Introduction

This book introduces the classical and modern image reconstruction technologies. It covers topics in two-dimensional (2D) parallel-beam and fan-beam imaging, three-dimensional (3D) parallel ray, parallel plane, and cone-beam imaging. Both analytical and iterative methods are presented. The applications in X-ray CT, SPECT (single photon emission computed tomography), PET (positron emission tomography), and MRI (magnetic resonance imaging) are discussed. Contemporary research results in exact region-of-interest (ROI) reconstruction with truncated projections, Katsevich’s cone-beam filtered backprojection algorithm, and reconstruction with highly under-sampled data are included.

Recent Research and Techniques

The last chapter of the book is devoted to the techniques of using a fast analytical algorithm to reconstruct an image that is equivalent to an iterative reconstruction. These techniques are the author’s most recent research results.

Intended Audience

This book is intended for students, engineers, and researchers who are interested in medical image reconstruction. Written in a non-mathematical way, this book provides an easy access to modern mathematical methods in medical imaging.

Table of Contents

Chapter 1 Basic Principles of Tomography
1.1 Tomography
1.2 Projection
1.3 Image Reconstruction
1.4 Backprojection
1.5 Mathematical Expressions
Problems
References

Chapter 2 Parallel-Beam Image Reconstruction
2.1 Fourier Transform
2.2 Central Slice Theorem
2.3 Reconstruction Algorithms
2.4 A Computer Simulation
2.5 ROI Reconstruction with Truncated Projections
2.6 Mathematical Expressions (The Fourier Transform and Convolution, The Hilbert Transform and the Finite Hilbert Transform, Proof of the Central Slice Theorem, Derivation of the Filtered Backprojection Algorithm, Expression of the Convolution Backprojection Algorithm, Expression of the Radon Inversion Formula, Derivation of the Backprojection-then-Filtering Algorithm)
Problems
References

Chapter 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 (Derivation of a Filtered Backprojection Fan-Beam Algorithm, A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform)
Problems
References

Chapter 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
Problems
References

Chapter 5 3D Image Reconstruction
5.1 Parallel Line-Integral Data
5.2 Parallel Plane-Integral Data
5.3 Cone-Beam Data (Feldkamp's Algorithm, Grangeat's Algorithm, Katsevich's Algorithm)
5.4 Mathematical Expressions (Backprojection-then-Filtering for Parallel Line-Integral Data, Filtered Backprojection Algorithm for Parallel Line-Integral Data, 3D Radon Inversion Formula, 3D Backprojection-then-Filtering Algorithm for Radon Data, Feldkamp's Algorithm, Tuy's Relationship, Grangeat's Relationship, Katsevich’s Algorithm)
Problems
References

Chapter 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 (Analytical Methods, Iterative Methods, Iterative Methods)
6.7 Noise Modeling as a Likelihood Function
6.8 Including Prior Knowledge
6.9 Mathematical Expressions (ART, Conjugate Gradient Algorithm, ML-EM, OS-EM, Green’s One-Step Late Algorithm, Matched and Unmatched Projector/Backprojector Pairs)
6.10 Reconstruction Using Highly Undersampled Data with l0 Minimization
Problems
References

Chapter 7 MRI Reconstruction
7.1 The 'M'
7.2 The 'R'
7.3 The 'I' (To Obtain z-Information, x-Information, y-Information)
7.4 Mathematical Expressions
Problems
References