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Advances in Sampling Theory and Techniques
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Book Description

This book provides both the most updated formulations of the sampling theory and practical algorithms of image sampling with sampling rates close to the theoretical minimum, as well interpolation-error-free methods of image resampling and the theory of discrete representation of signal integral transforms. Topics include classical sampling theory, compressed sensing, non-redundant sampling, fast signal resampling algorithms, the discrete uncertainty principle, digital convolution, and various versions of discrete Fourier transforms. Exercises based in MATLAB supplement the text throughout.

Book Details

Date Published: 4 March 2020
Pages: 214
ISBN: 9781510633834
Volume: PM315

Table of Contents
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Table of Contents

1 Introduction
1.1 A Historical Perspective of Sampling: From Ancient Mosaics to Computational Imaging
1.2 Book Overview

Part I: Signal Sampling

2 Sampling Theorems
2.1 Kotelnikov-Shannon Sampling Theorem: Sampling Band-Limited 1D Signals
2.2 Sampling 1D Band-Pass Signals
2.3 Sampling Band-Limited 2D Signals; Optimal Regular Sampling Lattices
2.4 Sampling Real Signals; Signal Reconstruction Distortions due to Spectral Aliasing
2.5 The Sampling Theorem in a Realistic Reformulation
2.6 Image Sampling with a Minimal Sampling Rate by Means of Image Sub-band Decomposition
2.7 The Discrete Sampling Theorem and Its Generalization to Continuous Signals
     2.7.1 Theorem formulation
     2.7.2 Discrete sampling theorem formulations for specific transforms
     2.7.3 The general sampling theorem
2.8 Exercises

3 Compressed Sensing Demystified
3.1 Redundancy of Regular Image Sampling and Image Spectra Sparsity
3.2 Compressed Sensing: Why and How It Is Possible to Precisely Reconstruct Signals Sampled with Aliasing
3.3 Compressed Sensing and the Problem of Minimizing the Signal Sampling Rate
3.4 Exercise

4 Image Sampling and Reconstruction with Sampling Rates Close to the Theoretical Minimum
4.1 The ASBSR Method of Image Sampling and Reconstruction
4.2 Experimental Verification of the Method
4.3 Some Practical Issues
4.4 Other Possible Applications of the ASBSR Method of Image Sampling and Reconstruction
     4.4.1 Image super-resolution from multiple chaotically sampled video frames
     4.4.2 Image reconstruction from their sparsely sampled or decimated projections
     4.4.3 Image reconstruction from sparsely sampled Fourier spectra
4.5 Exercises

5 Signal and Image Resampling, and Building Their Continuous Models
5.1 Signal/Image Resampling as an Interpolation Problem; Convolutional Interpolators
5.2 Discrete Sinc Interpolation: A Gold Standard for Signal Resampling
5.3 Fast Algorithms of Discrete Sinc Interpolation and Their Applications
     5.3.1 Signal sub-sampling with DFT or DCT spectral zero-padding
     5.3.2 Signal sub-sampling (zooming-in) by means of DFT- and DCT-based perfect fractional shift algorithms
     5.3.3 Quasi-continuous signal spectral and correlational analysis using the perfect fractional shift algorithm
     5.3.4 Fast image rotation using the fractional shift algorithm
     5.3.5 Signal and image resampling using scaled and rotated DFTs
5.4 Discrete Sinc Interpolation versus Other Interpolation Methods: Performance Comparison
5.5 Exercises

6 Discrete Sinc Interpolation in Other Applications and Implementations
6.1 Precise Numerical Differentiation and Integration of Sampled Signals
     6.1.1 Perfect digital differentiator and integrator
     6.1.2 Conventional numerical differentiation and integration algorithms versus perfect DFT/DCT versions: performance comparison
6.2 Local ("Elastic") Image Resampling: Sliding-Window Discrete Sinc Interpolation Algorithms
6.3 Image Data Resampling for Image Reconstruction from Projections
     6.3.1 Discrete Radon transform and filtered back-projection method for image reconstruction
     6.3.2 Direct Fourier method of image reconstruction
     6.3.3 Image reconstruction from fan-beam projections
6.4 Exercises

7 The Discrete Uncertainty Principle, Sinc-lets, and Other Peculiar Properties of Sampled Signals
7.1 The Discrete Uncertainty Principle
7.2 Sinc-lets: Sharply-Band-Limited Basis Functions with Sharply Limited Support
7.3 Exercises

Part II: Discrete Representation of Signal Transformations

8 Basic Principles of Discrete Representation of Signal Transformations

9 Discrete Representation of the Convolution Integral
9.1 Discrete Convolution
9.2 Point Spread Functions and Frequency Responses of Digital Filters
9.3 Treatment of Signal Borders in Digital Convolution

10 Discrete Representation of the Fourier Integral Transform
10.1 1D Discrete Fourier Transforms
10.2 2D Discrete Fourier Transforms
10.3 Discrete Cosine Transform
10.4 Boundary-Effect-Free Signal Convolution in the DCT Domain
10.5 DFT and Discrete Frequency Responses of Digital Filters
10.6 Exercises

Appendix 1 Fourier Series, Integral Fourier Transform, and Delta Function
A1.1 1D Fourier Series
A1.2 2D Fourier Series
A1.3 1D Integral Fourier Transform
A1.4 2D Integral Fourier Transform
A1.5 Delta Function, Sinc Function, and the Ideal Low-Pass Filter
A1.6 Poisson Summation Formula

Appendix 2 Discrete Fourier Transforms and Their Properties
A2.1 Invertibility of Discrete Fourier Transforms and the Discrete Sinc Function
A2.2 The Parseval's Relation for the DFT
A2.3 Cyclicity of the DFT
A2.4 Shift Theorem
A2.5 Convolution Theorem
A2.6 Symmetry Properties
A2.7 SDFT Spectra of Sinusoidal Signals
A2.8 Mutual Correspondence between the Indices of ShDFT Spectral Coefficients and Signal Frequencies
A2.9 DFT Spectra of Sparse Signals and Spectral Zero-Padding
A2.10 Invertibility of the Shifted DFT and Signal Resampling
A2.11 DFT as a Spectrum Analyzer
A2.12 Quasi-continuous Spectral Analysis
A2.13 Signal Resizing and Rotation Capability of the Rotated Scaled DFT
A2.14 Rotated and Scaled DFT as Digital Convolution

Preface

Signal sampling is the major method for converting analog signals into sets of numbers that form digital models of the signals. The key issues in the sampling theory and practice are

  • What is the minimal amount of numbers, or what is the minimal sampling rate, sufficient to represent analog signals with a given accuracy?
  • What kinds of signal distortions are caused by their sampling?
  • What signal attributes determine the minimal sampling rate?
  • How can one sample signals with sampling rates close to the theoretical minimum?
  • Is it possible to resample sampled signals without introducing additional distortions due to the resampling?
  • What are adequate discrete representations of signal transforms, such as convolution and Fourier transforms?

All of these issues are addresed in this book, supplemented by MATLAB® exercises, which you can download via the following link: http:/spie.org/ Samples/Pressbook_Supplemental/PM282_sup.zip.

Researchers, engineers, and students will benefit from the most updated formulations of the sampling theory, as well as practical algorithms of signal and image sampling with sampling rates close to the theoretical minimum and interpolation-error-free methods of signal/image resampling, geometrical transformations, differentiation, and integration.

Leonid P. Yaroslavsky
December 2019


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