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© 2011

Integral Geometry and Radon Transforms

Benefits

  • Presents material accessible to advanced undergraduates

  • Contains the required expository material on Lie group theory

  • Features self-contained chapters with bibliographical notes, exercises, and further results (with documentation)

Textbook

Table of contents

  1. Front Matter
    Pages 1-1
  2. Sigurdur Helgason
    Pages 1-62
  3. Sigurdur Helgason
    Pages 63-109
  4. Sigurdur Helgason
    Pages 111-169
  5. Sigurdur Helgason
    Pages 171-184
  6. Sigurdur Helgason
    Pages 209-219
  7. Sigurdur Helgason
    Pages 265-274
  8. Sigurdur Helgason
    Pages E1-E1
  9. Back Matter
    Pages 271-271

About this book

Introduction

In this text, integral geometry deals with Radon’s problem of representing a function on a manifold in terms of its integrals over certain submanifolds—hence the term the Radon transform. Examples and far-reaching generalizations lead to fundamental problems such as: (i) injectivity, (ii) inversion formulas, (iii) support questions, (iv) applications (e.g., to tomography, partial differential equations and group representations). For the case of the plane, the inversion theorem and the support theorem have had major applications in medicine through tomography and CAT scanning. While containing some recent research, the book is aimed at beginning graduate students for classroom use or self-study. A number of exercises point to further results with documentation. From the reviews: “Integral Geometry is a fascinating area, where numerous branches of mathematics meet together. The contents of the book is concentrated around the duality and double fibration, which is realized through the masterful treatment of a variety of examples. The book is written by an expert, who has made fundamental contributions to the area.” —Boris Rubin, Louisiana State University

Keywords

Homogeneous spaces in duality Manifolds and lie groups Radio astronomy Radon transform Spaces of constant curvature Topology of spaces X-ray tranform on symmetric spaces

Authors and affiliations

  1. 1., Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

About the authors

Dr. Sigurdur Helgason has authored or co-authered 9 books and 94 papers. He has received numerous honors and awards, including the Steele Prize, awarded by the American Mathematical Society in 1988. He has been a professor at MIT since 1960.

Bibliographic information

Reviews

From the reviews:

“Beginning graduate students and interested nonspecialists will gain from the book because of its clear exposition and comprehensive nature. Practitioners of integral geometry will find it a valuable reference with complete and clear proofs as well as specialized items of interest … . This book is a well-written and beautiful introduction to integral geometry from the perspective of group actions. It has valuable thought provoking exercises. … It should be read by anyone who would like to learn more about integral geometry.” (Fulton Gonzalez and Eric Todd Quinto, Bulletin of the American Mathematical Society, Vol. 50 (4), October, 2013)

“This specialist book in the field of geometric analysis is intended for graduate students and professionals. … The field has several applications including medical X-ray technology and tomography. The nine-chapter volume devotes a chapter to these technologies and other applications. The book is largely self-contained, and includes a chapter at the end with necessary background material … . The volume includes a substantial bibliography, and each chapter concludes with bibliographic notes. Summing Up: Recommended. Graduate students and above.” (D. Z. Spicer, Choice, Vol. 48 (11), August, 2011)

“Integral Geometry and Radon Transforms is truly a pedagogical contribution by a master in this field, aiming at providing proper background – and then some – for entry into real work on the topics he brings into the spotlight. … every page is filled with serious mathematics, and Helgason provides a lot of commentary and references that ought to be pursued by those wishing to enter the field. It is a marvelous example of fine scholarship.” (Michael Berg, The Mathematical Association of America, January, 2011)

“The book under review is devoted to integral operators of geometric nature. The monograph deals with the problem of representing a function on a manifold in terms of its integrals over certain submanifolds – hence the term Radon transform. … The book will be of interest for both students and specialists in integral geometry, analysis on homogeneous spaces and partial differential equations. It may serve as a source for further research in the area.” (Valery Vladimirovich Volchkov, Zentralblatt MATH, Vol. 1210, 2011)