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E-book: Multidimensional Signal and Color Image Processing Using Lattices

  • Format: PDF+DRM
  • Pub. Date: 12-Mar-2019
  • Publisher: John Wiley & Sons Inc
  • Language: eng
  • ISBN-13: 9781119111757
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Multidimensional Signal and Color Image Processing Using LatticesLarger Image
  • Format: PDF+DRM
  • Pub. Date: 12-Mar-2019
  • Publisher: John Wiley & Sons Inc
  • Language: eng
  • ISBN-13: 9781119111757
Other books in subject:

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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.

About the Companion Website xiii
1 Introduction
1(4)
2 Continuous-Domain Signals and Systems
5(36)
2.1 Introduction
5(2)
2.2 Multidimensional Signals
7(3)
2.2.1 Zero-One Functions
7(1)
2.2.2 Sinusoidal Signals
7(3)
2.2.3 Real Exponential Functions
10(1)
2.2 A Zone Plate
10(3)
2.2.5 Singularities
12(1)
2.2.6 Separable and Isotropic Functions
13(1)
2.3 Visualization of Two-Dimensional Signals
13(1)
2.4 Signal Spaces and Systems
14(1)
2.5 Continuous-Domain Linear Systems
15(7)
2.5.1 Linear Systems
15(4)
2.5.2 Linear Shift-Invariant Systems
19(1)
2.5.3 Response of a Linear System
20(1)
2.5.4 Response of a Linear Shift-Invariant System
20(2)
2.5.5 Frequency Response of an LSI System
22(1)
2.6 The Multidimensional Fourier Transform
22(11)
2.6.1 Fourier Transform Properties
23(4)
2.6.2 Evaluation of Multidimensional Fourier Transforms
27(3)
2.6.3 Two-Dimensional Fourier Transform of Polygonal Zero-One Functions
30(3)
2.6 A Fourier Transform of a Translating Still Image
33(1)
2.7 Further Properties of Differentiation and Related Systems
33(4)
2.7.1 Directional Derivative
34(1)
2.7.2 Laplacian
34(1)
2.7.3 Filtered Derivative Systems
35(2)
Problems
37(4)
3 Discrete-Domain Signals and Systems
41(28)
3.1 Introduction
41(1)
3.2 Lattices
42(4)
3.2.1 Basic Definitions
42(2)
3.2.2 Properties of Lattices
44(1)
3.2.3 Examples of 2D and 3D Lattices
44(2)
3.3 Sampling Structures
46(1)
3.4 Signals Denned on Lattices
47(1)
3.5 Special Multidimensional Signals on a Lattice
48(3)
3.5.1 Unit Sample
48(1)
3.5.2 Sinusoidal Signals
49(2)
3.6 Linear Systems Over Lattices
51(1)
3.6.1 Response of a Linear System
51(1)
3.6.2 Frequency Response
52(1)
3.7 Discrete-Domain Fourier Transforms Over a Lattice
52(7)
3.7.1 Definition of the Discrete-Domain Fourier Transform
52(1)
3.7.2 Properties of the Multidimensional Fourier Transform Over a Lattice Λ
53(4)
3.7.3 Evaluation of Forward and Inverse Discrete-Domain Fourier Transforms
57(2)
3.8 Finite Impulse Response (FIR) Filters
59(8)
3.8.1 Separable Filters
66(1)
Problems
67(2)
4 Discrete-Domain Periodic Signals
69(24)
4.1 Introduction
69(1)
4.2 Periodic Signals
69(3)
4.3 Linear Shift-Invariant Systems
72(1)
4.4 Discrete-Domain Periodic Fourier Transform
73(4)
4.5 Properties of the Discrete-Domain Periodic Fourier Transform
77(4)
4.6 Computation of the Discrete-Domain Periodic Fourier Transform
81(6)
4.6.1 Direct Computation
81(1)
4.6.2 Selection of Coset Representatives
82(5)
4.7 Vector Space Representation of Images Based on the Discrete-Domain Periodic Fourier Transform
87(3)
4.7.1 Vector Space Representation of Signals with Finite Extent
87(1)
4.7.2 Block-Based Vector-Space Representation
88(2)
Problems
90(3)
5 Continuous-Domain Periodic Signals
93(14)
5.1 Introduction
93(1)
5.2 Continuous-Domain Periodic Signals
93(1)
5.3 Linear Shift-Invariant Systems
94(2)
5.4 Continuous-Domain Periodic Fourier Transform
96(1)
5.5 Properties of the Continuous-Domain Periodic Fourier Transform
96(4)
5.6 Evaluation of the Continuous-Domain Periodic Fourier Transform
100(5)
Problems
105(2)
6 Sampling, Reconstruction and Sampling Theorems for Multidimensional Signals
107(18)
6.1 Introduction
107(1)
6.2 Ideal Sampling and Reconstruction of Continuous-Domain Signals
107(3)
6.3 Practical Sampling
110(2)
6.4 Practical Reconstruction
112(1)
6.5 Sampling and Periodization of Multidimensional Signals and Transforms
113(3)
6.6 Inverse Fourier Transforms
116(3)
6.6.1 Inverse Discrete-Domain Aperiodic Fourier Transform
117(1)
6.6.2 Inverse Continuous-Domain Periodic Fourier Transform
118(1)
6.6.3 Inverse Continuous-Domain Fourier Transform
119(1)
6.7 Signals and Transforms with Finite Support
119(2)
6.7.1 Continuous-Domain Signals with Finite Support
119(1)
6.7.2 Discrete-Domain Aperiodic Signals with Finite Support
120(1)
6.7.3 Band-Limited Continuous-Domain Γ-Periodic Signals
121(1)
Problems
121(4)
7 Light and Color Representation in Imaging Systems
125(38)
7.1 Introduction
125(1)
7.2 Light
125(3)
7.3 The Space of Light Stimuli
128(1)
7.4 The Color Vector Space
129(18)
7.4.1 Properties of Metamerism
130(2)
7.4.2 Algebraic Condition for Metameric Equivalence
132(3)
7.4.3 Extension of Metameric Equivalence to A
135(1)
7.4.4 Definition of the Color Vector Space
135(2)
7.4.5 Bases for the Vector Space C
137(1)
7.4.6 Transformation of Primaries
138(2)
7.4.7 The CIE Standard Observer
140(2)
7.4.8 Specification of Primaries
142(2)
7.4.9 Physically Realizable Colors
144(3)
7.5 Color Coordinate Systems
147(11)
7.5.1 Introduction
147(1)
7.5.2 Luminance and Chromaticity
147(6)
7.5.3 Linear Color Representations
153(2)
7.5.4 Perceptually Uniform Color Coordinates
155(2)
7.5.5 Display Referred Coordinates
157(1)
7.5.6 Luma-Color-Difference Representation
158(1)
Problems
158(5)
8 Processing of Color Signals
163(30)
8.1 Introduction
163(1)
8.2 Continuous-Domain Systems for Color Images
163(10)
8.2.1 Continuous-Domain Color Signals
163(3)
8.2.2 Continuous-Domain Systems for Color Signals
166(2)
8.2.3 Frequency Response and Fourier Transform
168(5)
8.3 Discrete-Domain Color Images
173(15)
8.3.1 Color Signals With All Components on a Single Lattice
173(2)
8.3.1.1 Sampling a Continuous-Domain Color Signal Using a Single Lattice
175(1)
8.3.1.2 S-CIELAB Error Criterion
175(5)
8.3.2 Color Signals With Different Components on Different Sampling Structures
180(8)
8.4 Color Mosaic Displays
188(5)
9 Random Field Models
193(22)
9.1 Introduction
193(1)
9.2 What is a Random Field?
194(1)
9.3 Image Moments
195(4)
9.3.1 Mean, Autocorrelation, Autocovariance
195(3)
9.3.2 Properties of the Autocorrelation Function
198(1)
9.3.3 Cross-Correlation
199(1)
9.4 Power Density Spectrum
199(3)
9.4.1 Properties of the Power Density Spectrum
200(1)
9.4.2 Cross Spectrum
201(1)
9.4.3 Spectral Density Matrix
201(1)
9.5 Filtering and Sampling of WSS Random Fields
202(5)
9.5.1 LSI Filtering of a Scalar WSS Random Field
202(2)
9.5.2 Why is Sƒ(u) Called a Power Density Spectrum?
204(1)
9.5.3 LSI Filtering of a WSS Color Random Field
205(1)
9.5.4 Sampling of a WSS Continuous-Domain Random Field
206(1)
9.6 Estimation of the Spectral Density Matrix
207(7)
Problems
214(1)
10 Analysis and Design of Multidimensional FIR Filters
215(22)
10.1 Introduction
215(1)
10.2 Moving Average Filters
215(2)
10.3 Gaussian Filters
217(3)
10.4 Band-pass and Band-stop Filters
220(5)
10.5 Frequency-Domain Design of Multidimensional FIR Filters
225(11)
10.5.1 FIR Filter Design Using Windows
226(3)
10.5.2 FIR Filter Design Using Least-pth Optimization
229(7)
Problems
236(1)
11 Changing the Sampling Structure of an Image
237(18)
11.1 Introduction
237(1)
11.2 Sublattices
237(2)
11.3 Upsampling
239(6)
11.4 Downsampling
245(3)
11.5 Arbitrary Sampling Structure Conversion
248(6)
11.5.1 Sampling Structure Conversion Using a Common Superlattice
248(3)
11.5.2 Polynomial Interpolation
251(3)
Problems
254(1)
12 Symmetry Invariant Signals and Systems
255(34)
12.1 LSI Systems Invariant to a Group of Symmetries
255(14)
12.1.1 Symmetries of a Lattice
255(3)
12.1.2 Symmetry-Group Invariant Systems
258(3)
12.1.3 Spaces of Symmetric Signals
261(8)
12.2 Symmetry-Invariant Discrete-Domain Periodic Signals and Systems
269(13)
12.2.1 Symmetric Discrete-Domain Periodic Signals
270(1)
12.2.2 Discrete-Domain Periodic Symmetry-Invariant Systems
271(2)
12.2.3 Discrete-Domain Symmetry-Invariant Periodic Fourier Transform
273(9)
12.3 Vector-Space Representation of Images Based on the Symmetry-Invariant Periodic Fourier Transform
282(7)
13 Lattices
289(22)
13.1 Introduction
289(1)
13.2 Basic Definitions
289(4)
13.3 Properties of Lattices
293(1)
13.4 Reciprocal Lattice
294(1)
13.5 Sublattices
295(1)
13.6 Cosets and the Quotient Group
296(2)
13.7 Basis Transformations
298(4)
13.7.1 Elementary Column Operations
299(1)
13.7.2 Hermite Normal Form
300(2)
13.8 Smith Normal Form
302(2)
13.9 Intersection and Sum of Lattices
304(7)
Appendix A Equivalence Relations 311(2)
Appendix B Groups 313(2)
Appendix C Vector Spaces 315(4)
Appendix D Multidimensional Fourier Transform Properties 319(4)
References 323(6)
Index 329
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.
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