This book covers facts and methods for the reconstruction of a function in a real affine or projective space from data of integrals, particularly over lines, planes, and spheres. Recent results stress explicit analytic methods. Coverage includes the relations between algebraic integral geometry and partial differential equations. The first half of the book includes the ray, the spherical mean transforms in the plane or in 3-space, and inversion from incomplete data.
Reconstructive Integral Geometry
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Editorial Reviews
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"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
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4.0 out of 5 stars
For those who want to look more deeply into tomography,
By San Diego Tomographer "TxBR" (San Diego, CA) - See all my reviews
This review is from: Reconstructive Integral Geometry (Monographs in Mathematics) (Hardcover)
This book is an excellent bridge to the more advanced mathematical literature, and a good follow up to Natterer's review of the subject. The contributions of Gelfand and his school to this beautiful subject are deep and wide ranging, and well worth pursuing.
Problems with typos in this edition, and skimpy detail on some of the proofs do subtract somewhat from the overall rating, however. I hope the author will produce a second edition.
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Inside This Book
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First Sentence:
The Riemann theory of integration in a real vector space of finite dimension is sufficient for our purposes except for Section 1.7, where we outline the Fourier-Plancherel theory. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
audible zone, nonlinear artifacts, singular fundamental solution, symmetric differential forms, spherical transform, ray transform, reconstruction formula, inversion formulae, integral geometry, spheres tangent, completeness condition, tangent field, line transform, compact support, exterior problem, integral transform, reconstruction problem Key Phrases - Capitalized Phrases (CAPs): (learn more)
Algebraic Integral Transform New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
The Riemann theory of integration in a real vector space of finite dimension is sufficient for our purposes except for Section 1.7, where we outline the Fourier-Plancherel theory. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
audible zone, nonlinear artifacts, singular fundamental solution, symmetric differential forms, spherical transform, ray transform, reconstruction formula, inversion formulae, integral geometry, spheres tangent, completeness condition, tangent field, line transform, compact support, exterior problem, integral transform, reconstruction problem Key Phrases - Capitalized Phrases (CAPs): (learn more)
Algebraic Integral Transform New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 6
books:
See all 6 books this book cites
1
book
cites this book:
- Methods of Mathematical Physics, Vol. 1 by Richard Courant in Back Matter
- Mathematical Problems of Tomography (Translations of Mathematical Monographs) by Gelfand in Back Matter
- Mathematical Methods in Image Reconstruction (Monographs on Mathematical Modeling and Computation) by Frank Natterer in Back Matter
- Introduction to the Mathematics of Medical Imaging by Charles L. Epstein in Back Matter
- Integral Geometry of Tensor Fields (Inverse and Ill-Posed Problems) by V. A. Sharafutdinov in Back Matter
See all 6 books this book cites
- The Radon Transform, Inverse Problems, and Tomography (Proceedings of Symposia in Applied Mathematics) by Gestur Olafsson and Eric Todd Quinto (Editor) on page 22
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