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Reconstructive Integral Geometry

  • Conference proceedings
  • © 2004

Overview

Authors:
  1. Victor Palamodov

    1. School of Mathematics, Tel Aviv University, Tel Aviv, Israel

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  • Covers a gap in the literature which was caused by the fast development in the field over the last 15-20 years
  • Addressed to researchers in both pure and applied mathematics

Part of the book series: Monographs in Mathematics (MMA, volume 98)

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Table of contents (9 papers)

  1. Front Matter

    Pages i-xii
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  2. Distributions and Fourier Transform

    • Victor Palamodov
    Pages 1-28

  3. Radon Transform

    • Victor Palamodov
    Pages 29-41

  4. The Funk Transform

    • Victor Palamodov
    Pages 43-64

  5. Reconstruction from Line Integrals

    • Victor Palamodov
    Pages 65-91

  6. Flat Integral Transform

    • Victor Palamodov
    Pages 93-104

  7. Incomplete Data Problems

    • Victor Palamodov
    Pages 105-113

  8. Spherical Transform and Inversion

    • Victor Palamodov
    Pages 115-134

  9. Algebraic Integral Transform

    • Victor Palamodov
    Pages 135-152

  10. Notes

    • Victor Palamodov
    Pages 153-156

  11. Back Matter

    Pages 157-164
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Keywords

About this book

One hundred years ago (1904) Hermann Minkowski [58] posed a problem: to re­ 2 construct an even function I on the sphere 8 from knowledge of the integrals MI (C) = fc Ids over big circles C. Paul Funk found an explicit reconstruction formula for I from data of big circle integrals. Johann Radon studied a similar problem for the Eu­ clidean plane and space. The interest in reconstruction problems like Minkowski­ Funk's and Radon's has grown tremendously in the last four decades, stimulated by the spectrum of new modalities of image reconstruction. These are X-ray, MRI, gamma and positron radiography, ultrasound, seismic tomography, electron mi­ croscopy, synthetic radar imaging and others. The physical principles of these methods are very different, however their mathematical models and solution meth­ ods have very much in common. The umbrella name reconstructive integral geom­ etryl is used to specify the variety of these problems and methods. The objective of this book is to present in a uniform way the scope of well­ known and recent results and methods in the reconstructive integral geometry. We do not touch here the problems arising in adaptation of analytic methods to numerical reconstruction algorithms. We refer to the books [61], [62] which are focused on these problems. Various aspects of interplay of integral geometry and differential equations are discussed in Chapters 7 and 8. The results presented here are partially new.

Reviews

"This book is an excellent overview of the field of integral geometry with emphasis on the functional analytic and differential geometric aspects. The author proves theorems for some of the most important Radon transforms, including transforms on hyperplanes, k-planes, lines, and spheres, and he investigates incomplete (limited) data problems including microlocal analytic issues…This book contains many treasures in integral geometry…and it belongs on the shelf of any analyst or geometer who would like to see how deep functional analysis and differential geometry are used to solve important problems in integral geometry." —Mathematical Reviews

Authors and Affiliations

  • School of Mathematics, Tel Aviv University, Tel Aviv, Israel

    Victor Palamodov

Bibliographic Information

  • Book Title: Reconstructive Integral Geometry

  • Authors: Victor Palamodov

  • Series Title: Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-0348-7941-5

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Basel AG 2004

  • Hardcover ISBN: 978-3-7643-7129-6Published: 20 August 2004

  • Softcover ISBN: 978-3-0348-9629-0Published: 14 October 2012

  • eBook ISBN: 978-3-0348-7941-5Published: 06 December 2012

  • Series ISSN: 1017-0480

  • Series E-ISSN: 2296-4886

  • Edition Number: 1

  • Number of Pages: XII, 164

  • Topics: Integral Transforms, Operational Calculus, Fourier Analysis

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