This innovative volume summarizes existing knowledge in the field, attempting to present as much data as possible about colour, accumulated in various branches of science (physics, phychophysics, colorimetry, physiology) from a unified theoretical position. Written by a colour specialist and a professional mathematician, the book offers a new theoretical framework based on functional analysis and convex analysis. Employing these branches of mathematics, instead of more conventional linear algebra, allows them to provide the knowledge required for developing techniques to measure colour appearance to the standards adopted in colorimetric measurements. The authors describe the mathematics in a language that is understandable for colour specialists and include a detailed overview of all chapters to help readers not familiar with colour science.
Divided into two parts, the book first covers various key aspects of light colour, such as colour stimulus space, colour mechanisms, colour detection and discrimination, light-colour perception typology, and light metamerism. The second part focuses on object colour, featuring detailed coverage of object-colour perception in single- and multiple-illuminant scenes, object-colour solid, colour constancy,
metamer mismatching, object-colour indeterminacy and more. Throughout the book, the authors combine differential geometry and topology with the scientific principles on which colour measurement and specification are currently based and applied in industrial applications. - Presents a unique compilation of the author’s substantial contributions to colour science - Offers a new approach to colour perception and measurement, developing the theoretical framework used in colorimetry - Bridges the gap between colour engineering and a coherent mathematical theory of colour - Outlines mathematical foundations applicable to the colour vision of humans and animals as well as technologies equipped with artificial photosensors - Contains algorithms for solving various problems in colour science, such as the mathematical problem of describing metameric lights - Formulates all results to be accessible to non-mathematicians and colour specialists
Foundations of Colour Science: From Colorimetry to Perception is an invaluable resource for academics, researchers, industry professionals and undergraduate and graduate students with interest in a mathematical approach to the science of colour.
Table of Contents
1 Outline for readers in a hurry 1
I Light colour 81
2 Colour stimulus space and colour mechanisms 85
2.1 Grassmann structures and Grassmann colour codes 89
2.2 Continuous Grassmann structures and continuous Grassmann colour codes 97
3 Identification of Grassmann structures based on metameric matching 101
3.1 Colourmatching functions 102
3.2 Monochromatic primaries and colour matching functions in the trichromatic case (=3) 109
3.3 Fundamental colour mechanisms in human colour vision 112
3.3.1 K¨onig’s approach to identification of the fundamental colourmechanisms 120
3.3.2 Some estimates of the cone fundamentals used in colour research and applications 123
4 Colour-signal cone 129
4.1 Strong colour-signal-cone-boundary hypothesis 133
4.2 Empirical status of the strong colour-signal-cone-boundary hypothesis 138
4.3 Colour-signal-cone-boundary hypothesis 145
4.4 The colour-signal cone of a 3-pigment Grassmann-Govardovskii structure 149
5 Colour stimulus manifold 153
5.1 Three-dimensional colour stimulusmanifold 155
5.2 Non-linear colour stimulus map Colour stimulus transformation caused by themedium 160
5.2.1 The colour stimulus shift caused by the medium variations 161
5.2.2 Colour robustness tomediumvariations 163
5.3 Causes of individual differences in trichromatic colour matching 165
5.3.1 Effect of the photopigment peak sensitivity on the-coordinates 166
5.3.2 Effect of the ocular media transmittance on -coordinates 171
5.3.3 Trade-off between the ocular media spectral transmittance and the photopigment peak sensitivity in their effect on colour 174
5.3.4 Dependence of the equivalent peak-wavelength shift on light Impossibility to overcome colour deficiency using a coloured filter 176
5.3.5 Parametric identification of fundamental colour mechanisms 180
6 Light metamerism 183
6.1 Metamer sets 184
6.2 Colour mechanisms’ transformations preserving light metamerism 188
6.3 Lightmetamerismindex 190
7 Light metamer mismatching 191
7.1 Metamer-mismatch regions 191
7.2 Indices of lightmetamer mismatching 197
7.3 Computing trichromaticmetamer-mismatch regions 202
7.3.1 Effect of the spectral positioning of photopigments onmetamer mismatching 206
7.3.2 Effect of the peak photopigment absorbance on metamer mismatching 210
7.3.3 Metamer mismatching depending on the position in the chromaticity diagram 211
7.3.4 Metamer mismatching induced by pre-receptoral filters 211
7.3.5 Differences between cone fundamentals as revealed bymetamer mismatching 217
7.3.6 Metamer mismatching for the 10◦ colour matching functions of Stiles and Burch 221
7.3.7 Metamer mismatching induced by neutral density filters 234
7.3.8 Metamer mismatching produced by camera sensors 238
8 Light-colour perception 243
8.1 Achromatic scales and achromatic codes 248
8.1.1 Ordinal brightness scales 249
8.1.2 Grassmann brightness code Luminance 254
8.2 Hue, purity, and brightness fibre bundles Cylindrical and psychophysical colour coordinates 262
8.3 Colour transformation caused by media and metamer mismatching, as expressed in the psychophysical colour coordinates 270
8.4 Light-colour perception in dichromats 273
8.5 Chromatic structures 280
8.5.1 Partial hue-matching 283
8.5.2 Experiment on partial hue-matching 289
8.5.3 Colour categories 292
8.5.4 Chromatically ordered structures 297
8.5.5 Chromatic scales and chromatic codes 299
8.5.6 Hue, purity and saturation in chromatic structures 301
8.6 Light-colour manifold 304
8.6.1 Hue cyclic order 305
8.6.2 Light-colour manifold 308
8.6.3 Circular Hering structures, their representation and experimental identification 311
8.6.4 Light-colour manifold vs colour stimulus manifold 321
9 Typology of light-colour perception Inter-individual differences 329
10 Colour matching structures and matching metamerism 341
10.1 Colourmatching structures 347
10.2 Matchingmetamerism 358
11 Identification of Grassmann structures induced by colour matching structures 363
11.1 Colour matching set, threshold set, and sensitivity function 364
11.2 Regular and strongly regular tolerance extensions 368
11.3 Identification of Grassmann structures induced by colour matching tolerance relations 371
11.3.1 Identification of the linear colour mechanism space as a subspace in the linear span of a given set of linearly independent functionals 372
11.3.2 Deriving the linear colour mechanism space from the colour matching set (the method of tangential hyperplane 378
11.3.3 Deriving the fundamental colour mechanisms from the colour matching set that they generate (the method of quadratic approximation) 383
12 Identification of indiscriminate relations Colour detection and discrimination 391
12.1 Colour detectionmodels 394
12.1.1 Single-channel detectionmodels 394
12.1.2 Fundamental colour mechanisms revisited 397
12.1.3 Multi-channel detectionmodels 399
12.2 Peak-detector model equivalent to a sublinear colour detectionmodel 400
12.2.1 Sublinear colour detectionmodels 401
12.2.2 Multi-channel sublinearmodels 402
12.2.3 Themost sensitive colour mechanisms 404
12.3 Colour discriminationmodels 409
13 In search of colour mechanisms in the eye and the brain 413
13.1 Do the cone photoreceptor responses encode the colour stimulus? 413
13.1.1 Local non-linearity of the photoreceptor response 414
13.1.2 Light adaptation in photoreceptors 415
13.1.3 Spatial interaction between the cone photoreceptors 417
13.1.4 Why the colour stimulus cannot be derived from the cone photoreceptor responses 417
13.2 Do cone-opponent neural cells encode the opponent chromatic codes? 418
13.3 Transition to a different paradigm 425
13.3.1 From symmetric to asymmetric colour matching 425
13.3.2 Fromlight stimulus to light-stimulus array 428
13.3.3 On the notion of ”neural image” 430
13.4 Spatio-chromatic processing in the visual cortex 436
13.4.1 Estimating luminance-pattern gradient using simple cortical cells 436
13.4.2 Directional gradient-encoding with double-opponent cells 446
13.4.3 Difference in spatial sensitivity of (M+L)-, (M-L)-, and S-(M+L)-cells, and its implication for colour perception 449
13.4.4 Representation of the colour-signal surface in the form of its tangent bundle 450
Object colour 458
14 Object-colour solid 465
14.1 General properties of the object-colour solid 466
14.2 Optimal object stimuli 468
14.3 Elementary step functions as optimal object stimuli 470
14.4 Optimal object stimuli for trichromatic human observers 472
14.5 Condition for all step functions of degree to be optimal object stimuli 472
15 Trichromatic regular object-colour solid 475
15.1 Meridians of the trichromatic regular object-colour solid 475
15.2 Equator of the trichromatic object-colour solid and strictly optimal object stimuli 481
16 Object-colour stimulus manifold 489
16.1 Objectmetamerism 489
16.2 Object atlas 493
16.3 Object-colour stimulus manifold Illuminant-induced nonlinear object-colour stimulusmap 496
16.4 Trichromatic object-colour stimulusmanifold 497
16.4.1 Trichromatic regular object-colour stimulus manifold and its spherical representation 497
16.4.2 Spherical representation of the trichromatic objectcolour stimulus manifold and the object-colour stimulus gamut 502
16.4.3 Object-colour stimulus shift induced by the illuminant change 504
17 Object-colour perception in a single-illuminant scene 507
17.1 Perceptual object-colour coordinates 513
17.2 Perceptual correlates of coordinates 516
17.3 Effect of illumination on object-colour in a single-illuminant scene: Object-colour shift induced by illumination 521
17.4 Object-colour perception by dichromats in a single-illuminant scene 524
18 Object metamer mismatching 535
18.1 Metamer-mismatch regions 535
18.2 Numerical evaluation ofmetamer-mismatch regions 539
18.3 Indices of objectmetamer mismatching 542
18.4 Object-metamerism-preserving transformations of colour mechanisms 545
19 Object-colour perception in a multiple-illuminant scene 549
19.1 Object/light colour equivalence and its inseparability 554
19.2 Object/light atlas 556
19.3 Object/light colour stimulusmanifold 557
19.3.1 Asymmetric colourmatching 557
19.3.2 Material colour 561
19.3.3 Lighting colour 562
19.3.4 Object/light colour stimulus manifold Material and lighting components of object/light colour stimulus manifold Material- and lighting-colour coordinates 564
19.4 Material colour shift induced by illumination change Implication for the problemof ”colour constancy” 569
20 Object-colour indeterminacy 573
20.1 Trade-off between object and light components 573
20.2 Trade-off betweenmaterial and lighting colours 579
20.2.1 Invariant relationship between lightness and lighting brightness 581
20.2.2 Invariant relationship between lightness, lighting brightness and shading brightness 586
20.2.3 Shading as a sensory basis of shape 588
20.2.4 Invariant relationship between material-colour image and lighting-colour image in the chromatic domain 590
20.3 Object-colour indeterminacy in variegated scenes Impact of articulation 591
20.4 Implication for measuring object-colour 594
21 On perception in general: An outline of an alternative approach 601
21.1 What is colour for? 603
21.2 The need for a new approach to perception: Linguistic metaphor 607
22 Epilogue 619
References 623
A Some auxiliary facts from functional analysis 649
A.1 Banach spaces of measures and functions, and stimulus spaces 649
A.2 Convex analysis 652
B Proofs 657
