Filtering for Parallel Line-Integral Data	p. 102
Filtered Backprojection Algorithm for Parallel Line-Integral Data	p. 103
3D Radon Inversion Formula	p. 104
3D Backprojection-then-Filtering Algorithm for Radon Data	p. 104
Feldkamp's Algorithm	p. 105
Tuy's Relationship	p. 106
Grangeat's Relationship	p. 108
Katsevich's Algorithm	p. 111
Worked Examples	p. 117
Summary	p. 119
Problems	p. 120
References	p. 121
Iterative Reconstruction	p. 125
Solving a System of Linear Equations	p. 125
Algebraic Reconstruction Technique	p. 130
Gradient Descent Algorithms	p. 131
Maximum-Likelihood Expectation-Maximization Algorithms	p. 134
Ordered-Subset Expectation-Maximization Algorithm	p. 135
Noise Handling	p. 136
Analytical Methods-Windowing	p. 136
Iterative Methods-Stopping Early	p. 137
Iterative Methods-Choosing Pixels	p. 138
Iterative Methods-Accurate Modeling	p. 140
Noise Modeling as a Likelihood Function	p. 141
Including Prior Knowledge	p. 143
Mathematical Expressions	p. 145
Art	p. 145
Conjugate Gradient Algorithm	p. 146
ML-EM	p. 148
OS-EM	p. 151
Green's One-Step Late Algorithm	p. 151
Matched and Unmatched Projector/Backprojector Pairs	p. 151
Reconstruction Using Highly Undersampled Data with l₀ Minimization	p. 153
Worked Examples	p. 156
Summary	p. 167
Problems	p. 168
References	p. 170
MRI Reconstruction	p. 175
The "M"	p. 175
The "R"	p. 177
The "I"	p. 180
To Obtain z-Information-Slice Selection	p. 180
To Obtain x-Information-Frequency Encoding	p. 182
To Obtain y-Information-Phase Encoding	p. 183
Mathematical Expressions	p. 185
Worked Examples	p. 188
Summary	p. 190
Problems	p. 191
References	p. 192
Index	p. 193
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