Henry Kang provides the fundamental color principles and mathematical tools to prepare the reader for a new era of color reproduction, and for subsequent applications in multispectral imaging, medical imaging, remote sensing, and machine vision. This book is intended to bridge the gap between color science and computational color technology, putting color adaptation, color constancy, color transforms, color display, and color rendition in the domain of vector-matrix representations and theories. *Computational Color Technology* deals with color digital images on the spectral level using vector-matrix representations so that the reader can learn to process digital color images via linear algebra and matrix theory.

Pages: 524

ISBN: 9780819461193

Volume: PM159

- Preface xv
- Acknowledgments xix
- 1 Tristimulus Specification 1
- 1.1 Definitions of CIE Tristimulus Values 1
- 1.2 Vector-Space Representations of Tristimulus Values 3
- 1.3 Object Spectrum 5
- 1.4 Color-Matching Functions 5
- 1.5 CIE Standard Illuminants 10
- 1.5.1 Standard viewing conditions 13
- 1.6 Effect of Illuminant 14
- 1.7 Stimulus Function 15
- 1.8 Perceived Object 15
- 1.9 Remarks 16
- References 16
- 2 Color Principles and Properties 17
- 2.1 Visual Sensitivity and Color-Matching Functions 17
- 2.2 Identity Property 19
- 2.3 Color Match 20
- 2.4 Transitivity Law 21
- 2.5 Proportionality Law 21
- 2.6 Additivity Law 21
- 2.7 Dependence of Color-Matching Functions on Choice of Primaries 22
- 2.8 Transformation of Primaries 22
- 2.9 Invariant of Matrix A (Transformation of Tristimulus Vectors) 23
- 2.10 Constraints on the Image Reproduction 23
- References 24
- 3 Metamerism 27
- 3.1 Types of Metameric Matching 27
- 3.1.1 Metameric illuminants 28
- 3.1.2 Metameric object spectra 28
- 3.1.3 Metameric stimulus functions 28
- 3.2 Matrix R Theory 29
- 3.3 Properties of Matrix R 31
- 3.4 Metamers Under Different Illuminants 37
- 3.5 Metameric Correction 39
- 3.5.1 Additive correction 39
- 3.5.2 Multiplicative correction 39
- 3.5.3 Spectral correction 39
- 3.6 Indices of Metamerism 39
- 3.6.1 Index of metamerism potential 40
- References 40
- 4 Chromatic Adaptation 43
- 4.1 Von Kries Hypothesis 43
- 4.2 Helson-Judd-Warren Transform 46
- 4.3 Nayatani Model 47
- 4.4 Bartleson Transform 48
- 4.5 Fairchild Model 49
- 4.6 Hunt Model 51
- 4.7 BFD Transform 52
- 4.8 Guth Model 53
- 4.9 Retinex Theory 53
- 4.10 Remarks 54
- References 54
- 5 CIE Color Spaces 57
- 5.1 CIE 1931 Chromaticity Coordinates 57
- 5.1.1 Color gamut boundary of CIEXYZ 57
- 5.2 CIELUV Space 59
- 5.2.1 Color gamut boundary of CIELUV 60
- 5.3 CIELAB Space 60
- 5.3.1 CIELAB to CIEXYZ transform 62
- 5.3.2 Color gamut boundary of CIELAB 62
- 5.4 Modifications 65
- 5.5 CIE Color Appearance Model 69
- 5.6 S-CIELAB 73
- References 73
- 6 RGB Color Spaces 77
- 6.1 RGB Primaries 77
- 6.2 Transformation of RGB Primaries 80
- 6.2.1 Conversion formula 81
- 6.2.2 Conversion formula between RGB primaries 83
- 6.3 RGB Color-Encoding Standards 84
- 6.3.1 Viewing conditions 84
- 6.3.2 Digital representation 84
- 6.3.3 Optical-electronic transfer function 85
- 6.4 Conversion Mechanism 86
- 6.5 Comparisons of RGB Primaries and Encoding Standards 86
- 6.6 Remarks 99
- References 99
- 7 Device-Dependent Color Spaces 103
- 7.1 Red-Green-Blue (RGB) Color Space 103
- 7.2 Hue-Saturation-Value (HSV) Space 104
- 7.3 Hue-Lightness-Saturation (HLS) Space 105
- 7.4 Lightness-Saturation-Hue (LEF) Space 106
- 7.5 Cyan-Magenta-Yellow (CMY) Color Space 107
- 7.6 Ideal Block-Dye Model 108
- 7.6.1 Ideal color conversion 108
- 7.7 Color Gamut Boundary of Block Dyes 111
- 7.7.1 Ideal primary colors of block dyes 112
- 7.7.2 Additive color mixing of block dyes 115
- 7.7.3 Subtractive color mixing of block dyes 115
- 7.8 Color Gamut Boundary of Imaging Devices 120
- 7.8.1 Test target of color gamut 122
- 7.8.2 Device gamut model and interpolation method 122
- 7.9 Color Gamut Mapping 124
- 7.9.1 Color-mapping algorithm 125
- 7.9.2 Directional strategy 126
- 7.9.3 Criteria of gamut mapping 129
- 7.10 CIE Guidelines for Color Gamut Mapping 129
- References 130
- 8 Regression 135
- 8.1 Regression Method 135
- 8.2 Forward Color Transformation 139
- 8.3 Inverse Color Transformation 141
- 8.4 Extension to Spectral Data 142
- 8.5 Results of Forward Regression 143
- 8.6 Results of Inverse Regression 146
- 8.7 Remarks 148
- References 149
- 9 Three-Dimensional Lookup Table with Interpolation 151
- 9.1 Structure of 3D Lookup Table 151
- 9.1.1 Packing 151
- 9.1.2 Extraction 152
- 9.1.3 Interpolation 153
- 9.2 Geometric Interpolations 153
- 9.2.1 Bilinear interpolation 154
- 9.2.2 Trilinear interpolation 155
- 9.2.3 Prism interpolation 157
- 9.2.4 Pyramid interpolation 159
- 9.2.5 Tetrahedral interpolation 161
- 9.2.6 Derivatives and extensions 163
- 9.3 Cellular Regression 164
- 9.4 Nonuniform Lookup Table 165
- 9.5 Inverse Color Transform 166
- 9.6 Sequential Linear Interpolation 168
- 9.7 Results of Forward 3D Interpolation 170
- 9.8 Results of Inverse 3D Interpolation 177
- 9.9 Remarks 180
- References 180
- 10 Metameric Decomposition and Reconstruction 183
- 10.1 Metameric Spectrum Decomposition 183
- 10.2 Metameric Spectrum Reconstruction 189
- 10.2.1 Spectrum reconstruction from the fundamental
- and metameric black 189
- 10.2.2 Spectrum reconstruction from tristimulus values 191
- 10.2.3 Error measures 194
- 10.3 Results of Spectrum Reconstruction 194
- 10.3.1 Results from average fundamental and metameric black 194
- 10.3.2 Results of spectrum reconstruction from tristimulus values 199
- 10.4 Application 200
- 10.5 Remarks 201
- References 202
- 11 Spectrum Decomposition and Reconstruction 203
- 11.1 Spectrum Reconstruction 203
- 11.2 General Inverse Method 204
- 11.2.1 Spectrum reconstruction via orthogonal projection 205
- 11.2.2 Spectrum reconstruction via smoothing inverse 205
- 11.2.3 Spectrum reconstruction via Wiener inverse 209
- 11.3 Spectrum Decomposition and Reconstruction Methods 212
- 11.4 Principal Component Analysis 212
- 11.5 Basis Vectors 214
- 11.6 Spectrum Reconstruction from the Input Spectrum 220
- 11.7 Spectrum Reconstruction from Tristimulus Values 223
- 11.8 Error Metrics 224
- 11.9 Results and Discussions 224
- 11.9.1 Spectrum reconstruction from the object spectrum 225
- 11.9.2 Spectrum reconstruction from the tristimulus values 228
- 11.10 Applications 229
- References 230
- 12 Computational Color Constancy 233
- 12.1 Image Irradiance Model 233
- 12.1.1 Reflection phenomenon 234
- 12.2 Finite-Dimensional Linear Models 236
- 12.3 Three-Two Constraint 240
- 12.4 Three-Three Constraint 242
- 12.4.1 Gray world assumption 243
- 12.4.2 Saellstroen-Buchsbaum model 244
- 12.4.3 Dichromatic reflection model 245
- 12.4.4 Estimation of illumination 246
- 12.4.5 Other dichromatic models 250
- 12.4.6 Volumetric model 253
- 12.5 Gamut-Mapping Approach 255
- 12.6 Lightness/Retinex Model 256
- 12.7 General Linear Transform 258
- 12.8 Spectral Sharpening 259
- 12.8.1 Sensor-based sharpening 260
- 12.8.2 Data-based sharpening 261
- 12.8.3 Perfect sharpening 264
- 12.8.4 Diagonal transform of the 3-2 world 266
- 12.9 Von Kries Color Prediction 266
- 12.10 Remarks 268
- References 268
- 13 White-Point Conversion 273
- 13.1 White-Point Conversion via RGB Space 273
- 13.2 White-Point Conversion via Tristimulus Ratios of Illuminants 283
- 13.3 White-Point Conversion via Difference in Illuminants 286
- 13.4 White-Point Conversion via Polynomial Regression 295
- 13.5 Remarks 298
- References 299
- 14 Multispectral Imaging 301
- 14.1 Multispectral Irradiance Model 303
- 14.2 Sensitivity and Uniformity of a Digital Camera 305
- 14.2.1 Spatial uniformity of a digital camera 306
- 14.2.2 Spectral sensitivity of a digital camera 308
- 14.3 Spectral Transmittance of Filters 308
- 14.3.1 Design of optimal filters 309
- 14.3.2 Equal-spacing filter set 310
- 14.3.3 Selection of optimal filters 311
- 14.4 Spectral Radiance of Illuminant 311
- 14.5 Determination of Matrix ' AE 312
- 14.6 Spectral Reconstruction 314
- 14.6.1 Tristimulus values using PCA 314
- 14.6.2 Pseudo-inverse estimation 315
- 14.6.3 Smoothing inverse estimation 316
- 14.6.4 Wiener estimation 316
- 14.7 Multispectral Image Representation 317
- 14.8 Multispectral Image Quality 319
- References 320
- 15 Densitometry 325
- 15.1 Densitometer 326
- 15.1.1 Precision of density measurements 327
- 15.1.2 Applications 329
- 15.2 Beer-Lambert-Bouguer Law 331
- 15.3 Proportionality 332
- 15.3.1 Density ratio measurement 334
- 15.4 Additivity 334
- 15.5 Proportionality and Additivity Failures 335
- 15.5.1 Filter bandwidth 335
- 15.5.2 First-surface reflection 335
- 15.5.3 Multiple internal reflections 335
- 15.5.4 Opacity 335
- 15.5.5 Halftone pattern 336
- 15.5.6 Tone characteristics of commercial printers 336
- 15.6 Empirical Proportionality Correction 338
- 15.7 Empirical Additivity Correction 341
- 15.8 Density-Masking Equation 342
- 15.9 Device-Masking Equation 343
- 15.9.1 Single-step conversion of the device-masking equation 344
- 15.9.2 Multistep conversion of the device-masking equation 345
- 15.9.3 Intuitive approach 346
- 15.10 Performance of the Device-Masking Equation 347
- 15.11 Gray Balancing 347
- 15.12 Gray-Component Replacement 349
- 15.13 Digital Implementation 350
- 15.13.1 Results of the integer masking equation 351
- 15.14 Remarks 353
- References 354
- 16 Kubelka-Munk Theory 355
- 16.1 Two-Constant Kubelka-Munk Theory 356
- 16.2 Single-Constant Kubelka-Munk theory 357
- 16.3 Determination of the Single Constant 360
- 16.4 Derivation of Saunderson's Correction 360
- 16.5 Generalized Kubelka-Munk Model 362
- 16.6 Cellular Extension of the Kubelka-Munk Model 365
- 16.7 Applications 365
- 16.7.1 Applications to multispectral imaging 366
- References 366
- 17 Light-Reflection Model 369
- 17.1 Three-Primary Neugebauer Equations 369
- 17.2 Demichel Dot-Overlap Model 370
- 17.3 Simplifications 371
- 17.4 Four-Primary Neugebauer Equation 373
- 17.5 Cellular Extension of the Neugebauer Equations 375
- 17.6 Spectral Extension of the Neugebauer Equations 376
- References 382
- 18 Halftone Printing Models 385
- 18.1 Murray-Davies Equation 385
- 18.1.1 Spectral extension of the Murray-Davies equation 387
- 18.1.2 Expanded Murray-Davies model 388
- 18.2 Yule-Nielsen Model 388
- 18.2.1 Spectral extension of Yule-Nielsen model 390
- 18.3 Area Coverage-Density Relationship 392
- 18.4 Clapper-Yule Model 393
- 18.4.1 Spectral extension of the Clapper-Yule model 394
- 18.5 Hybrid Approaches 394
- 18.6 Cellular Extension of Color-Mixing Models 395
- 18.7 Dot Gain 396
- 18.8 Comparisons of Halftone Models 400
- References 402
- 19 Issues of Digital Color Imaging 407
- 19.1 Human Visual Model 407
- 19.1.1 Contrast sensitivity function 409
- 19.1.2 Color visual model 410
- 19.2 Color Appearance Model 412
- 19.3 Integrated Spatial-Appearance Model 413
- 19.4 Image Quality 413
- 19.5 Imaging Technology 415
- 19.5.1 Device characteristics 415
- 19.5.2 Measurement-based tone correction 416
- 19.5.3 Tone level 417
- 19.6 Device-Independent Color Imaging 418
- 19.7 Device Characterization 421
- 19.8 Color Spaces and Transforms 423
- 19.8.1 Color-mixing models 424
- 19.9 Spectral Reproduction 425
- 19.10 Color-Gamut Mapping 425
- 19.11 Color Measurement 426
- 19.12 Color-Imaging Process 426
- 19.12.1 Performance 427
- 19.12.2 Cost 428
- 19.13 Color Architecture 428
- 19.14 Transformations between sRGB and Internet FAX Color Standard 430
- 19.15 Modular Implementation 434
- 19.15.1 SRGB-to-CIEXYZ transformation 434
- 19.15.2 Device/RGB-to-CIEXYZ transformation 436
- 19.15.3 CIEXYZ-to-CIELAB transformation 436
- 19.15.4 CIELAB-to-CIEXYZ transformation 437
- 19.15.5 CIEXYZ-to-colorimetric RGB transformation 438
- 19.15.6 CIEXYZ-to-Device/RGB transformation 438
- 19.16 Results and Discussion 439
- 19.16.1 SRGB-to-CIEXYZ transformation 439
- 19.16.2 Device/RGB-to-CIEXYZ transformation 440
- 19.16.3 CIEXYZ-to-CIELAB transformation 440
- 19.16.4 CIELAB-to-CIEXYZ transformation 441
- 19.16.5 CIEXYZ-to-sRGB transformation 441
- 19.16.6 Combined computational error 442
- 19.17 Remarks 443
- References 444

#### Appendices

- A1 Conversion Matrices 449
- A2 Conversion Matrices from RGB to ITU-R.BT.709/RGB 471
- A3 Conversion Matrices from RGB to ROMM/RGB 475
- A4 RGB Color-Encoding Standards 479
- A4.1 SMPTE-C/RGB 479
- A4.2 European TV Standard (EBU) 480
- A4.3 American TV YIQ Standard 481
- A4.4 PhotoYCC 482
- A4.5 SRGB Encoding Standards 483
- A4.6 E-sRGB Encoding Standard 484
- A4.7 Kodak ROMM/RGB Encoding Standard 485
- A4.8 Kodak RIMM/RGB 486
- References 487
- A5 Matrix Inversion 489
- A5.1 Triangularization 489
- A5.2 Back Substitution 491
- References 492
- A6 Color Errors of Reconstructed CRI Spectra with Respect to Measured Values 493
- A7 Color Errors of Reconstructed CRI Spectra with Respect to Measured Values Using Tristimulus Inputs 497
- A8 White-Point Conversion Accuracies Using Polynomial Regression 499
- A9 Digital Implementation of the Masking Equation 503
- A9.1 Integer Implementation of Forward Conversion 503
- A9.2 Integer Implementation of Inverse Conversion 506
- Index 509

### Preface

Recent developments in color imaging have evolved from the classical broadband description to a spectral representation. Color reproductions were attempted with spectral matching, and image capture via digital camera has extended to multispectral recording. These topics have appeared in a couple of books and scattered across several digital imaging journals. However, there is no integrated view or consistent representation of spectral color imaging. This book is intended to fill that void and bridge the gap between color science and computational color technology, putting color adaptation, color constancy, color transforms, color display, and color rendition in the domain of vector-matrix representations and theories. The aim of this book is to deal with color digital images in the spectral level using vector-matrix representations so that one can process digital color images by employing linear algebra and matrix theory.

This is the onset of a new era of color reproduction. Spectral reconstruction provides the means for the highest level of color matching. As pointed out by Dr. R. W. G. Hunt, spectral color matching gives color fidelity under any viewing conditions. However, current color technology and mathematical tools are still insufficient for giving accurate spectral reconstructions (and may never be sufficient because of device variations and color measurement uncertainties). Nevertheless, this book provides the fundamental color principles and mathematical tools to prepare one for this new era and for subsequent applications in multispectral imaging, medical imaging, remote sensing, and machine vision. The intent is to bridge color science, mathematical formulations, psychophysical phenomena, physical models, and practical implementations all in one work.

The contents of this book are primarily aimed at digital color imaging professionals for research and development purposes. This book can also be used as a textbook for undergraduate and graduate students in digital imaging, printing, and graphic arts. The book is organized into five parts. The first part, Chapters 1-7, is devoted to the fundamentals of color science such as the CIE tristimulus specifications, principles of color matching, metamerism, chromatic adaptation, and color spaces. These topics are presented in vector-matrix forms, giving a new flavor to old material and, in many cases, revealing new perspectives and insights. This is because the representation of the spectral sensitivity of human vision and related visual phenomena in vector-matrix form provide the foundation for computational color technology. The vector-space representation makes possible the use of the well-developed fields of linear algebra and matrix theory.

Chapter 1 gives the definitions of CIE tristimulus values. Each component, such as color matching function, illuminant, and object spectrum, is given in vector-matrix notation under several different vector associations of components. This sets the stage for subsequent computations. Chapter 2 presents the fundamental principles governing color matching such as the identity, proportionality, and additivity laws. Based on these laws, the conversion of primaries is simply a linear transform. Chapter 3 discusses the metameric matching from the perspective of the vector-matrix representation, which allows the derivation of matrix R, the orthogonal projection of the tristimulus color space. The properties of matrix R are discussed in detail. Several levels of the metameric matching are discussed and metameric corrections are provided. Chapter 4 presents various models of the chromatic adaptation from the fundamental von Kries hypothesis to complex retinex theory. Chapter 5 presents CIE color spaces and their relationships. Color gamut boundaries for CIELAB are derived, and a spatial extension of CIELAB is given. The most recent color appearance model, CIE CAM2000, is also included. Chapter 6 gives a comprehensive collection of RGB primaries and encoding standards and derives the conversion formula between RGB primaries. These standards are compared and their advantages and disadvantages are discussed. Chapter 7 presents the device-dependent color spaces based on the ideal block dye model. The methods of obtaining the color gamut boundary of imaging devices and color gamut mapping are provided. They are the essential parts of color rendering at the system level.

The second part of the book, Chapters 8-11, provides tools for color transformation and spectrum reconstruction. These empirical methods are developed purely on mathematical grounds and are formulated in the vector-matrix forms to enable matrix computations. In Chapter 8, the least-square minimization regression technique is given, and the vector-matrix formulation of the forward and inverse color transformations are derived and extended to the spectral domain. To test the quality of the regression technique, real-world color conversion data are used. Chapter 9 focuses on lookup-table techniques, and the structure of the 3D lookup table and geometric interpolations are discussed in detail. Several extensions and improvements are also provided, and real data are used to test the value of the 3DLUT technique. Chapter 10 shows the simplest spectrum reconstruction method by using the metameric decomposition of the matrix R theory. Two methods are developed for spectrum reconstruction; one using the sum of metameric black and fundamental spectra, and the other using tristimulus values without spectral information. The methods are tested by using CIE illuminants and spectra of the "Color Rendering Index" (CRI). Chapter 11 provides several sophisticated methods of the spectrum reconstruction, including the general inverse methods such as the smoothing inverse and Wiener inverse and the principal component analysis. Again, these methods are tested by using CRI spectra because spectrum reconstruction is the foundation for color spectral imaging, utilizing the vector-matrix representations.

The third part, Chapters 12-14, shows applications of spectral reconstruction to color science and technology, such as color constancy, white-point conversion, and multispectral imaging. This part deals with the psychophysical aspect of the surface reflection, considering signals reflected into the human visual pathway from the object surface under certain kinds of illumination. We discuss the topics of surface illumination and reflection, including metameric black, color constancy, the finite-dimensional linear model, white-point conversion (illuminant mapping), and multispectral image processing. These methods can be used to estimate (or recover) surface and illuminant spectra, and can be applied to remote sensing and machine vision. Chapter 12 discusses computational color constancy, which estimates the surface spectrum and illumination simultaneously. The image irradiance model and finite-dimensional linear models for approximating the color constancy phenomenon are presented, and various constraints are imposed in order to solve the finitedimensional linear equations. Chapter 13 describes the application of fundamental color principles to white-point conversion. Several methods are developed and the conversion accuracy is compared. Chapter 14 discusses the applications of spectrum reconstruction for multispectral imaging. Multispectral images are acquired by digital cameras, and the camera characteristics with respect to color image quality are discussed. For device compatibility and cross-media rendering, a proposed multispectral image representation is given. Finally, the multispectral image quality is discussed.

The fourth part, Chapters 15-18, deals with the physical model accounting for the intrinsic physical and chemical interactions occurring in the colorants and substrates. This is mainly applied to the printing process, halftone printing in particular. In this section, physical models of the Neugebauer equations, the Murray-Davies equation, the Yule-Nielsen model, the Clapper-Yule model, the Beer-Lambert-Bouguer law, the density-masking equation, and the Kubelka-Munk theory are discussed. These equations are then reformulated in the vector-matrix notation and expanded in both spectral and spatial domains with the help of the vector-matrix theory in order to derive new insights and develop new ways of employing these equations. It is shown that this spectral extension has applications in the spectral color reproduction that greatly improve the color image quality. Chapter 15 describes densitometry beginning with the Beer-Lambert-Bouguer law and its proportionality and additivity failures. Empirical corrections for proportionality and additivity failures are then developed. The density masking equation is then presented and extended to the device masking equation, which can be applied to gray balancing, gray component replacement, and maximum ink loading. Chapter 16 reformulates the Kubelka-Munk theory in the vector-matrix form. A general Kubelka-Munk model is presented using four fluxes that can be reduced to other halftone printing models. Chapter 17 presents the Neugebauer equations, extending them to spectral domain by using the vector-matrix notation. This notation provides the means to finding the inverse Neugebauer equations and to obtaining the amounts of primary inks. Finally, Chapter 18 contains various halftone printing models such as the Murray-Davies equation, the Yule-Nielsen model, and the Clapper-Yule model. Chapter 18 also discusses dot gain and describes a physical model that takes the optical and spatial components into account. The last part, Chapter 19, expresses my view of the salient issues in digital color imaging. Digital color imaging is an extremely complex phenomenon, involving the human visual model, the color appearance model, image quality, imaging technology, device characterization and calibration, color space transformation, color gamut mapping, and color measurement. The complexity can be reduced and image quality improved by a proper color architecture design. A simple transformation between sRGB and Internet FAX is used to illustrate this point.

Henry R. Kang

March, 2006

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