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.

Table of Contents:
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 Λ 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 Γ-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