
Multidimensional Signal and Color Image Processing Using Lattices
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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.
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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.
Content
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 Zero-One 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 Zero-One 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 Transform 118
6.6.3 Inverse Continuous-Domain Fourier Transform 119
6.7 Signals and Transforms with Finite Support 119
6.7.1 Continuous-Domain Signals with Finite Support 119
6.7.2 Discrete-Domain Aperiodic Signals with Finite Support 120
6.7.3 Band-Limited Continuous-Domain G-Periodic Signals 121
Problems 121
7 Light and Color Representation in Imaging Systems 125
7.1 Introduction 125
7.2 Light 125
7.3 The Space of Light Stimuli 128
7.4 The Color Vector Space 129
7.4.1 Properties of Metamerism 130
7.4.2 Algebraic Condition for Metameric Equivalence 132
7.4.3 Extension of Metameric Equivalence to A 135
7.4.4 Definition of the Color Vector Space 135
7.4.5 Bases for the Vector Space C 137
7.4.6 Transformation of Primaries 138
7.4.7 The CIE Standard Observer 140
7.4.8 Specification of Primaries 142
7.4.9 Physically Realizable Colors 144
7.5 Color Coordinate Systems 147
7.5.1 Introduction 147
7.5.2 Luminance and Chromaticity 147
7.5.3 Linear Color Representations 153
7.5.4 Perceptually Uniform Color Coordinates 155
7.5.5 Display Referred Coordinates 157
7.5.6 Luma-Color-Difference Representation 158
Problems 158
8 Processing of Color Signals 163
8.1 Introduction 163
8.2 Continuous-Domain Systems for Color Images 163
8.2.1 Continuous-Domain Color Signals 163
8.2.2 Continuous-Domain Systems for Color Signals 166
8.2.3 Frequency Response and Fourier Transform 168
8.3 Discrete-Domain Color Images 173
8.3.1 Color Signals With All Components on a Single Lattice 173
8.3.1.1 Sampling a Continuous-Domain Color Signal Using a Single Lattice 175
8.3.1.2 S-CIELAB Error Criterion 175
8.3.2 Color Signals With Different Components on Different Sampling Structures 180
8.4 Color Mosaic Displays 188
9 Random Field Models 193
9.1 Introduction 193
9.2 What is a Random Field? 194
9.3 Image Moments 195
9.3.1 Mean, Autocorrelation, Autocovariance 195
9.3.2 Properties of the Autocorrelation Function 198
9.3.3 Cross-Correlation 199
9.4 Power Density Spectrum 199
9.4.1 Properties of the Power Density Spectrum 200
9.4.2 Cross Spectrum 201
9.4.3 Spectral Density Matrix 201
9.5 Filtering and Sampling of WSS Random Fields 202
9.5.1 LSI Filtering of a Scalar WSS Random Field 202
9.5.2 Why is Sf(u) Called a Power Density Spectrum? 204
9.5.3 LSI Filtering of a WSS Color Random Field 205
9.5.4 Sampling of a WSS Continuous-Domain Random Field 206
9.6 Estimation of the Spectral Density Matrix 207
Problems 214
10 Analysis and Design of Multidimensional FIR Filters 215
10.1 Introduction 215
10.2 Moving Average Filters 215
10.3 Gaussian Filters 217
10.4 Band-pass and Band-stop Filters 220
10.5 Frequency-Domain Design of Multidimensional FIR Filters 225
10.5.1 FIR Filter Design Using Windows 226
10.5.2 FIR Filter Design Using Least-pth Optimization 229
Problems 236
11 Changing the Sampling Structure of an Image 237
11.1 Introduction 237
11.2 Sublattices 237
11.3 Upsampling 239
11.4 Downsampling 245
11.5 Arbitrary Sampling Structure Conversion 248
11.5.1 Sampling Structure Conversion Using a Common Superlattice 248
11.5.2 Polynomial Interpolation 251
Problems 254
12 Symmetry Invariant Signals and Systems 255
12.1 LSI Systems Invariant to a Group of Symmetries 255
12.1.1 Symmetries of a Lattice 255
12.1.2 Symmetry-Group Invariant Systems 258
12.1.3 Spaces of Symmetric Signals 261
12.2 Symmetry-Invariant Discrete-Domain Periodic Signals and Systems 269
12.2.1 Symmetric Discrete-Domain Periodic Signals 270
12.2.2 Discrete-Domain Periodic Symmetry-Invariant Systems 271
12.2.3 Discrete-Domain Symmetry-Invariant Periodic Fourier Transform 273
12.3 Vector-Space Representation of Images Based on the Symmetry-Invariant Periodic Fourier Transform 282
13 Lattices 289
13.1 Introduction 289
13.2 Basic Definitions 289
13.3 Properties of Lattices 293
13.4 Reciprocal Lattice 294
13.5 Sublattices 295
13.6 Cosets and the Quotient Group 296
13.7 Basis Transformations 298
13.7.1 Elementary Column Operations 299
13.7.2 Hermite Normal Form 300
13.8 Smith Normal Form 302
13.9 Intersection and Sum of Lattices 304
Appendix A: Equivalence Relations 311
Appendix B: Groups 313
Appendix C: Vector Spaces 315
Appendix D: Multidimensional Fourier Transform Properties 319
References 323
Index 329
1
Introduction
This book presents the theory of multidimensional (multiD) signals and systems, primarily in the context of image and video processing. MultiD signals are considered to be functions defined on some domain of dimension two or higher with values belonging to a set , the range. These values may represent the brightness or color of an image or some other type of measurement at each point in the domain. With this interpretation, a multiD signal is represented as
(1.1)i.e. each element of the domain is mapped to the value belonging to the range . In conventional continuous-time one-dimensional (1D) signals and systems theory [Oppenheim and Willsky (1997)], and are both the set of real numbers , and so a real 1D signal would be written , . MultiD signals arise when the domain is a space with two or more dimensions. The domain can be continuous, as in the case of real-world still and time-varying images, or discrete, as in the case of sampled images. In addition, the range can also be a higher-dimensional space, for example the three-dimensional color space of human vision.
In this book, we are mainly concerned with examples from conventional still and time-varying images, although the theory has broader applicability. A conventional planar image is written , where lies in a planar region associated with the Euclidean space . Here, denotes the horizontal spatial position and is the vertical spatial position while denotes image brightness or color. The domain can be itself, or a discrete subset in the case of sampled images. Similarly, a conventional time-varying image is written , where lies in a subset (possibly discrete) of , which may also be written . Here, and are as above, and represents time. Higher-dimensional cases also exist, for example time-varying volumetric images with and where denotes some measurement taken at location at time . The domain can also be a more complicated manifold such as a cylinder or a sphere, as in panoramic imaging.
MultiD signal processing has been an active area of study for over fifty years. Early work was in optics and the continuous domain. Papoulis's classic text on Systems and Transforms with Applications in Optics appeared in 1968 [Papoulis (1968)]. Soon after, work on two-dimensional digital filtering started to appear, for example [Hu and Rabiner (1972)]. Over the years, there have been several books devoted to multiD digital signal processing and numerous books on image and video processing. The present book is distinguished from these works in a number of aspects. The book is mainly concerned with the theory of discrete-domain processing of real- or vector-valued multiD signals. The application examples are drawn from grayscale and color image processing and video processing. In particular, the book is not intended to present the state-of-the-art algorithms for particular image processing tasks.
Most previous books on multiD signals considered rectangularly sampled signals for the main development and presented non-rectangular sampling on a lattice as a subsidiary extension. A lattice, as in crystal lattice, is a mathematical structure from which we can construct more general sampling structures. In this book, the theory is developed on lattices from the beginning, and rectangular sampling is considered a special case. Another difference is that most books use normalized representations for non-rectangular sampling that are dependent on the lattice basis. Although this may be convenient for certain manipulations, this approach obscures much of the geometrical and scale information of the signal. We prefer to use basis-independent representations as much as possible, and introduce the basis where needed to perform calculations or implementations. Thus, we do not use such normalized representations but rather use the actual units of space-time and frequency.
Another distinguishing feature of this book is the treatment of color. Color signals are viewed as multiD signals with values in a vector space, in this case the vector space of human color vision, and color signal processing is viewed as vector-valued signal processing. Most multiD signal processing books deal mainly with scalar signals, representing a grayscale or brightness value. If color models are introduced, color signal processing generally involves separate processing of three color channels. Here we present the theory of multiD signal processing where the input and output are vector-valued signals, further developing the theory introduced in Dubois (2010).
In general, multiD signals in the real world, such as still and time-varying images, are functions of the continuous space and time variables. Consider for example a light signal falling on a camera sensor or emanating from a motion-picture screen. These multiD signals are converted to discrete-domain signals for digital processing, storage, and transmission. They may eventually be converted back to continuous-domain signals, for example for viewing on a display device. Thus, we begin with an overview of scalar-valued continuous-domain multiD signals and systems, i.e. the domain is for some integer . In particular we introduce the concepts of signal space, linear shift-invariant systems and the continuous-domain multiD Fourier transform, develop properties of the Fourier transform and present some examples. Continuous-domain signal spaces and transforms involve advanced mathematical analysis to provide a general theory for arbitrary signal spaces. We do not attempt to provide a rigorous analysis. We assume that signals belong to a suitable signal space for which transforms are well defined and the properties hold. For example, a space of tempered distributions would be satisfactory. However, we do not develop the theory of distributions and take an informal approach to the Dirac delta and related singularities. We refer the reader to references for a rigorous analysis, e.g., [Stein and Weiss (1971), Richards and Youn (1990)].
There are many possible domains for multiD signals, generally subsets of for some . These domains can be continuous or discrete, or a hybrid that is continuous in some dimensions and discrete in others, like in analog TV scanning. The domain can also correspond to one period of a periodic signal, whether continuous or discrete. Among the possible domains, certain of them allow for the possibility of linear shift-invariant filtering. These domains have the algebraic structure of a locally-compact Abelian (LCA) group. While we cannot go into the detail of such structures, their main feature is that the concept of shift is well defined and commutative. is an example, as is any lattice in . The LCA group is the classical setting for abstract harmonic analysis, e.g., as presented in Rudin (1962). An early work on signal processing in this setting is the Ph.D. thesis of Rudolph Seviora [Seviora (1971)]), which considered generalized digital filtering on LCA groups. More recently Cariolaro has developed signal processing on LCA groups in a comprehensive book [Cariolaro (2011)].
In this book, we have elected to provide a separate development for the cases of continuous-domain aperiodic signals, discrete-domain aperiodic signals, discrete-domain periodic signals, and continuous-domain periodic signals (Chapters 02-05). Each case has its own sphere of application, and while the development may be redundant from an abstract mathematical perspective, the concrete details are sufficiently different to warrant their own presentation. Each of these chapters follows a similar roadmap, presenting concepts of signal space, linear shift-invariant (LSI) systems, Fourier transforms and their properties. For discrete-domain signals, we use lattices to describe the sampling structure. For periodic signals, we use lattices to describe the periodicity. Since lattices form an underlying tool used throughout the book, we have chosen to gather all definitions and results about lattices that we need for this work in Chapter 13, which may be consulted any time as needed. We prefer not to interrupt the flow of the book at the beginning with this material, and we wish to give it a higher status than an appendix. This is why we have chosen to include it as the last chapter in the book.
In Chapter 6 we see the relationship between the four representations. Discrete-domain aperiodic and periodic signals can be obtained for the corresponding continuous-domain signals by a sampling operation. This is shown to induce a periodization in the frequency domain. In another view, discrete and continuous-domain periodic signals can be obtained by periodization of corresponding aperiodic signals, resulting in sampling in the frequency domain. These results are all explored in Chapter 06and various sampling theorems are presented. We do not explicitly explore hybrid signals, which may correspond to a different one of the above types in different dimensions. This extension is usually straightforward; many examples are given in Cariolaro ( 2011).
Having developed the theory of processing of multiD scalar signals, we address the nature of the signal range in Chapter 7 , specifically for color image signals. Here we take up the vector-space view of color spaces as presented in Dubois (2010), where colors are viewed as equivalence classes...
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