Sampling, Wavelets, and Tomography

• 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 Weyl—Heisenberg Frames.- 5 General Irregular Weyl—Heisenberg 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.

"The book places emphasis on all three themes it considers. It presents applications with a broad perspective.... The book, containing contributions of some of the renowned scientists in the field, is interesting and informative, and presents an insightful approach towards wavelets, sampling theory, and tomography. To conclude, the book is of immense value to the mathematically inclined researcher whose work centers arround the application of sampling theory."   —Journal of the Indian Institute of Science"This volume has been compiled by the editors with a readership of mathematicians, scientists, and engineers working in signal and image processing. It is a collection of invited articles on sampling, wavelets, and tomography, three active areas in contemporary mathematics having Fourier analysis as a common root. This state-of-the-art book not only presents new results in these research areas, but also demonstrates the role of sampling in both wavelet theory and tomography. One key feature of the book is an introductory chapter stressing the interdependence of the three main areas covered." —Int. Mathem. Nachrichten"The aim of the book is to highlight the interconnections between sampling, wavelets and tomography, and to present new results. The first chapter of the book...provides the reader with an exceptional overview, and also with an illuminating review of the other chapters, pointing out the interconnections between them, and giving concise introduction into the three fundamental topics in the title, including fundamental definitions and results.... The book makes useful reading and gives clear nontedious presentation of the mathematical ideas involved." —Revue Roumaine de Mathématiques Pures et Appliquées"This state-of-the-art book not only presents new results… but it also demonstrates the role of sampling in both wavelet theory and tomography.… The topicscovered (from frame theory, time-frequency analysis, to tomography or new results on wavelets) show the wide range in the high level of activity in this area." —Monatshefte für Mathematik