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，Katsevichs cone-beam filtered backprojection algorithm， and reconstruction with highly undersampled data with/o-minimization are also included.
　　This book is written for engineers and researchers in the field of biomedical engineering specializing in medical imaging and image processing with image reconstruction.

　　The first time I heard about image reconstruction was twenty years ago Icame to the University of Utah as a post-doctoral fellow in the Departmentof Radiology. Dr. Grant Gullberg and Dr. Rolf Clackdoyle gave many lec-tures on image reconstruction and I took notes. Even today I still go backto those notes from time to time. I benefit from those notes significantly.This book is complied together with parts of those notes and some currentresearch papers with most mathematical proofs removed. I am grateful toDr. Gullberg and Dr. Clackdoyle for introducing me to the wonderful worldof image reconstruction. I appreciate Dr. Michel Defrise， Dr. Ge Wang， andDr. Guang-Hong Chen for their helpful suggestions. I also like to thank mycolleagues in the department and in other institutions. I would especially liketo thank Kathy Gullberg and Jacob Piatt for proof-reading the drafts.
　　This tutorial text introduces the classical and modern image reconstruc-tion technologies to the general audience. It covers the topics in two-dimensional （2D） parallel-beam and fan-beam imaging， three-dimensional（3D） parallel ray， parallel plane， and cone-beam imaging. Both analyticaland iterative methods are presented. The applications in X-ray CT， SPECT（single photon emission computed tomography）， PET （positron emissiontomography）， and MRI （magnetic resonance imaging） are also discussed.Contemporary research results in exact ROI （region-of-interest） reconstruc-tion with truncated projections， Katsevichs cone-beam filtered backprojec-tion algorithm， and reconstruction with highly undersampled data with/0-minimization are also included in this book.
　　This book is written in an easy-to-read style， which lets the diagrams dothe most talking. The readers who intend to get into medical image recon-struction will gain the general knowledge of the field in a painless way. I hopeyou enjoy reading it as much as I enjoy writing （and drawing） it. The firsttime reader can skip the more challenging materials marked by the \"*\" signwithout interrupting the flow of this book.

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

1 Basic Principles of Tomography
1.1 Tomography
1.2 Projection
1.3 Image Reconstruction
1.4 Backprojection
* 1.5 Mathematical Expressions
1.5.1 Projection
1.5.2 Backprojection
1.5.3 The Dirac δ-function
1.6 Worked Examples
1.7 Summary
Problems
References

2 Parallel-Beam Image Reconstruction
2.1 Fourier Transform
2.2 Central Slice Theorem
2.3 Reconstruction Algorithms
2.3.1 Method 1
2.3.2 Method 2
2.3.3 Method 3
2.3.4 Method 4
2.3.5 Method 5
2.4 A Computer Simulation
*2.5 ROI Reconstruction with Truncated Projections
*2.6 Mathematical Expressions
2.6.1 The Fourier Transform and Convolution
2.6.2 The Hilbert Transform and the Finite Hilbert Transform
2.6.3 Proof of the Central Slice Theorem
2.6.4 Derivation of the Filtered Backprojection Algorithm
2.6.5 Expression of the Convolution Backprojection Algorithm
2.6.6 Expression of the Radon Inversion Formula
2.6.7 Derivation of the Backprojection-then-Filtering Algorithm
2.7 Worked Examples
2.8 Summary
Problems
References

3 Fan-Beam Image Reconstruction
3.1 Fan-Beam Geometry and Point Spread Function
3.2 Parallel-Beam to Fan-Beam Algorithm Conversion
3.3 Short Scan
*3.4 Mathematical Expressions
3.4.1 Derivation of a Filtered Backprojection Fan-Beam Algorithm
3.4.2 A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform
3.5 Worked Examples
3.6 Summary
Problems
References

4 Transmission and Emission Tomography
4.1 X-Ray Computed Tomography
4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography
4.3 Attenuation Correction for Emission Tomography
*4.4 Mathematical Expressions
4.5 Worked Examples
4.6 Summary
Problems
References

5 3D Image Reconstruction
5.1 Parallel Line-Integral Data
5.1.1 Backprojection-then-Filtering
5.1.2 Filtered Backprojection
5.2 Parallel Plane-Integral Data
5.3 Cone-Beam Data
5.3.1 Feldkamp\\s Algorithm
5.3.2 Grangeat\\s Algorithm
5.3.3 Katsevich\\s Algorithm
*5.4 Mathematical Expressions
5.4.1 Backprojection-then-Filtering for Parallel Line-Integral Data
5.4.2 Filtered Backprojection Algorithm for Parallel Line-Integral Data
5.4.3 3D Radon Inversion Formula
5.4.4 3D Backprojection-then-Filtering Algorithm for Radon Data
5.4.5 Feldkamp\\s Algorithm
5.4.6 Tuy\\s Relationship
5.4.7 Grangeat\\s Relationship
5.4.8 Katsevieh\\s Algorithm
5.5 Worked Examples
5.6 Summary
Problems
References

6 Iterative Reconstruction
6.1 Solving a System of Linear Equations
6.2 Algebraic Reconstruction Technique
6.3 Gradient Descent Algorithms
6.4 Maximum-Likelihood Expectation-Maximization Algorithms
6.5 Ordered-Subset Expectation-Maximization Algorithm
6.6 Noise Handling
6.6.1 Analytical Methods——Windowing
6.6.2 Iterative Methods——Stopping Early
6.6.3 Iterative Methods——Choosing Pixels
6.6.4 Iterative Methods——Accurate Modeling
6.7 Noise Modeling as a Likelihood Function
6.8 Including Prior Knowledge
*6.9 Mathematical Expressions
6.9.1 ART
6.9.2 Conjugate Gradient Algorithm
6.9.3 ML-EM
6.9.4 OS-EM
6.9.5 Green\\s One-Step Late Algorithm
6.9.6 Matched and Unmatched Projector/Backprojector Pairs
*6.10 Reconstruction Using Highly Undersampled Data with 10 Minimization
6.11 Worked Examples
6.12 Summary
Problems
References

7 MRI Reconstruction
7.1 The \\\"M\\\"
7.2 The \\\"R\\\"
7.3 The \\\"T\\\"
7.3.1 To Obtain z-Information——Slice Selection
7.3.2 To Obtain x-Information——Frequency Encoding
7.3.3 To Obtain y-Information——Phase Encoding
*7.4 Mathematical Expressions
7.5 Worked Examples
7.6 Summary
Problems
References
Index