O'Reilly logo

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Image Reconstruction

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

  1. Cover
  2. Title Page
  3. Copyright
  4. Preface
  5. Contents
  6. 1 Basic principles of tomography
    1. 1.1 Tomography
    2. 1.2 Projection
    3. 1.3 Image reconstruction
    4. 1.4 Backprojection
    5. 1.5 Mathematical expressions
      1. 1.5.1 Projection
      2. 1.5.2 Backprojection
      3. 1.5.3 The Dirac δ-function
    6. 1.6 Worked examples
    7. 1.7 Summary
    8. Problems
    9. Bibliography
  7. 2 Parallel-beam image reconstruction
    1. 2.1 Fourier transform
    2. 2.2 Central slice theorem
    3. 2.3 Reconstruction algorithms
      1. 2.3.1 Method 1
      2. 2.3.2 Method 2
      3. 2.3.3 Method 3
      4. 2.3.4 Method 4
      5. 2.3.5 Method 5
      6. 2.3.6 Method 6
    4. 2.4 A computer simulation
    5. 2.5 ROI reconstruction with truncated projections
    6. 2.6 Mathematical expressions
      1. 2.6.1 The Fourier transform and convolution
      2. 2.6.2 The Hilbert transform and the finite Hilbert transform
      3. 2.6.3 Proof of the central slice theorem
      4. 2.6.4 Derivation of the FBP algorithm
      5. 2.6.5 Expression of the convolution backprojection algorithm
      6. 2.6.6 Expression of the Radon inversion formula
      7. 2.6.7 Derivation of the backprojection-then-filtering algorithm
      8. 2.6.8 Expression of the derivative–backprojection–Hilbert transform algorithm
      9. 2.6.9 Derivation of the backprojection–derivative–Hilbert transform algorithm
    7. 2.7 Worked examples
    8. 2.8 Summary
    9. Problems
    10. Bibliography
  8. 3 Fan-beam image reconstruction
    1. 3.1 Fan-beam geometry and the point spread function
    2. 3.2 Parallel-beam to fan-beam algorithm conversion
    3. 3.3 Short scan
    4. 3.4 Mathematical expressions
      1. 3.4.1 Derivation of a filtered backprojection fan-beam algorithm
      2. 3.4.2 A fan-beam algorithm using the derivative and the Hilbert transform
      3. 3.4.3 Expression for the Parker weights
      4. 3.4.4 Errors caused by finite bandwidth implementation
    5. 3.5 Worked examples
    6. 3.6 Summary
    7. Problems
    8. Bibliography
  9. 4 Transmission and emission tomography
    1. 4.1 X-ray computed tomography
    2. 4.2 Positron emission tomography and single-photon emission computed tomography
    3. 4.3 Noise propagation in reconstruction
      1. 4.3.1 Noise variance of emission data
      2. 4.3.2 Noise variance of transmission data
      3. 4.3.3 Noise propagation in an FBP algorithm
    4. 4.4 Attenuation correction for emission tomography
      1. 4.4.1 PET
      2. 4.4.2 SPECT: Tretiak–Metz FBP algorithm for uniform attenuation
      3. 4.4.3 SPECT: Inouye’s algorithm for uniform attenuation
    5. 4.5 Mathematical expressions
      1. 4.5.1 Expression for Tretiak–Metz FBP algorithm
      2. 4.5.2 Derivation for Inouye’s algorithm
      3. 4.5.3 Rullgård’s derivative-then-backprojection algorithm for uniform attenuation
      4. 4.5.4 Novikov–Natterer FBP algorithm for nonuniform attenuation SPECT
    6. 4.6 Worked examples
    7. 4.7 Summary
    8. Problems
    9. Bibliography
  10. 5 Three-dimensional image reconstruction
    1. 5.1 Parallel line-integral data
      1. 5.1.1 Backprojection-then-filtering
      2. 5.1.2 Filtered backprojection
    2. 5.2 Parallel plane-integral data
    3. 5.3 Cone-beam data
      1. 5.3.1 Feldkamp’s algorithm
      2. 5.3.2 Grangeat’s algorithm
      3. 5.3.3 Katsevich’s algorithm
    4. 5.4 Mathematical expressions
      1. 5.4.1 Backprojection-then-filtering for parallel line-integral data
      2. 5.4.2 FBP algorithm for parallel line-integral data
      3. 5.4.3 Three-dimensional Radon inversion formula (FBP algorithm)
      4. 5.4.4 Three-dimensional backprojection-then-filtering algorithm for Radon data
      5. 5.4.5 Feldkamp’s algorithm
      6. 5.4.6 Tuy’s relationship
      7. 5.4.7 Grangeat’s relationship
      8. 5.4.8 Katsevich’s algorithm
    5. 5.5 Worked examples
    6. 5.6 Summary
    7. Problems
    8. Bibliography
  11. 6 Iterative reconstruction
    1. 6.1 Solving a system of linear equations
    2. 6.2 Algebraic reconstruction technique
    3. 6.3 Gradient descent algorithms
      1. 6.3.1 The gradient descent algorithm
      2. 6.3.2 The Landweber algorithm
      3. 6.3.3 The conjugate gradient algorithm
    4. 6.4 ML-EM algorithms
    5. 6.5 OS-EM algorithm
    6. 6.6 Noise handling
      1. 6.6.1 Analytical methods – windowing
      2. 6.6.2 Iterative methods – stopping early
      3. 6.6.3 Iterative methods – choosing pixels
      4. 6.6.4 Iterative methods – accurate modeling
    7. 6.7 Noise modeling as a likelihood function
    8. 6.8 Including prior knowledge (Bayesian)
    9. 6.9 Mathematical expressions
      1. 6.9.1 ART
      2. 6.9.2 The Landweber algorithm
      3. 6.9.3 CG algorithm
      4. 6.9.4 ML-EM
      5. 6.9.5 OS-EM
      6. 6.9.6 MAP (Green’s one-step late algorithm)
      7. 6.9.7 Matched and unmatched projector/backprojector pairs
    10. 6.10 Reconstruction using highly undersampled data
    11. 6.11 Worked examples
    12. 6.12 Summary
    13. Problems
    14. Bibliography
  12. 7 MRI reconstruction
    1. 7.1 The “M”
    2. 7.2 The “R”
    3. 7.3 The “I”
      1. 7.3.1 To obtain z-information: slice selection
      2. 7.3.2 To obtain x-information: frequency encoding
      3. 7.3.3 To obtain y-information: phase encoding
    4. 7.4 Mathematical expressions
    5. 7.5 Image reconstruction for MRI
      1. 7.5.1 Fourier reconstruction
      2. 7.5.2 Iterative reconstruction
    6. 7.6 Worked examples
    7. 7.7 Summary
    8. Problems
    9. Bibliography
  13. 8 Using FBP to perform iterative reconstruction
    1. 8.1 The Landweber algorithm: From recursive form to non-recursive form
    2. 8.2 The Landweber algorithm: From non-recursive form to closed form
    3. 8.3 The Landweber algorithm: From closed form to backprojection-then-filtering algorithm
      1. 8.3.1 Implementation of (ATA)–1 in the Fourier domain
      2. 8.3.2 Implementation of I –(I – αATA)k in the Fourier domain
      3. 8.3.3 Landweber algorithm: Backprojection-then-filtering algorithm
      4. 8.3.4 Numerical examples of the window function
    4. 8.4 The Landweber algorithm: The weighted FBP algorithm
      1. 8.4.1 Landweber algorithm: FBP without noise weighting
      2. 8.4.2 Landweber algorithm: FBP with view-based noise weighting
      3. 8.4.3 Landweber algorithm: FBP with ray-based noise weighting
    5. 8.5 FBP algorithm with quadratic constraints
      1. 8.5.1 Example of minimum norm-constrained FBP
      2. 8.5.2 Example of reference image-constrained FBP
    6. 8.6 Convolution backprojection
    7. 8.7 Non-quadratic constraints
    8. 8.8 A viewpoint from calculus of variations
    9. 8.9 Summary
    10. Problems
    11. Bibliography
  14. Index