The Universality of the Radon Transform

Hardcover | May 11, 2004

byLeon Ehrenpreis

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Written by a leading scholar in mathematics, this monograph discusses the Radon transform, a field that has wide ranging applications to X-ray technology, partial differential equations, nuclear magnetic resonance scanning, and tomography. In this book, Ehrenpreis focuses on recent researchand highlights the strong relationship between high-level pure mathematics and applications of the Radon transform to areas such as medical imaging.The first part of the book discusses parametric and nonparametric Radon transforms, Harmonic Functions and Radon transform on Algebraic Varieties, nonlinear Radon and Fourier transforms, Radon transform on groups, and Radon transform as the interrelation of geometry and analysis. The later partsdiscuss the extension of solutions of differential equations, Periods of Eisenstein and Poincare, and some problems of integral geometry arising in tomography. Examples and proofs are provided throughout the book to aid the reader's understanding.This is the latest title in the Oxford Mathematical Monographs, which includes texts and monographs covering many topics of current research interest in pure and applied mathematics. Other titles include: Carbone and Semmes: A graphic apology for symmetry and implicitness; Higson and Roe: AnalyticK-Homology; Iwaniec and Martin: Geometric Function Theory and Nonlinear Analysis; Lyons and Qian: System Control and Rough Paths. Also new in paperback Johnson and Lapidus: The Feynman Integral and Feynman's Operational Calculus; Donaldson and Kronheimer: The geometry of four-manifolds.

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Written by a leading scholar in mathematics, this monograph discusses the Radon transform, a field that has wide ranging applications to X-ray technology, partial differential equations, nuclear magnetic resonance scanning, and tomography. In this book, Ehrenpreis focuses on recent researchand highlights the strong relationship betwee...

Leon Ehrenpreis is a Professor of Mathematics, Temple University, USA.

other books by Leon Ehrenpreis

Format:HardcoverDimensions:740 pages, 9.21 × 6.14 × 1.61 inPublished:May 11, 2004Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198509782

ISBN - 13:9780198509783

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Table of Contents

PrefaceLeon Ehrenpreis: Chapters I-XLeon Ehrenpreis: I. IntroductionI.1 Functions, Geometry and SpacesI.2 Parametric Radon transformI.3 Geometry of the nonparametric Radon transformI.4 Parametrization problemsI.5 Differential equationsI.6 Lie groupsI.7 Fourier transform on varieties: The projection slice theorem and the Poisson summation FormulaI.8 Tensor products and direct integralsII. The nonparametric Radon transformII.1 Radon transform and Fourier transformII.2 Tensor products and their topologyII.3 Support conditionsIII. Harmonic functions in RnIII.1 Algebraic theoryIII.2 Analytic theoryIII.3 Fourier series expansions on spheresIII.4 Fourier expansions on hyperbolasIII.5 Deformation theoryIV. Harmonic functions and Radon transform on algebraic varietiesIV.1 Algebraic theory and finite Cauchy problemIV.2 The compact Watergate problemIV.3 The noncompact Watergate problemV. The nonlinear Radon and Fourier transformsV.1 Nonlinear Radon transformV.2 Nonconvex support and regularityV.3 Wave front setV.4 Microglobal analysisVI. The parametric Radon transformVI.1 The John and invariance equationsVI.2 Characterization by John equationsVI.3 Non-Fourier analysis approachVI.4 Some other parametric linear Radon transformsVII. Radon transform on groupsVII.1 Affine and projection methodsVII.2 The nilpotent (horocyclic) Radon transform on G/KVIII. Radon transform as the interrelation of geometry and analysisVIII.1 Integral geometry and differential equationsVIII.2 The Poisson summation formula and exotic intertwiningVIII.3 The Euler-MacLaurin summation formulaIX. Extension of solutions of differential equationsIX.1 Formulation of the problemIX.2 Hartogs-Lewy extensionIX.3 Wave front sets and the Caucy problemX. Periods of Eisenstein and Poincare seriesX.1 The Lorentz group, Minowski geometry and a nonlinear projection-slice theoremX.2 Spreads and cylindrical coordinates in Minowski geometryX.3 Eisenstein series and their periodsX.4 Poincareseries and their periodsX.5 Hyperbolic Eisenstein and Poincare seriesX.6 The four dimensional representationX.7 Higher dimensional groupsBibiliography of Chapters I-XXI. Peter Kuchment and Eric Todd Quinto: Some problems of integral geometry arising in tomographyXI.1 IntroductionXI.2 X-ray tomographyXI.3 Attenuated and exponential Radon transformsXI.4 Hyperbolic integral geometry and electrical impedance tomographyIndex