Reconstruction from Integral Data: 1st Edition (Hardback) book cover

Reconstruction from Integral Data

1st Edition

By Victor Palamodov

Chapman and Hall/CRC

172 pages | 16 B/W Illus.

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Hardback: 9781498710107
pub: 2016-05-02
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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.

Table of Contents

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


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


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


Cormack’s Curves

Confocal Paraboloids

Cassini Ovals and Ovaloids

Applications to the Spherical Mean Transform

Oscillatory Sets



Time Reversal Structure

Boundary Isometry for Waves in a Cavity

Range Conditions

Spheres Tangent to a Hyperplane

Summary of Spherical Mean Transform


Bibliographic notes appear at the end of each chapter.

About the Author

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.

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
MATHEMATICS / Graphic Methods
MATHEMATICS / Functional Analysis