Advanced Topics in Shannon Sampling and Interpolation Theory

1 Gabor's Signal Expansion and Its Relation to Sampling of the Sliding-Window Spectrum.- 1.1 Introduction.- 1.2 Sliding-Window Spectrum.- 1.2.1 Inversion Formulas.- 1.2.2 Space and Frequency Shift.- 1.2.3 Some Integrals Concerning the Sliding-Window Spectrum.- 1.2.4 Discrete-Time Signals.- 1.3 Sampling Theorem for the Sliding-Window Spectrum.- 1.3.1 Discrete-Time Signals.- 1.4 Examples of Window Functions.- 1.4.1 Gaussian Window Function.- 1.4.2 Discrete-Time Signals.- 1.5 Gabor's Signal Expansion.- 1.5.1 Discrete-Time Signals.- 1.6 Examples of Elementary Signals.- 1.6.1 Rect Elementary Signal.- 1.6.2 Sine Elementary Signal.- 1.6.3 Gaussian Elementary Signal.- 1.6.4 Discrete-Time Signals.- 1.7 Degrees of Freedom of a Signal.- 1.8 Optical Generation of Gabor's Expansion Coefficients for Rastered Signals.- 1.9 Conclusion.- 2 Sampling in Optics.- 2.1 Introduction.- 2.2 Historical Background.- 2.3 The von Laue Analysis.- 2.4 Degrees of Freedom of an Image.- 2.4.1 Use of the Sampling Theorem.- 2.4.2 Some Objections.- 2.4.3 The Eigenfmiction Technique.- 2.4.4 The Gerchberg Method.- 2.5 Superresolving Pupils.- 2.5.1 Singular Value Analysis.- 2.5.2 Incoherent Imaging.- 2.5.3 Survey of Extensions.- 2.6 Fresnel SampHng.- 2.7 Exponential SampHng.- 2.8 Partially Coherent Fields.- 2.9 Optical Processing.- 2.10 Conclusion.- 3 A Multidimensional Extension of Papoulis' Generalized Sampling Expansion with the Application in Minimum Density Sampling.- I: A Multidimensional Extension of Papoulis' Generalized Sampling Expansion.- 3.1 Introduction.- 3.2 GSE Formulation.- 3.3 M-D Extension.- 3.3.1 M-D Sampling Theorem.- 3.3.2 M-D GSE Formulation.- 3.3.3 Examples.- 3.4 Extension Generalization.- 3.5 Conclusion.- II: Sampling Multidimensional Band-Limited Functions At Minimum Densities.- 3.6 Sample Interdependency.- 3.7 Sampling Density Reduction Using M-D GSE.- 3.7.1 Sampling Decimation.- 3.7.2 A Second Formulation for Sampling Decimation.- 3.8 Computational Complexity of the Two Formulations.- 3.8.1 Gram-Schmidt Searching Algorithm.- 3.9 Sampling at the Minimum Density.- 3.10 Discussion.- 3.11 Conclusion.- 4 Nonuniform Sampling.- 4.1 Preliminary Discussions.- 4.2 General Nonuniform Sampling Theorems.- 4.2.1 Lagrange Interpolation.- 4.2.2 Interpolation from Nonuniform Samples of a Signal and Its Derivatives.- 4.2.3 Nonuniform Sampling for Nonband-Limited Signals.- 4.2.4 Jittered Sampling.- 4.2.5 Past Sampling.- 4.2.6 Stability of Nonuniform Sampling Interpolation.- 4.2.7 Interpolation Viewed as a Time Varying System.- 4.2.8 Random Samphng.- 4.3 Spectral Analysis of Nonuniform Samples and Signal Recovery.- 4.3.1 Extension of the Parseval Relationship to Nonuniform Samples.- 4.3.2 Estimating the Spectrum of Nonuniform Samples.- 4.3.3 Spectral Analysis of Random Sampling.- 4.4 Discussion on Reconstruction Methods.- 4.4.1 Signal Recovery Through Non-Linear Methods.- 4.4.2 Iterative Methods for Signal Recovery.- 5 Linear Prediction by Samples from the Past.- 5.1 Preliminaries.- 5.2 Prediction of Deterministic Signals.- 5.2.1 General Results.- 5.2.2 Specific Prediction Sums.- 5.2.3 An Inverse Result.- 5.2.4 Prediction of Derivatives f(s) by Samples of.- 5.2.5 Round-OfF and Time Jitter Errors.- 5.3 Prediction of Random Signals.- 5.3.1 Continuous and Differentiable Stochastic Processes.- 5.3.2 Prediction of Weak Sense Stationary Stochastic Processes.- 6 Polar, Spiral, and Generalized Sampling and Interpolation.- 6.1 Introduction.- 6.2 Sampling in Polar Coordinates.- 6.2.1 Sampling of Periodic Functions.- 6.2.2 A Formula for Interpolating from Samples on a Uniform Polar Lattice.- 6.2.3 Applications in Computer Tomography (CT).- 6.3 Spiral Sampling.- 6.3.1 Linear Spiral Sampling Theorem.- 6.3.2 Reconstruction from Samples on Expanding Spirals.- 6.4 Reconstruction from Non-Uniform Samples by Convex Projections.- 6.4.1 The Method of Projections onto Convex Sets.- 6.4.2 Iterative Reconstruction by POCS.- 6.5 Experimental Results.- 6.5.1 Reconstruction of One-Dimensional Signals.- 6.5.2 Reconstruction of Images.- 6.6 Conclusions.- Appendix A.- A.1 Derivation of Projections onto Convex Sets Ci.- Appendix B.- B. 1 Derivation of the Projection onto the Set C0= ?iCi.- 7 Error Analysis in Application of Generalizations of the Sampling Theorem.- Foreword: Welcomed General Sources for the Sampling Theorems.- 7.1 Introduction - Sampling Theorems.- 7.1.1 The Shannon Sampling Theorem - A Brief Introduction and History.- 7.1.2 The Generalized Transform Sampling Theorem.- 7.1.3 System Interpretation of the Sampling Theorems.- 7.1.4 Self-Truncating Sampling Series for Better Truncation Error Bound.- 7.1.5 A New Impulse Train-The Extended Poisson Sum Formula.- 7.2 Error Bounds of the Present Extension of the Sampling Theorem.- 7.2.1 The Aliasing Error Bound.- 7.2.2 The Truncation Error Bound.- 7.3 Applications.- 7.3.1 Optics-Integral Equations Representation for Circular Aperture.- 7.3.2 The Gibbs' Phenomena of the General Orthogonal Expansion-A Possible Remedy.- 7.3.3 Boundary-Value Problems.- 7.3.4 Other Apphcations and Suggested Extensions.- Appendix A.- A.1 Analysis of Gibbs' Phenomena.