Integral Geometry and Radon Transforms
Title | Integral Geometry and Radon Transforms PDF eBook |
Author | Sigurdur Helgason |
Publisher | Springer Science & Business Media |
Pages | 309 |
Release | 2010-11-17 |
Genre | Mathematics |
ISBN | 1441960546 |
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
Stochastic and Integral Geometry
Title | Stochastic and Integral Geometry PDF eBook |
Author | R.V. Ambartzumian |
Publisher | Springer Science & Business Media |
Pages | 135 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 9400939213 |
Reconstructive Integral Geometry
Title | Reconstructive Integral Geometry PDF eBook |
Author | Victor Palamodov |
Publisher | Springer Science & Business Media |
Pages | 184 |
Release | 2004-08-20 |
Genre | Mathematics |
ISBN | 9783764371296 |
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.
Geometric Tomography
Title | Geometric Tomography PDF eBook |
Author | Richard J. Gardner |
Publisher | Cambridge University Press |
Pages | 7 |
Release | 2006-06-19 |
Genre | Mathematics |
ISBN | 0521866804 |
Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. It overlaps with convex geometry, and employs many tools from that area including integral geometry. It also has connections to geometric probing in robotics and to stereology. The main text contains a rigorous treatment of the subject starting from basic concepts and moving up to the research frontier: seventy-two unsolved problems are stated. Each chapter ends with extensive notes, historical remarks, and some biographies. This comprehensive work will be invaluable to specialists in geometry and tomography; the opening chapters can also be read by advanced undergraduate students.
Integral Geometry and Convolution Equations
Title | Integral Geometry and Convolution Equations PDF eBook |
Author | V.V. Volchkov |
Publisher | Springer Science & Business Media |
Pages | 466 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 9401000239 |
Integral geometry deals with the problem of determining functions by their integrals over given families of sets. These integrals de?ne the corresponding integraltransformandoneofthemainquestionsinintegralgeometryaskswhen this transform is injective. On the other hand, when we work with complex measures or forms, operators appear whose kernels are non-trivial but which describe important classes of functions. Most of the questions arising here relate, in one way or another, to the convolution equations. Some of the well known publications in this ?eld include the works by J. Radon, F. John, J. Delsarte, L. Zalcman, C. A. Berenstein, M. L. Agranovsky and recent monographs by L. H ̈ ormander and S. Helgason. Until recently research in this area was carried out mostly using the technique of the Fourier transform and corresponding methods of complex analysis. In recent years the present author has worked out an essentially di?erent methodology based on the description of various function spaces in terms of - pansions in special functions, which has enabled him to establish best possible results in several well known problems.
Integral Geometry of Tensor Fields
Title | Integral Geometry of Tensor Fields PDF eBook |
Author | V. A. Sharafutdinov |
Publisher | Walter de Gruyter |
Pages | 277 |
Release | 2012-01-02 |
Genre | Mathematics |
ISBN | 3110900092 |
The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology.
Integral Geometry and Valuations
Title | Integral Geometry and Valuations PDF eBook |
Author | Semyon Alesker |
Publisher | Springer |
Pages | 121 |
Release | 2014-10-09 |
Genre | Mathematics |
ISBN | 3034808747 |
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.