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Spie Press Book

Computational Color Technology
Author(s): Henry R. Kang
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Book Description

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.

Book Details

Date Published: 17 May 2006
Pages: 524
ISBN: 9780819461193
Volume: PM159

Table of Contents
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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


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


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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