Multidimensional Signal and Color Image Processing Using Lattices

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.

Autorentext

PROFESSOR ERIC DUBOIS is Emeritus Professor at the University of Ottawa, Canada, a Life Fellow of the Institute of Electrical and Electronic Engineers and a Fellow of the Engineering Institute of Canada. He is a recipient of the 2013 George S. Glinski Award for Excellence in Research from the Faculty of Engineering at the University of Ottawa. His current research is focused on stereoscopic and multiview imaging, image sampling theory, image-based virtual environments and color signal processing.

Inhalt
About the Companion Website xiii

1 Introduction 1

2 Continuous-Domain Signals and Systems 5

2.1 Introduction 5

2.2 Multidimensional Signals 7

2.2.1 ZeroOne Functions 7

2.2.2 Sinusoidal Signals 7

2.2.3 Real Exponential Functions 10

2.2.4 Zone Plate 10

2.2.5 Singularities 12

2.2.6 Separable and Isotropic Functions 13

2.3 Visualization of Two-Dimensional Signals 13

2.4 Signal Spaces and Systems 14

2.5 Continuous-Domain Linear Systems 15

2.5.1 Linear Systems 15

2.5.2 Linear Shift-Invariant Systems 19

2.5.3 Response of a Linear System 20

2.5.4 Response of a Linear Shift-Invariant System 20

2.5.5 Frequency Response of an LSI System 22

2.6 The Multidimensional Fourier Transform 22

2.6.1 Fourier Transform Properties 23

2.6.2 Evaluation of Multidimensional Fourier Transforms 27

2.6.3 Two-Dimensional Fourier Transform of Polygonal ZeroOne Functions 30

2.6.4 Fourier Transform of a Translating Still Image 33

2.7 Further Properties of Differentiation and Related Systems 33

2.7.1 Directional Derivative 34

2.7.2 Laplacian 34

2.7.3 Filtered Derivative Systems 35

Problems 37

3 Discrete-Domain Signals and Systems 41

3.1 Introduction 41

3.2 Lattices 42

3.2.1 Basic Definitions 42

3.2.2 Properties of Lattices 44

3.2.3 Examples of 2D and 3D Lattices 44

3.3 Sampling Structures 46

3.4 Signals Defined on Lattices 47

3.5 Special Multidimensional Signals on a Lattice 48

3.5.1 Unit Sample 48

3.5.2 Sinusoidal Signals 49

3.6 Linear Systems Over Lattices 51

3.6.1 Response of a Linear System 51

3.6.2 Frequency Response 52

3.7 Discrete-Domain Fourier Transforms Over a Lattice 52

3.7.1 Definition of the Discrete-Domain Fourier Transform 52

3.7.2 Properties of the Multidimensional Fourier Transform Over a Lattice 53

3.7.3 Evaluation of Forward and Inverse Discrete-Domain Fourier Transforms 57

3.8 Finite Impulse Response (FIR) Filters 59

3.8.1 Separable Filters 66

Problems 67

4 Discrete-Domain Periodic Signals 69

4.1 Introduction 69

4.2 Periodic Signals 69

4.3 Linear Shift-Invariant Systems 72

4.4 Discrete-Domain Periodic Fourier Transform 73

4.5 Properties of the Discrete-Domain Periodic Fourier Transform 77

4.6 Computation of the Discrete-Domain Periodic Fourier Transform 81

4.6.1 Direct Computation 81

4.6.2 Selection of Coset Representatives 82

4.7 Vector Space Representation of Images Based on the Discrete-Domain Periodic Fourier Transform 87

4.7.1 Vector Space Representation of Signals with Finite Extent 87

4.7.2 Block-Based Vector-Space Representation 88

Problems 90

5 Continuous-Domain Periodic Signals 93

5.1 Introduction 93

5.2 Continuous-Domain Periodic Signals 93

5.3 Linear Shift-Invariant Systems 94

5.4 Continuous-Domain Periodic Fourier Transform 96

5.5 Properties of the Continuous-Domain Periodic Fourier Transform 96

5.6 Evaluation of the Continuous-Domain Periodic Fourier Transform 100

Problems 105

6 Sampling, Reconstruction and Sampling Theorems for Multidimensional Signals 107

6.1 Introduction 107

6.2 Ideal Sampling and Reconstruction of Continuous-Domain Signals 107

6.3 Practical Sampling 110

6.4 Practical Reconstruction 112

6.5 Sampling and Periodization of Multidimensional Signals and Transforms 113

6.6 Inverse Fourier Transforms 116

6.6.1 Inverse Discrete-Domain Aperiodic Fourier Transform 117 6.6.2 Inverse Continuous-Domain Periodic Fourier Transfor...