Sampling, Wavelets, and Tomography

Sampling, wavelets, and tomography are three active areas of contemporary mathematics sharing common roots that lie at the heart of harmonic and Fourier analysis. The advent of new techniques in mathematical analysis has strengthened their interdependence and led to some new and interesting results in the field.

This state-of-the-art book not only presents new results in these research areas, but it also demonstrates the role of sampling in both wavelet theory and tomography. Specific topics covered include:

• Robustness of Regular Sampling in Sobolev Algebras Irregular and Semi-Irregular Weyl-Heisenberg Frames Adaptive Irregular Sampling in Meshfree Flow Simulation Sampling Theorems for Non-Bandlimited Signals Polynomial Matrix Factorization, Multidimensional Filter Banks, and Wavelets Generalized Frame Multiresolution Analysis of Abstract Hilbert Spaces Sampling Theory and Parallel-Beam Tomography Thin-Plate Spline Interpolation in Medical Imaging Filtered Back-Projection Algorithms for Spiral Cone Computed Tomography

  Aimed at mathematicians, scientists, and engineers working in signal and image processing and medical imaging, the work is designed to be accessible to an audience with diverse mathematical backgrounds. Although the volume reflects the contributions of renowned mathematicians and engineers, each chapter has an expository introduction written for the non-specialist. One of the key features of the book is an introductory chapter stressing the interdependence of the three main areas covered. A comprehensive index completes the work.

  Contributors: J.J. Benedetto, N.K. Bose, P.G. Casazza, Y.C. Eldar, H.G. Feichtinger, A. Faridani, A. Iske, S. Jaffard, A. Katsevich, S. Lertrattanapanich, G. Lauritsch, B. Mair, M. Papadakis, P.P. Vaidyanathan, T. Werther, D.C. Wilson, A.I. Zayed

One of the key features of the book is an introductory chapter, written by the editors, to stress the interdependence of the three areas covered

Contenu
1 A Prelude to Sampling, Wavelets, and Tomography.- 1.1 Introduction.- 1.2 Sampling.- 1.3 Sampling and Frames.- 1.4 Wavelets and Multiresolution Analysis.- 1.5 Tomography, Spline Interpolation, and Medical Imaging.- References.- 2 Sampling Without Input Constraints: Consistent Reconstruction in Arbitrary Spaces.- 1 Introduction.- 2 Consistent Reconstruction.- 3 Reconstruction Scheme.- 4 Stability and Performance Analysis.- 5 Reconstruction from Nonredundant Measurements.- 6 Reconstruction from Redundant Measurements.- 7 Constructing Signals with Prescribed Properties.- References.- 3 An Introduction to Irregular Weyl-Heisenberg Frames.- 1 Introduction.- 2 Preliminaries.- 3 Density.- 4 Semi-Irregular WeylHeisenberg Frames.- 5 General Irregular WeylHeisenberg Frames.- 6 Concluding Remarks.- References.- 4 Robustness of Regular Sampling in Sobolev Algebras.- 1 Introduction.- 2 Wiener Amalgam Spaces W(B, LP) = W(B, ?P).- 3 Generalities on Spline-type Spaces.- 4 Sobolev Spaces ?2S.- 5 LP-Theory.- 6 Changing the Smoothness Parameter.- 7 Changing the Sampling Lattice.- 8 Jitter Stability.- References.- 5 Sampling Theorems for Nonbandlimited Signals.- 1 Introduction.- 2 Signal Models.- 3 Biorthogonal Partners.- 4 Nonuniform Sampling.- 5 Derivative Sampling.- 6 Discrete-Time Case.- 7 Concluding Remarks.- References.- 6 Polynomial Matrix Factorization, Multidimensional Filter Banks, and Wavelets.- 1 Introduction.- 2 Status of Results in Multivariate Polynomial Matrix Factorization.- 3 Ladder Structure of Biorthogonal Multidimensional Multiband.- 4 Sampling in Wavelet Subspaces and Multiresolution Analysis.- 5 Wavelet Superresolution.- 6 Conclusions.- References.- 7 Function Spaces Based on Wavelet Expansions.- 1 Introduction.- 2 Wavelet expansions and sparseness.- 3Robustness criteria.- 4 Distributions of wavelet coefficients.- 5 Spaces Osps' and contour-type functions.- References.- 8 Generalized Frame Multiresolution Analysis of Abstract Hilbert Spaces.- 1 Introduction and Preliminaries.- 2 Construction and Characterization of the Frame Multiwavelet Vector Sets Associated with a GFMRA.- 3 Examples of GFMRAs.- References.- 9 Sampling Theory and Parallel-Beam Tomography.- 1 Introduction.- 2 The Two-Dimensional Radon Transform.- 3 Sampling Lattices for the Radon Transform.- 4 The Support of (Rf).- 5 Sampling Conditions.- 6 The Filtered Backprojection Algorithm.- 7 Analysis of the Effects of Undersampling.- 8 Further Developments.- References.- 10 Filtered Backprojection Algorithms for Spiral Cone Beam CT.- 1 Introduction.- 2 PI Line and PI Window.- 3 An Approximate Algorithm for Cone Beam CT.- 4 An Exact Algorithm for Cone Beam CT.- 5 Practical Implementation and Numerical Experiments.- 6 Selected Proofs.- References.- 11 Adaptive Irregular Sampling in Meshfree Flow Simulation.- 1 Introduction.- 2 Meshfree Particle Methods for Flow Simulation.- 3 Polyharmonic Splines.- 4 Adaptive Irregular Sampling.- 5 Meshfree Flow Simulation.- References.- 12 Thin-Plate Spline Interpolation.- 1 Introduction.- 2 A Brief Review of One-Dimensional Interpolation Techniques.- 3 Interpolating with Trigonometric Polynomials.- 4 Piecewise Linear Interpolation.- 5 Cubic Spline Interpolation.- 6 The Two-Dimensional Thin-Plate Spline Transformation.- References.