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.

• About the Companion Website xiii1 Introduction 12 Continuous-Domain Signals and Systems 52.1 Introduction 52.2 Multidimensional Signals 72.2.1 Zero–One Functions 72.2.2 Sinusoidal Signals 72.2.3 Real Exponential Functions 102.2.4 Zone Plate 102.2.5 Singularities 122.2.6 Separable and Isotropic Functions 132.3 Visualization of Two-Dimensional Signals 132.4 Signal Spaces and Systems 142.5 Continuous-Domain Linear Systems 152.5.1 Linear Systems 152.5.2 Linear Shift-Invariant Systems 192.5.3 Response of a Linear System 202.5.4 Response of a Linear Shift-Invariant System 202.5.5 Frequency Response of an LSI System 222.6 The Multidimensional Fourier Transform 222.6.1 Fourier Transform Properties 232.6.2 Evaluation of Multidimensional Fourier Transforms 272.6.3 Two-Dimensional Fourier Transform of Polygonal Zero–One Functions 302.6.4 Fourier Transform of a Translating Still Image 332.7 Further Properties of Differentiation and Related Systems 332.7.1 Directional Derivative 342.7.2 Laplacian 342.7.3 Filtered Derivative Systems 35Problems 373 Discrete-Domain Signals and Systems 413.1 Introduction 413.2 Lattices 423.2.1 Basic Definitions 423.2.2 Properties of Lattices 443.2.3 Examples of 2D and 3D Lattices 443.3 Sampling Structures 463.4 Signals Defined on Lattices 473.5 Special Multidimensional Signals on a Lattice 483.5.1 Unit Sample 483.5.2 Sinusoidal Signals 493.6 Linear Systems Over Lattices 513.6.1 Response of a Linear System 513.6.2 Frequency Response 523.7 Discrete-Domain Fourier Transforms Over a Lattice 523.7.1 Definition of the Discrete-Domain Fourier Transform 523.7.2 Properties of the Multidimensional Fourier Transform Over a Lattice Λ 533.7.3 Evaluation of Forward and Inverse Discrete-Domain Fourier Transforms 573.8 Finite Impulse Response (FIR) Filters 593.8.1 Separable Filters 66Problems 674 Discrete-Domain Periodic Signals 694.1 Introduction 694.2 Periodic Signals 694.3 Linear Shift-Invariant Systems 724.4 Discrete-Domain Periodic Fourier Transform 734.5 Properties of the Discrete-Domain Periodic Fourier Transform 774.6 Computation of the Discrete-Domain Periodic Fourier Transform 814.6.1 Direct Computation 814.6.2 Selection of Coset Representatives 824.7 Vector Space Representation of Images Based on the Discrete-Domain Periodic Fourier Transform 874.7.1 Vector Space Representation of Signals with Finite Extent 874.7.2 Block-Based Vector-Space Representation 88Problems 905 Continuous-Domain Periodic Signals 935.1 Introduction 935.2 Continuous-Domain Periodic Signals 935.3 Linear Shift-Invariant Systems 945.4 Continuous-Domain Periodic Fourier Transform 965.5 Properties of the Continuous-Domain Periodic Fourier Transform 965.6 Evaluation of the Continuous-Domain Periodic Fourier Transform 100Problems 1056 Sampling, Reconstruction and Sampling Theorems for Multidimensional Signals 1076.1 Introduction 1076.2 Ideal Sampling and Reconstruction of Continuous-Domain Signals 1076.3 Practical Sampling 1106.4 Practical Reconstruction 1126.5 Sampling and Periodization of Multidimensional Signals and Transforms 1136.6 Inverse Fourier Transforms 1166.6.1 Inverse Discrete-Domain Aperiodic Fourier Transform 1176.6.2 Inverse Continuous-Domain Periodic Fourier Transform 1186.6.3 Inverse Continuous-Domain Fourier Transform 1196.7 Signals and Transforms with Finite Support 1196.7.1 Continuous-Domain Signals with Finite Support 1196.7.2 Discrete-Domain Aperiodic Signals with Finite Support 1206.7.3 Band-Limited Continuous-Domain Γ-Periodic Signals 121Problems 1217 Light and Color Representation in Imaging Systems 1257.1 Introduction 1257.2 Light 1257.3 The Space of Light Stimuli 1287.4 The Color Vector Space 1297.4.1 Properties of Metamerism 1307.4.2 Algebraic Condition for Metameric Equivalence 1327.4.3 Extension of Metameric Equivalence to A 1357.4.4 Definition of the Color Vector Space 1357.4.5 Bases for the Vector Space C 1377.4.6 Transformation of Primaries 1387.4.7 The CIE Standard Observer 1407.4.8 Specification of Primaries 1427.4.9 Physically Realizable Colors 1447.5 Color Coordinate Systems 1477.5.1 Introduction 1477.5.2 Luminance and Chromaticity 1477.5.3 Linear Color Representations 1537.5.4 Perceptually Uniform Color Coordinates 1557.5.5 Display Referred Coordinates 1577.5.6 Luma-Color-Difference Representation 158Problems 1588 Processing of Color Signals 1638.1 Introduction 1638.2 Continuous-Domain Systems for Color Images 1638.2.1 Continuous-Domain Color Signals 1638.2.2 Continuous-Domain Systems for Color Signals 1668.2.3 Frequency Response and Fourier Transform 1688.3 Discrete-Domain Color Images 1738.3.1 Color Signals With All Components on a Single Lattice 1738.3.1.1 Sampling a Continuous-Domain Color Signal Using a Single Lattice 1758.3.1.2 S-CIELAB Error Criterion 1758.3.2 Color Signals With Different Components on Different Sampling Structures 1808.4 Color Mosaic Displays 1889 Random Field Models 1939.1 Introduction 1939.2 What is a Random Field? 1949.3 Image Moments 1959.3.1 Mean, Autocorrelation, Autocovariance 1959.3.2 Properties of the Autocorrelation Function 1989.3.3 Cross-Correlation 1999.4 Power Density Spectrum 1999.4.1 Properties of the Power Density Spectrum 2009.4.2 Cross Spectrum 2019.4.3 Spectral Density Matrix 2019.5 Filtering and Sampling of WSS Random Fields 2029.5.1 LSI Filtering of a Scalar WSS Random Field 2029.5.2 Why is Sf(u) Called a Power Density Spectrum? 2049.5.3 LSI Filtering of a WSS Color Random Field 2059.5.4 Sampling of a WSS Continuous-Domain Random Field 2069.6 Estimation of the Spectral Density Matrix 207Problems 21410 Analysis and Design of Multidimensional FIR Filters 21510.1 Introduction 21510.2 Moving Average Filters 21510.3 Gaussian Filters 21710.4 Band-pass and Band-stop Filters 22010.5 Frequency-Domain Design of Multidimensional FIR Filters 22510.5.1 FIR Filter Design Using Windows 22610.5.2 FIR Filter Design Using Least-pth Optimization 229Problems 23611 Changing the Sampling Structure of an Image 23711.1 Introduction 23711.2 Sublattices 23711.3 Upsampling 23911.4 Downsampling 24511.5 Arbitrary Sampling Structure Conversion 24811.5.1 Sampling Structure Conversion Using a Common Superlattice 24811.5.2 Polynomial Interpolation 251Problems 25412 Symmetry Invariant Signals and Systems 25512.1 LSI Systems Invariant to a Group of Symmetries 25512.1.1 Symmetries of a Lattice 25512.1.2 Symmetry-Group Invariant Systems 25812.1.3 Spaces of Symmetric Signals 26112.2 Symmetry-Invariant Discrete-Domain Periodic Signals and Systems 26912.2.1 Symmetric Discrete-Domain Periodic Signals 27012.2.2 Discrete-Domain Periodic Symmetry-Invariant Systems 27112.2.3 Discrete-Domain Symmetry-Invariant Periodic Fourier Transform 27312.3 Vector-Space Representation of Images Based on the Symmetry-Invariant Periodic Fourier Transform 28213 Lattices 28913.1 Introduction 28913.2 Basic Definitions 28913.3 Properties of Lattices 29313.4 Reciprocal Lattice 29413.5 Sublattices 29513.6 Cosets and the Quotient Group 29613.7 Basis Transformations 29813.7.1 Elementary Column Operations 29913.7.2 Hermite Normal Form 30013.8 Smith Normal Form 30213.9 Intersection and Sum of Lattices 304Appendix A: Equivalence Relations 311Appendix B: Groups 313Appendix C: Vector Spaces 315Appendix D: Multidimensional Fourier Transform Properties 319References 323Index 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.

• About the Companion Website xiii1 Introduction 12 Continuous-Domain Signals and Systems 52.1 Introduction 52.2 Multidimensional Signals 72.2.1 Zero–One Functions 72.2.2 Sinusoidal Signals 72.2.3 Real Exponential Functions 102.2.4 Zone Plate 102.2.5 Singularities 122.2.6 Separable and Isotropic Functions 132.3 Visualization of Two-Dimensional Signals 132.4 Signal Spaces and Systems 142.5 Continuous-Domain Linear Systems 152.5.1 Linear Systems 152.5.2 Linear Shift-Invariant Systems 192.5.3 Response of a Linear System 202.5.4 Response of a Linear Shift-Invariant System 202.5.5 Frequency Response of an LSI System 222.6 The Multidimensional Fourier Transform 222.6.1 Fourier Transform Properties 232.6.2 Evaluation of Multidimensional Fourier Transforms 272.6.3 Two-Dimensional Fourier Transform of Polygonal Zero–One Functions 302.6.4 Fourier Transform of a Translating Still Image 332.7 Further Properties of Differentiation and Related Systems 332.7.1 Directional Derivative 342.7.2 Laplacian 342.7.3 Filtered Derivative Systems 35Problems 373 Discrete-Domain Signals and Systems 413.1 Introduction 413.2 Lattices 423.2.1 Basic Definitions 423.2.2 Properties of Lattices 443.2.3 Examples of 2D and 3D Lattices 443.3 Sampling Structures 463.4 Signals Defined on Lattices 473.5 Special Multidimensional Signals on a Lattice 483.5.1 Unit Sample 483.5.2 Sinusoidal Signals 493.6 Linear Systems Over Lattices 513.6.1 Response of a Linear System 513.6.2 Frequency Response 523.7 Discrete-Domain Fourier Transforms Over a Lattice 523.7.1 Definition of the Discrete-Domain Fourier Transform 523.7.2 Properties of the Multidimensional Fourier Transform Over a Lattice Λ 533.7.3 Evaluation of Forward and Inverse Discrete-Domain Fourier Transforms 573.8 Finite Impulse Response (FIR) Filters 593.8.1 Separable Filters 66Problems 674 Discrete-Domain Periodic Signals 694.1 Introduction 694.2 Periodic Signals 694.3 Linear Shift-Invariant Systems 724.4 Discrete-Domain Periodic Fourier Transform 734.5 Properties of the Discrete-Domain Periodic Fourier Transform 774.6 Computation of the Discrete-Domain Periodic Fourier Transform 814.6.1 Direct Computation 814.6.2 Selection of Coset Representatives 824.7 Vector Space Representation of Images Based on the Discrete-Domain Periodic Fourier Transform 874.7.1 Vector Space Representation of Signals with Finite Extent 874.7.2 Block-Based Vector-Space Representation 88Problems 905 Continuous-Domain Periodic Signals 935.1 Introduction 935.2 Continuous-Domain Periodic Signals 935.3 Linear Shift-Invariant Systems 945.4 Continuous-Domain Periodic Fourier Transform 965.5 Properties of the Continuous-Domain Periodic Fourier Transform 965.6 Evaluation of the Continuous-Domain Periodic Fourier Transform 100Problems 1056 Sampling, Reconstruction and Sampling Theorems for Multidimensional Signals 1076.1 Introduction 1076.2 Ideal Sampling and Reconstruction of Continuous-Domain Signals 1076.3 Practical Sampling 1106.4 Practical Reconstruction 1126.5 Sampling and Periodization of Multidimensional Signals and Transforms 1136.6 Inverse Fourier Transforms 1166.6.1 Inverse Discrete-Domain Aperiodic Fourier Transform 1176.6.2 Inverse Continuous-Domain Periodic Fourier Transform 1186.6.3 Inverse Continuous-Domain Fourier Transform 1196.7 Signals and Transforms with Finite Support 1196.7.1 Continuous-Domain Signals with Finite Support 1196.7.2 Discrete-Domain Aperiodic Signals with Finite Support 1206.7.3 Band-Limited Continuous-Domain Γ-Periodic Signals 121Problems 1217 Light and Color Representation in Imaging Systems 1257.1 Introduction 1257.2 Light 1257.3 The Space of Light Stimuli 1287.4 The Color Vector Space 1297.4.1 Properties of Metamerism 1307.4.2 Algebraic Condition for Metameric Equivalence 1327.4.3 Extension of Metameric Equivalence to A 1357.4.4 Definition of the Color Vector Space 1357.4.5 Bases for the Vector Space C 1377.4.6 Transformation of Primaries 1387.4.7 The CIE Standard Observer 1407.4.8 Specification of Primaries 1427.4.9 Physically Realizable Colors 1447.5 Color Coordinate Systems 1477.5.1 Introduction 1477.5.2 Luminance and Chromaticity 1477.5.3 Linear Color Representations 1537.5.4 Perceptually Uniform Color Coordinates 1557.5.5 Display Referred Coordinates 1577.5.6 Luma-Color-Difference Representation 158Problems 1588 Processing of Color Signals 1638.1 Introduction 1638.2 Continuous-Domain Systems for Color Images 1638.2.1 Continuous-Domain Color Signals 1638.2.2 Continuous-Domain Systems for Color Signals 1668.2.3 Frequency Response and Fourier Transform 1688.3 Discrete-Domain Color Images 1738.3.1 Color Signals With All Components on a Single Lattice 1738.3.1.1 Sampling a Continuous-Domain Color Signal Using a Single Lattice 1758.3.1.2 S-CIELAB Error Criterion 1758.3.2 Color Signals With Different Components on Different Sampling Structures 1808.4 Color Mosaic Displays 1889 Random Field Models 1939.1 Introduction 1939.2 What is a Random Field? 1949.3 Image Moments 1959.3.1 Mean, Autocorrelation, Autocovariance 1959.3.2 Properties of the Autocorrelation Function 1989.3.3 Cross-Correlation 1999.4 Power Density Spectrum 1999.4.1 Properties of the Power Density Spectrum 2009.4.2 Cross Spectrum 2019.4.3 Spectral Density Matrix 2019.5 Filtering and Sampling of WSS Random Fields 2029.5.1 LSI Filtering of a Scalar WSS Random Field 2029.5.2 Why is Sf(u) Called a Power Density Spectrum? 2049.5.3 LSI Filtering of a WSS Color Random Field 2059.5.4 Sampling of a WSS Continuous-Domain Random Field 2069.6 Estimation of the Spectral Density Matrix 207Problems 21410 Analysis and Design of Multidimensional FIR Filters 21510.1 Introduction 21510.2 Moving Average Filters 21510.3 Gaussian Filters 21710.4 Band-pass and Band-stop Filters 22010.5 Frequency-Domain Design of Multidimensional FIR Filters 22510.5.1 FIR Filter Design Using Windows 22610.5.2 FIR Filter Design Using Least-pth Optimization 229Problems 23611 Changing the Sampling Structure of an Image 23711.1 Introduction 23711.2 Sublattices 23711.3 Upsampling 23911.4 Downsampling 24511.5 Arbitrary Sampling Structure Conversion 24811.5.1 Sampling Structure Conversion Using a Common Superlattice 24811.5.2 Polynomial Interpolation 251Problems 25412 Symmetry Invariant Signals and Systems 25512.1 LSI Systems Invariant to a Group of Symmetries 25512.1.1 Symmetries of a Lattice 25512.1.2 Symmetry-Group Invariant Systems 25812.1.3 Spaces of Symmetric Signals 26112.2 Symmetry-Invariant Discrete-Domain Periodic Signals and Systems 26912.2.1 Symmetric Discrete-Domain Periodic Signals 27012.2.2 Discrete-Domain Periodic Symmetry-Invariant Systems 27112.2.3 Discrete-Domain Symmetry-Invariant Periodic Fourier Transform 27312.3 Vector-Space Representation of Images Based on the Symmetry-Invariant Periodic Fourier Transform 28213 Lattices 28913.1 Introduction 28913.2 Basic Definitions 28913.3 Properties of Lattices 29313.4 Reciprocal Lattice 29413.5 Sublattices 29513.6 Cosets and the Quotient Group 29613.7 Basis Transformations 29813.7.1 Elementary Column Operations 29913.7.2 Hermite Normal Form 30013.8 Smith Normal Form 30213.9 Intersection and Sum of Lattices 304Appendix A: Equivalence Relations 311Appendix B: Groups 313Appendix C: Vector Spaces 315Appendix D: Multidimensional Fourier Transform Properties 319References 323Index 329