The Radon Transform and Local Tomography

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ISBN: 9780849394928 | 0849394929
Cover: Hardcover
Copyright: 2/6/1996

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Over the past decade, the field of image processing has made tremendous advances. One type of image processing that is currently of particular interest is "tomographic imaging," a technique for computing the density function of a body, or discontinuity surfaces of this function. Today, tomography is widely used, and has applications in such fields as medicine, engineering, physics, geophysics, and security. The Radon Transform and Local Tomography clearly explains the theoretical, computational, and practical aspects of applied tomography. It includes sufficient background information to make it essentially self-contained for most readers.

Introduction
Brief Description of New Results and the Aims of the Book
Review of Some Applications of the Radon Transform
Properties of the Radon Transform and Inversion Formulas
Definitions and Properties of the Radon Transform and Related Transforms
Inversion Formulas for R
Singular Value Decomposition of the Radon Transform
Estimates in Sobolev Spaces
Inversion Formulas for the Backprojection Operator
Inversion Formulas for X-Ray Transform
Uniqueness Theorems for the Radon and X-Ray Transforms
Attenuated and Exponential Radon Transforms
Convergence Properties of the Inversion Formulas on Various Classes of Functions
Range Theorems and Reconstruction Algorithms
Range Functions for R on Smooth Functions
Range Functions for R on Sobolev Spaces
Range Theorems for R*
Range Theorem for X-Ray Transform
Numerical Solution of the Equation Rf = g with Noisy Data
Filtered Backprojection Algorithm
Other Reconstruction Algorithms
Singularities of the Radon Transform
Introduction
Singular Support of the Radon Transform
The Relation Between S and S (WE NEED A "HAT" OVER THE LAST S. See hard copy of toc for details)
The Envelopes and the Duality Law
Asymptotics of Rf Near S
Singularities of the Radon Transform: An Alternative Approach
Asymptotics of the Fourier Transform
Wave Front Sets
Singularities of X-Ray Transform
Stable Calculation of the Legendre Transform
Local Tomography
Introduction
A Family of Local Tomography Functions
Optimization of Noise Stability
Algorithm for Finding Values of Jumps of a Function Using Local Tomography
Numerical Implementation
Local Tomography for the Exponential Radon Transform
Local Tomography for the Generalized Radon Transform
Local Tomography for the Limited-Angle Data
Asymptotics of Pseudodifferential Operators, Acting on a Piecewise-Smooth Function, f, Near the Singular Support of f
Pseudolocal Tomography
Introduction
Definition of a Pseudolocal Tomography Function
Investigation of the Convergence frc(x) ¦ f(x) as r ¦ 0
More Results on Functions frc, fr, and on convergence frc ¦ f
A Family of Pseudolocal Tomography Functions
Numerical Implementation of Pseudolocal Tomography
Pseudolocal Tomography for the Exponential Radon Transform
Geometric Tomography
Basic Idea
Description of the Algorithm and Numerical Experiments
Inversion of Incomplete Tomographic Data
Inversion of Incomplete Fourier Transform Data
Filtered Backprojection Method for Inversion of the Limited-Angle Tomographic Data
The Extrapolation Problem
The Davison-Grnnbaum Algorithm
Inversion of Cone-Beam Data
Inversion of the Complete Cone-Beam Data
Inversion of Incomplete Cone-Beam Data
An Exact Algorithm for the Cone-Beam Circle Geometry
g-Ray Tomography
Radon Transform of Distributions
Main Definitions
Properties of the Test Function Spaces
Examples
Range Theorem for the Radon Transform on e'
A Definition Based on Spherical Harmonics Expansion
When Does the Radon Transform on Distributions Coincide with the Classical Radon Transform?
The Dual Radon Transform on Distributions
Abel-Type Integral Equation
The Classical Abel Equation
Abel-Type Equations
Reduction of the Equation to a More Stable One
Finding Locations and Values of Jumps of the Solution to the Abel Equation
Multidimensional Algorithm for Finding Discontinuities of Signals from Noisy Discrete Data
Introduction
Edge Detection Algorithm
Thin Line Detection Algorithm
Generalization of the Algorithms
Justification of the Edge Detection Algorithm
Justification of the Algorithm for Thin Line Detection
Justification of the General Scheme
Numerical Experiments
Proof of Auxiliary Results
Test of Randomness and Its Applications
Introduction
Consistency of Rank Test Against Change Points (Change Surfaces) Alternative
Consistency of Rank Test Against Trend in Location
Auxiliary Results
Abstract and Functional Spaces
Distribution Theory
Pseudodifferential and Fourier Integral Operators
Special Functions
Asymptotic Expansions
Linear Equations in Banach Spaces
Ill-Posed Problems
Examples of Regularization of Ill-Posed Problems
Radon Transform and PDE
Statistics
Research Problems
Bibliographical Notes
References
Index
List of Notations

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The Radon Transform and Local Tomography

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