Index of Peter Toft's Ph.D. thesis

Keywords: Radon transform, Hough transform, Reconstruction, PET, Linux.


Resume paa dansk (Abstract in Danish)



Style Conventions

PART I Curve Parameter Detection using the Radon Transform

chapter 1 Slant Stacking

section 1.1 Why Consider the Radon Transform
section 1.2 Defining the (p,tau) Radon Transform
section 1.3 Basic Properties of the (p,tau) Radon Transform
subsection 1.3.1 Linearity
subsection 1.3.2 Shifting
subsection 1.3.3 Scaling
subsection 1.3.4 The Point Source
subsection 1.3.5 The Line
section 1.4 Discrete Slant Stacking
subsection 1.4.1 Nearest Neighbour Interpolation
subsection 1.4.2 Linear Interpolation
subsection 1.4.3 Sinc Interpolation
subsection 1.4.4 Sampling Properties of the Discrete Radon Transform
section 1.5 Discrete Radon Transform of a Discrete Line
subsection 1.5.1 Comparison of Different Interpolation Methods
section 1.6 Discrete Radon Transform of Points
section 1.7 Slant Stacking and Images with Steep Lines
section 1.8 Detection of Lines Convolved with a Wavelet
section 1.9 Summary

chapter 2 The Normal Radon Transform

section 2.1 Defining the (rho,theta) Radon Transform
subsection 2.1.1 The Point Source
subsection 2.1.2 The (rho,theta) Radon Transform of a Line
section 2.2 The Discrete (rho,theta) Radon Transform
subsection 2.2.1 Sampling Properties of the (rho,theta) Radon Transform
subsection 2.2.2 The Discrete (rho,theta) Radon Transform of Several Lines
subsection 2.2.3 The Discrete (rho,theta) Radon Transform of Points
section 2.3 Summary

chapter 3 The Hough Transform

section 3.1 The (p,tau) Hough Transform
subsection 3.1.1 Line Detection Using The Hough Transform
subsection 3.1.2 Choosing Sampling Parameters with the (p,tau) Hough Transform
section 3.2 The (rho,theta) Hough Transform
subsection 3.2.1 Choosing sampling parameters with the (rho,theta) Hough Transform
subsection 3.2.2 Comparison Between Different Optimization Strategies
subsection 3.2.3 Other Hough-like Algorithms
section 3.3 Summary

chapter 4 The FCE-Algorithm

section 4.1 The Generalized Radon Transform
subsection 4.1.1 The Continuous Generalized Radon Transform
subsection 4.1.2 The Discrete Generalized Radon Transform
section 4.2 Image Point Mapping
section 4.3 Parameter Domain Samplinm
section 4.4 Parameter Domain Blurring
section 4.5 The Fast Curve Estimation Algorithm
section 4.6 The Hyperbolic Transformation Curve
subsection 4.6.1 Clusters in the Hyperbolic case
subsection 4.6.2 An Example with Eight Hyperbolas
subsection 4.6.3 An Example with a Noise Corrupted Synthetic CMP-gather
section 4.7 Summary

chapter 5 Curve Parameter Estimation in Noisy Images

section 5.1 Lines with Wiggles
section 5.2 A ``Fuzzy'' Radon Transform
section 5.3 Detection of Curves in Noisy Images
subsection 5.3.1 The Generalized Radon Transform
subsection 5.3.2 Curve Detection using the Generalized Radon Transform
subsection 5.3.3 Discussion
subsection 5.3.4 An Example of Line Detection in a very Noisy Image
section 5.4 Summary

PART II The Inverse Radon Transform and PET

chapter 6 Introduction to Computerized Tomography

section 6.1 Fundamental Theory of the CT-Scanner
section 6.2 The PET Scanner
subsection 6.2.1 Correction for Attenuation in PET
section 6.3 Summary

chapter 7 Inversion of the Radon Transform

section 7.1 The Fourier Slice Theorem
section 7.2 Filtered Backprojection
section 7.3 Filtering after Backprojection
section 7.4 Calculation using Operators
subsection 7.4.1 The Zero Frequency Problem
section 7.5 Sampling Considerations
section 7.6 Inversion of the (p,tau) Radon Transform
subsection 7.6.1 Fourier Slice Theorem
subsection 7.6.2 Filtered Backprojection
subsection 7.6.3 An Inversion Formula using the Hilbert Transform
subsection 7.6.4 Filtering after Backprojection
section 7.7 Summary

chapter 8 Numerical Implementation of Direct Reconstruction Algorithms

section 8.1 Using the DFT to Approximate the Fourier Transformation
subsection 8.1.1 The FFT Applied for Filtering
section 8.2 Discrete Implementation of Backprojection
section 8.3 Implementation of Filtering after Backprojection
section 8.4 Implementation of The Fourier Slice Theorem
subsection 8.4.1 Non-linear sampling of the Radon domain
section 8.5 Examples Using Direct Reconstruction Algorithms
subsection 8.5.1 Reconstruction using Different Methods
subsection 8.5.2 Sinogram with very few Samples in the Angular Direction
subsection 8.5.3 Reconstruction with Varying Image Size
subsection 8.5.4 Reconstruction into a Oversampled Image
subsection 8.5.5 Noise in the Sinogram
section 8.6 Summary

chapter 9 Reconstruction Algorithms Based on Linear Algebra

section 9.1 From the Radon Transform to Linear Algebra based Reconstruction
section 9.2 The Calculation of Matrix Elements
subsection 9.2.1 Pixel Oriented Nearest Neighbour Approximation
subsection 9.2.2 Discrete Radon Transform
subsection 9.2.3 First Order Pixel Oriented Interpolation Strategy
subsection 9.2.4 The Sinc Interpolation Strategy
section 9.3 Duality between Matrix Operations and the Radon Transform
section 9.4 Regularization and Constraints
section 9.5 Singular Value Decomposition
section 9.6 Iterative Reconstruction using ART
subsection 9.6.1 ART with Constraints
subsection 9.6.2 Initialization
section 9.7 Multiplicative ART
section 9.8 The EM algorithm
section 9.9 The Conjugate Gradient Method
section 9.10 Accelerated Iterative Reconstruction
subsection 9.10.1 A Fast 2D Iterative Reconstruction Package
section 9.11 Examples Using Iterative Reconstruction Algorithms
subsection 9.11.1 A Small Reconstruction Example
subsection 9.11.2 A Larger Reconstruction Example
subsection 9.11.3 Comparison with Different Noise Levels
subsection 9.11.4 Reconstruction Using Constraints
subsection 9.11.5 Reconstruction Using Regularization
section 9.12 Summary

chapter 10 The 3D Radon Transform for Lines

section 10.1 Lines in a Three Dimensional Space
subsection 10.1.1 Limiting the 3D line parameters
section 10.2 Fourier Slice Reconstruction in 3D
section 10.3 Backprojection Based Inversion of Line Integrals in 3D
section 10.4 Filtering after Backprojection of Line Integrals in 3D
section 10.5 Filtered Backprojection of Line Integrals in 3D
section 10.6 Reconstruction Scheme for 3D Multi Ring PET Scanners
section 10.7 A 3D Reconstruction Package
section 10.8 Implementation of the 3D Reconstruction Methods
subsection 10.8.1 Implementation of the Backprojection Operator
subsection 10.8.2 Implementation of the Radon Transform Operator
section 10.9 Examples of 3D Reconstructed Volumes
subsection 10.9.1 Reconstruction of a Ball
subsection 10.9.2 Reconstruction of the Mickey Phantom
section 10.10 Summary

chapter 11 Noise Contributions from Blank, Transmission and Emission Scans in PET

section 11.1 Introduction
section 11.2 Theory
subsection 11.2.1 One Emission and One Transmission Scan
subsection 11.2.2 Two Emission Scans
subsection 11.2.3 Two Emission and Two Transmission Scans
subsection 11.2.4 Zeroes in the Transmission Sinogram
subsection 11.2.5 Limitations
section 11.3 Overview of the Measurements
subsection 11.3.1 Phantom studies
subsection 11.3.2 Human studies
subsection 11.3.3 Fitting data
section 11.4 Results
section 11.5 Optimization
section 11.6 Discussion of the Results
section 11.7 Summary

Conclusion and Topics for Further Research

PART III Appendices

chapter A The Dirac Delta Function

chapter B Properties of the Normal Radon Transformation

section B.1 Basic Properties of the Radon Transform
subsection B.1.1 Linearity
subsection B.1.2 Shifting
subsection B.1.3 Rotation
subsection B.1.4 Scaling
subsection B.1.5 Convolution
section B.2 The Shepp-Logan Phantom Brain
section B.3 Analytical Radon Transform of Primitives
subsection B.3.1 The Circular Disc
subsection B.3.2 The Square
subsection B.3.3 The Triangle
subsection B.3.4 The Gaussian Bell
subsection B.3.5 The Pyramid

chapter C Usage of the 2D Program Packages

section C.1 The Analytical Sinogram Program ``RadonAna''
section C.2 The Direct Reconstruction Program ``iradon''
section C.3 The Fast Iterative Reconstruction Program ``it''

chapter D The Three Dimensional Radon Transformation

section D.1 The Three Dimensional Fourier Slice Theorem
section D.2 Filtered Backprojection in 3D
section D.3 Connection between the 3D plane integrals and 3D line integrals

chapter E Properties of the 3D line Radon Transform

section E.1 Basic Properties of the 3D line Radon Transform
subsection E.1.1 Linearity
subsection E.1.2 Translation
subsection E.1.3 Rotation and Scaling
section E.2 Analytical Radon Transformation of Primitives
subsection E.2.1 The ball
subsection E.2.2 The Gaussian bell

chapter F Usage of the 3D Reconstruction Tools

section F.1 The 3D Reconstruction Program ``Recon3D''
section F.2 The Analytical Sinogram Program ``3D_RadonAna''

PART IV Papers

chapter G Fast Radon Transform for Detection of Seismic Reflections

chapter H Fast Curve Estimation Using Pre-Conditioned Generalized Radon Transform

chapter I Using the Generalized Radon Transform for Detection of Curves in Noisy Images

chapter J Estimation of the Noise Contributions from Blank, Transmission and Emission Scans in PET

chapter K Estimation of the Noise Contributions from Blank, Transmission and Emission Scans in PET

chapter L A very fast Implementation of 2D Iterative Reconstruction Algorithms

chapter M Accelerated 2D Iterative Reconstruction

chapter N Mean Field Reconstruction with Snaky Edge Hints

chapter O Detection of Lines with Wiggles using the Radon Transform

Thesis Bibliography