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Book Description

An Innovative Approach to Multidimensional Signals and Systems Theory for Image and Video Processing

In this volume, Eric Dubois further develops the theory of multi-D signal processing wherein input and output are vector-value signals. With this framework, he introduces the reader to crucial concepts in signal processing such as continuous- and discrete-domain signals and systems, discrete-domain periodic signals, sampling and reconstruction, light and color, random field models, image representation and more. 

While most treatments use normalized representations for non-rectangular sampling, this approach obscures much of the geometrical and scale information of the signal. In contrast, Dr. Dubois uses actual units of space-time and frequency. Basis-independent representations appear as much as possible, and the basis is introduced where needed to perform calculations or implementations. Thus, lattice theory is developed from the beginning and rectangular sampling is treated as a special case. This is especially significant in the treatment of color and color image processing and for discrete transform representations based on symmetry groups, including fast computational algorithms. Other features include:

  • An entire chapter on lattices, giving the reader a thorough grounding in the use of lattices in signal processing
  • Extensive treatment of lattices as used to describe discrete-domain signals and signal periodicities
  • Chapters on sampling and reconstruction, random field models, symmetry invariant signals and systems and multidimensional Fourier transformation properties
  • Supplemented throughout with MATLAB examples and accompanying downloadable source code

Graduate and doctoral students as well as senior undergraduates and professionals working in signal processing or video/image processing and imaging will appreciate this fresh approach to multidimensional signals and systems theory, both as a thorough introduction to the subject and as inspiration for future research.

Table of Contents

  1. Cover
  2. Dedication
  3. About the Companion Website
  4. 1 Introduction
  5. 2 Continuous‐Domain Signals and Systems
    1. 2.1 Introduction
    2. 2.2 Multidimensional Signals
    3. 2.3 Visualization of Two‐Dimensional Signals
    4. 2.4 Signal Spaces and Systems
    5. 2.5 Continuous‐Domain Linear Systems
    6. 2.6 The Multidimensional Fourier Transform
    7. 2.7 Further Properties of Differentiation and Related Systems
  6. 3 Discrete‐Domain Signals and Systems
    1. 3.1 Introduction
    2. 3.2 Lattices
    3. 3.3 Sampling Structures
    4. 3.4 Signals Defined on Lattices
    5. 3.5 Special Multidimensional Signals on a Lattice
    6. 3.6 Linear Systems Over Lattices
    7. 3.7 Discrete‐Domain Fourier Transforms Over a Lattice
    8. 3.8 Finite Impulse Response (FIR) Filters
  7. 4 Discrete‐Domain Periodic Signals
    1. 4.1 Introduction
    2. 4.2 Periodic Signals
    3. 4.3 Linear Shift‐Invariant Systems
    4. 4.4 Discrete‐Domain Periodic Fourier Transform
    5. 4.5 Properties of the Discrete‐Domain Periodic Fourier Transform
    6. 4.6 Computation of the Discrete‐Domain Periodic Fourier Transform
    7. 4.7 Vector Space Representation of Images Based on the Discrete‐Domain Periodic Fourier Transform
  8. 5 Continuous‐Domain Periodic Signals
    1. 5.1 Introduction
    2. 5.2 Continuous‐Domain Periodic Signals
    3. 5.3 Linear Shift‐Invariant Systems
    4. 5.4 Continuous‐Domain Periodic Fourier Transform
    5. 5.5 Properties of the Continuous‐Domain Periodic Fourier Transform
    6. 5.6 Evaluation of the Continuous‐Domain Periodic Fourier Transform
  9. 6 Sampling, Reconstruction and Sampling Theorems for Multidimensional Signals
    1. 6.1 Introduction
    2. 6.2 Ideal Sampling and Reconstruction of Continuous‐Domain Signals
    3. 6.3 Practical Sampling
    4. 6.4 Practical Reconstruction
    5. 6.5 Sampling and Periodization of Multidimensional Signals and Transforms
    6. 6.6 Inverse Fourier Transforms
    7. 6.7 Signals and Transforms with Finite Support
  10. 7 Light and Color Representation in Imaging Systems
    1. 7.1 Introduction
    2. 7.2 Light
    3. 7.3 The Space of Light Stimuli
    4. 7.4 The Color Vector Space
    5. 7.5 Color Coordinate Systems
  11. 8 Processing of Color Signals
    1. 8.1 Introduction
    2. 8.2 Continuous‐Domain Systems for Color Images
    3. 8.3 Discrete‐Domain Color Images
    4. 8.4 Color Mosaic Displays
  12. 9 Random Field Models
    1. 9.1 Introduction
    2. 9.2 What is a Random Field?
    3. 9.3 Image Moments
    4. 9.4 Power Density Spectrum
    5. 9.5 Filtering and Sampling of WSS Random Fields
    6. 9.6 Estimation of the Spectral Density Matrix
  13. 10 Analysis and Design of Multidimensional FIR Filters
    1. 10.1 Introduction
    2. 10.2 Moving Average Filters
    3. 10.3 Gaussian Filters
    4. 10.4 Band‐pass and Band‐stop Filters
    5. 10.5 Frequency‐Domain Design of Multidimensional FIR Filters
  14. 11 Changing the Sampling Structure of an Image
    1. 11.1 Introduction
    2. 11.2 Sublattices
    3. 11.3 Upsampling
    4. 11.4 Downsampling
    5. 11.5 Arbitrary Sampling Structure Conversion
  15. 12 Symmetry Invariant Signals and Systems
    1. 12.1 LSI Systems Invariant to a Group of Symmetries
    2. 12.2 Symmetry‐Invariant Discrete‐Domain Periodic Signals and Systems
    3. 12.3 Vector‐Space Representation of Images Based on the Symmetry‐Invariant Periodic Fourier Transform
  16. 13 Lattices
    1. 13.1 Introduction
    2. 13.2 Basic Definitions
    3. 13.3 Properties of Lattices
    4. 13.4 Reciprocal Lattice
    5. 13.5 Sublattices
    6. 13.6 Cosets and the Quotient Group
    7. 13.7 Basis Transformations
    8. 13.8 Smith Normal Form
    9. 13.9 Intersection and Sum of Lattices
  17. Appendix A Equivalence Relations
  18. Appendix B Groups
  19. Appendix C Vector Spaces
  20. Appendix D Multidimensional Fourier Transform Properties
  21. References
  22. Index
  23. End User License Agreement
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