Reconstructive Integral Geometry

Victor Palamodov

ISBN 10: 3764371293 / ISBN 13: 9783764371296

Published by Birkhäuser Basel, 2004

About this Item

Druck auf Anfrage Neuware - Printed after ordering - 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.

Seller Inventory # 9783764371296

Bibliographic Details

• Title: Reconstructive Integral Geometry
• Publisher: Birkhäuser Basel
• Publication Date: 2004
• Language: English
• ISBN 10: 3764371293
• ISBN 13: 9783764371296
• Binding: Buch
• Condition: Neu
• Dimensions: 241W x 160H x 16D millimeters 241 millimeters width by 160 millimeters height by 16 millimeters depth
• Weight: 442 grams

About this title

Synopsis

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.

Review

"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