Abel transform

Abstract: We report a new variation of the velocity map ion imaging method that allows the central section of the photofragment ion cloud to be recorded exclusively. The relevant speed and angular distributions for a molecular photodissociation or scattering event may therefore be obtained without need to utilize inversion methods such as the inverse Abel transform. In contrast to the recently reported “slicing” technique of Kitsopoulos and co-workers [C. R. Gebhardt et al., Rev. Sci. Instrum. 72, 3848 (2001)], our method makes no use of grids or pulsed electric fields which can distort the photofragment cloud and therefore compromise the resolution of velocity mapping. We find that by operating a multilens velocity mapping assembly at low voltages, the ion cloud stretches in the acceleration region owing to the kinetic energy release in the fragments. Furthermore, this inherent stretching is sufficient to allow the central section of the distribution to be recorded exclusively by application of a narrow time gate ...

Abstract: 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) Ae f(x) as r Ae 0 More Results on Functions frc, fr, and on convergence frc Ae 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-Grunbaum 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

Abstract: Basic theory and representation formulas.- Applications of Abel's original integral equation: Determination of potentials.- Applications of a transformed abel integral equation.- Smoothing properties of the abel operators.- Existence and uniqueness theorems.- Relations between abel transform and other integral transforms.- Nonlinear abel integral equations of second kind.- Illposedness and stabilization of linear abel integral equations of first kind.- On numerical treatment of first kind abel integral equations.