Reconstruction from Integral Data

Victor Palamodov



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
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.