Book description
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.
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.
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 Content:
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
Indexing
Table of contents
- Cover
- Title Page
- Copyright
- Preface
- Contents
- 1 Basic principles of tomography
-
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
- 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 FBP 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
- 2.6.8 Expression of the derivative–backprojection–Hilbert transform algorithm
- 2.6.9 Derivation of the backprojection–derivative–Hilbert transform algorithm
- 2.7 Worked examples
- 2.8 Summary
- Problems
- Bibliography
- 3 Fan-beam image reconstruction
- 4 Transmission and emission tomography
-
5 Three-dimensional image reconstruction
- 5.1 Parallel line-integral data
- 5.2 Parallel plane-integral data
- 5.3 Cone-beam data
-
5.4 Mathematical expressions
- 5.4.1 Backprojection-then-filtering for parallel line-integral data
- 5.4.2 FBP algorithm for parallel line-integral data
- 5.4.3 Three-dimensional Radon inversion formula (FBP algorithm)
- 5.4.4 Three-dimensional 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 Katsevich’s algorithm
- 5.5 Worked examples
- 5.6 Summary
- Problems
- Bibliography
-
6 Iterative reconstruction
- 6.1 Solving a system of linear equations
- 6.2 Algebraic reconstruction technique
- 6.3 Gradient descent algorithms
- 6.4 ML-EM algorithms
- 6.5 OS-EM algorithm
- 6.6 Noise handling
- 6.7 Noise modeling as a likelihood function
- 6.8 Including prior knowledge (Bayesian)
- 6.9 Mathematical expressions
- 6.10 Reconstruction using highly undersampled data
- 6.11 Worked examples
- 6.12 Summary
- Problems
- Bibliography
- 7 MRI reconstruction
-
8 Using FBP to perform iterative reconstruction
- 8.1 The Landweber algorithm: From recursive form to non-recursive form
- 8.2 The Landweber algorithm: From non-recursive form to closed form
- 8.3 The Landweber algorithm: From closed form to backprojection-then-filtering algorithm
- 8.4 The Landweber algorithm: The weighted FBP algorithm
- 8.5 FBP algorithm with quadratic constraints
- 8.6 Convolution backprojection
- 8.7 Non-quadratic constraints
- 8.8 A viewpoint from calculus of variations
- 8.9 Summary
- Problems
- Bibliography
- Index
Product information
- Title: Image Reconstruction
- Author(s):
- Release date: March 2017
- Publisher(s): De Gruyter
- ISBN: 9783110498028
You might also like
book
Multimodal Scene Understanding
Multimodal Scene Understanding: Algorithms, Applications and Deep Learning presents recent advances in multi-modal computing, with a …
book
Biomedical Imaging
Covering both physical as well as mathematical and algorithmic foundations, this graduate textbook provides the reader …
book
Biomedical Imaging
Biomedical Imaging: Applications and Advances discusses the technologies and latest developments in the increasingly important field …
book
Computational Vision and Medical Image Processing IV
Computational Vision and Medical Image Processing. VIPIMAGE 2013 contains invited lectures and full papers presented at …