Reconstructive Integral Geometry

Author : Victor Palamodov
Publisher : Birkhäuser
ISBN 13 : 3034879415
Total Pages : 171 pages
Book Rating : 4.15/5 ( download)

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Download or read book Reconstructive Integral Geometry written by Victor Palamodov and published by Birkhäuser. This book was released on 2012-12-06 with total page 171 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Victor Palamodov
Publisher : CRC Press
ISBN 13 : 1498710115
Total Pages : 172 pages
Book Rating : 4.14/5 ( download)

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Download or read book Reconstruction from Integral Data written by Victor Palamodov and published by CRC Press. This book was released on 2016-04-27 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reconstruction of a function from data of integrals is used for problems arising in diagnostics, including x-ray, positron radiography, ultrasound, scattering, sonar, seismic, impedance, wave tomography, crystallography, photo-thermo-acoustics, photoelastics, and strain tomography. Reconstruction from Integral Data presents both long-standing and recent mathematical results from this field in a uniform way. The book focuses on exact analytic formulas for reconstructing a function or a vector field from data of integrals over lines, rays, circles, arcs, parabolas, hyperbolas, planes, hyperplanes, spheres, and paraboloids. It also addresses range characterizations. Coverage is motivated by both applications and pure mathematics. The book first presents known facts on the classical and attenuated Radon transform. It then deals with reconstructions from data of ray (circle) integrals. The author goes on to cover reconstructions in classical and new geometries. The final chapter collects necessary definitions and elementary facts from geometry and analysis that are not always included in textbooks.

Author : Sigurdur Helgason
Publisher : Springer Science & Business Media
ISBN 13 : 1441960554
Total Pages : 309 pages
Book Rating : 4.59/5 ( download)

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Download or read book Integral Geometry and Radon Transforms written by Sigurdur Helgason and published by Springer Science & Business Media. This book was released on 2010-10-27 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 di erential 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 vibration, 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

Author : Valery V. Volchkov
Publisher : Springer Science & Business Media
ISBN 13 : 3034805721
Total Pages : 596 pages
Book Rating : 4.28/5 ( download)

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Download or read book Offbeat Integral Geometry on Symmetric Spaces written by Valery V. Volchkov and published by Springer Science & Business Media. This book was released on 2013-01-30 with total page 596 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book demonstrates the development of integral geometry on domains of homogeneous spaces since 1990. It covers a wide range of topics, including analysis on multidimensional Euclidean domains and Riemannian symmetric spaces of arbitrary ranks as well as recent work on phase space and the Heisenberg group. The book includes many significant recent results, some of them hitherto unpublished, among which can be pointed out uniqueness theorems for various classes of functions, far-reaching generalizations of the two-radii problem, the modern versions of the Pompeiu problem, and explicit reconstruction formulae in problems of integral geometry. These results are intriguing and useful in various fields of contemporary mathematics. The proofs given are “minimal” in the sense that they involve only those concepts and facts which are indispensable for the essence of the subject. Each chapter provides a historical perspective on the results presented and includes many interesting open problems. Readers will find this book relevant to harmonic analysis on homogeneous spaces, invariant spaces theory, integral transforms on symmetric spaces and the Heisenberg group, integral equations, special functions, and transmutation operators theory.

Author : Andrew Markoe
Publisher : American Mathematical Soc.
ISBN 13 : 0821837559
Total Pages : 176 pages
Book Rating : 4.59/5 ( download)

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Download or read book Integral Geometry and Tomography written by Andrew Markoe and published by American Mathematical Soc.. This book was released on 2006 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume consists of a collection of papers that brings together fundamental research in Radon transforms, integral geometry, and tomography. It grew out of the Special Session at a Sectional Meeting of the American Mathematical Society in 2004. The book contains very recent work of some of the top researchers in the field. The articles in the book deal with the determination of properties of functions on a manifold by integral theoretic methods, or by determining the geometricstructure of subsets of a manifold by analytic methods. Of particular concern are ways of reconstructing an unknown function from some of its projections. Radon transforms were developed at the beginning of the twentieth century by researchers who were motivated by problems in differential geometry,mathematical physics, and partial differential equations. Later, medical applications of these transforms produced breakthroughs in imaging technology that resulted in the 1979 Nobel Prize in Physiology and Medicine for the development of computerized tomography. Today the subject boasts substantial cross-disciplinary interactions, both in pure and applied mathematics as well as medicine, engineering, biology, physics, geosciences, and industrial testing. Therefore, this volume should be ofinterest to a wide spectrum of researchers both in mathematics and in other fields.

Author : Semyon Alesker
Publisher : Springer
ISBN 13 : 3034808747
Total Pages : 121 pages
Book Rating : 4.43/5 ( download)

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Download or read book Integral Geometry and Valuations written by Semyon Alesker and published by Springer. This book was released on 2014-10-09 with total page 121 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last years there has been significant progress in the theory of valuations, which in turn has led to important achievements in integral geometry. This book originated from two courses delivered by the authors at the CRM and provides a self-contained introduction to these topics, covering most of the recent advances. The first part, by Semyon Alesker, provides an introduction to the theory of convex valuations with emphasis on recent developments. In particular, it presents the new structures on the space of valuations discovered after Alesker's irreducibility theorem. The newly developed theory of valuations on manifolds is also described. In the second part, Joseph H. G. Fu gives a modern introduction to integral geometry in the sense of Blaschke and Santaló. The approach is new and based on the notions and tools presented in the first part. This original viewpoint not only enlightens the classical integral geometry of euclidean space, but it also allows the computation of kinematic formulas in other geometries, such as hermitian spaces. The book will appeal to graduate students and interested researchers from related fields including convex, stochastic, and differential geometry. ​

Author : Izrailʹ Moiseevich Gelʹfand
Publisher : American Mathematical Soc.
ISBN 13 : 9780821829325
Total Pages : 136 pages
Book Rating : 4.27/5 ( download)

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Download or read book Selected Topics in Integral Geometry written by Izrailʹ Moiseevich Gelʹfand and published by American Mathematical Soc.. This book was released on 2003 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: The miracle of integral geometry is that it is often possible to recover a function on a manifold just from the knowledge of its integrals over certain submanifolds. The founding example is the Radon transform, introduced at the beginning of the 20th century. Since then, many other transforms were found, and the general theory was developed. Moreover, many important practical applications were discovered. The best known, but by no means the only one, being to medical tomography. This book is a general introduction to integral geometry, the first from this point of view for almost four decades. The authors, all leading experts in the field, represent one of the most influential schools in integral geometry. The book presents in detail basic examples of integral geometry problems, such as the Radon transform on the plane and in space, the John transform, the Minkowski-Funk transform, integral geometry on the hyperbolic plane and in the hyperbolic space, the horospherical transform and its relation to representations of $SL(2,\mathbb C)$, integral geometry on quadrics, etc. The study of these examples allows the authors to explain important general topics of integral geometry, such as the Cavalieri conditions, local and nonlocal inversion formulas, and overdetermined problems in integral geometry. Many of the results in the book were obtained by the authors in the course of their career-long work in integral geometry. This book is suitable for graduate students and researchers working in integral geometry and its applications.

Author : Izrail_ Moiseevich Gel_fand
Publisher : American Mathematical Soc.
ISBN 13 : 9780821898048
Total Pages : 192 pages
Book Rating : 4.43/5 ( download)

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Download or read book Selected Topics in Integral Geometry written by Izrail_ Moiseevich Gel_fand and published by American Mathematical Soc.. This book was released on 2003-09-02 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: The miracle of integral geometry is that it is often possible to recover a function on a manifold just from the knowledge of its integrals over certain submanifolds. The founding example is the Radon transform, introduced at the beginning of the 20th century. Since then, many other transforms were found, and the general theory was developed. Moreover, many important practical applications were discovered, the best known, but by no means the only one, being to medical tomography. The present book is a general introduction to integral geometry, the first from this point of view for almost four decades. The authors, all leading experts in the field, represent one of the most influential schools in integral geometry. The book presents in detail basic examples of integral geometry problems, such as the Radon transform on the plane and in space, the John transform, the Minkowski-Funk transform, integral geometry on the hyperbolic plane and in the hyperbolic space, the horospherical transform and its relation to representations of $SL(2,\mathbb C)$, integral geometry on quadrics, etc. The study of these examples allows the authors to explain important general topics of integral geometry, such as the Cavalieri conditions, local and nonlocal inversion formulas, and overdetermined problems. Many of the results in the book were obtained by the authors in the course of their career-long work in integral geometry.

Author : Massimo Picardello
Publisher : CRC Press
ISBN 13 : 148228569X
Total Pages : 180 pages
Book Rating : 4.97/5 ( download)

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Download or read book Harmonic Analysis and Integral Geometry written by Massimo Picardello and published by CRC Press. This book was released on 2019-05-08 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: Comprising a selection of expository and research papers, Harmonic Analysis and Integral Geometry grew from presentations offered at the July 1998 Summer University of Safi, Morocco-an annual, advanced research school and congress. This lively and very successful event drew the attendance of many top researchers, who offered both individual lecture

Author : De-lin Ren
Publisher : World Scientific
ISBN 13 : 9789810211073
Total Pages : 260 pages
Book Rating : 4.74/5 ( download)

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Download or read book Topics in Integral Geometry written by De-lin Ren and published by World Scientific. This book was released on 1994 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: Essentials of integral geometry in a homogenous space are presented and the focus is on the basic results and applications. This book provides the readers with new findings, some being published for the first time and serves as an excellent graduate text.