1st Edition

Reconstruction from Integral Data

By Victor Palamodov Copyright 2016
    184 Pages 16 B/W Illustrations
    by Chapman & Hall

    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.

    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.

    Biography

    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.