Palamodov Reconstruction from Integral Data

Erscheinungsjahr 2016
ISBN: 978-1-4987-1011-4
Verlag: CRC Press
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 184 Seiten

Reihe: Monographs and Research Notes in Mathematics

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.

Zielgruppe

This book is intended for professional mathematicians, researchers, and students interested in geometry. It will also be useful to professional physicists, researchers, and students.

Autoren/Hrsg.

Palamodov, Victor

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

Radon Transform

Radon Transform and Inversion

Range Conditions and Frequency Analysis

Support Theorem

Reconstruction of Functions from Attenuated Integrals

Reconstruction of Differential Forms

Ray and Line Integral Transforms

Introduction

Reconstruction from Line Integrals

Range Conditions

Shift-Invariant FBP Reconstruction

Backprojection Filtration Method

Tuy’s Regularized Method

Ray Integrals of Differential Forms

Symmetric Tensors and Differentials

Reconstruction from Ray Integrals

Factorization Method

Factorable Maps

Spaces of Constant Curvature

Funk Transform on the Orthogonal Group

Reconstruction from Non-Redundant Data

Range Conditions

General Method of Reconstruction

Geometric Integral Transforms

Reconstruction
Integral Transforms with Weights

Resolved Generating Functions

Analysis of Convergence

Wave Front of Integral Transform

Applications to Classical Geometries

Minkowski–Funk Transform

Nongeodesic Hyperplane Sections of a Sphere

Totally Geodesic Transform in Hyperbolic Spaces

Horospherical Transform

Hyperboloids

Cormack’s Curves

Confocal Paraboloids

Cassini Ovals and Ovaloids

Applications to the Spherical Mean Transform

Oscillatory Sets

Reconstruction

Examples

Time Reversal Structure

Boundary Isometry for Waves in a Cavity

Range Conditions

Spheres Tangent to a Hyperplane

Summary of Spherical Mean Transform

Appendix

Bibliographic notes appear at the end of each chapter.

Über Autor(innen)

Victor Palamodov is a professor in the School of Mathematical Sciences at Tel-Aviv University. His research interests include mathematical and algebraic analysis and applications to physics and medical diagnostics.