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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Presents the science of colour from new perspectives and outlines results obtained from the authors' work in the mathematical theory of colour 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.
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Table of Contents
Cover
Title Page
Copyright
Preface
1 Outline for Readers in a Hurry
Notes
Part I: Light Colour
2 Colour Stimulus Space and Colour Mechanisms
2.1 Grassmann structures and Grassmann colour codes
2.2 Continuous Grassmann structures and continuous Grassmann colour codes
Notes
3 Identification of Grassmann Structures Based on Metameric Matching
3.1 Colour matching functions
3.2 Monochromatic primaries and colour matching functions in the trichromatic case (
)
3.3 Fundamental colour mechanisms in human colour vision
Notes
4 Colour‐Signal Cone
4.1 Strong colour‐signal‐cone‐boundary hypothesis
4.2 Empirical status of the strong colour‐signal‐cone‐boundary hypothesis
4.3 Colour‐signal‐cone‐boundary hypothesis
4.4 The colour‐signal cone of a 3‐pigment Grassmann–Govardovskii structure
Notes
5 Colour Stimulus Manifold
5.1 Three‐dimensional colour stimulus manifold
5.2 Non‐linear colour stimulus map. Colour stimulus transformation caused by the medium
5.3 Causes of individual differences in trichromatic colour matching
Notes
6 Light Metamerism
6.1 Metamer sets
6.2 Colour mechanisms' transformations preserving light metamerism
6.3 Light metamerism index
Notes
7 Light Metamer Mismatching
7.1 Metamer‐mismatch regions
7.3 Computing trichromatic metamer‐mismatch regions
Notes
8 Light‐Colour Perception
8.1 Achromatic scales and achromatic codes
8.2 Hue, purity, and brightness fibre bundles. Cylindrical and psychophysical colour coordinates
8.3 Colour transformation caused by media and metamer mismatching, as expressed in the psychophysical colour coordinates
8.4 Light‐colour perception in dichromats
8.5 Chromatic structures
8.6 Light‐colour manifold
Notes
9 Typology of Light‐Colour Perception. Inter‐Individual Differences
Notes
10 Colour Matching Structures and Matching Metamerism
10.1 Colour matching structures
10.2 Matching metamerism
Notes
11 Identification of Grassmann Structures Induced by Colour Matching Structures
11.1 Colour matching set, threshold set, and sensitivity function
11.2 Regular and strongly regular tolerance extensions
11.3 Identification of Grassmann structures induced by colour matching tolerance relations
Notes
12 Identification of Indiscriminate Relations. Colour Detection and Discrimination
12.1 Colour detection models
12.2 Peak‐detector model equivalent to a sublinear colour detection model
12.3 Colour discrimination models
Notes
13 In Search of Colour Mechanisms in the Eye and the Brain
13.1 Do the cone photoreceptor responses encode the colour stimulus?
13.2 Do cone‐opponent neural cells encode the opponent chromatic codes?
13.3 Transition to a different paradigm
13.4 Spatio‐chromatic processing in the visual cortex
Notes
Part II: Object Colour
14 Object‐Colour Solid
14.1 General properties of the object‐colour solid
14.2 Optimal object stimuli
14.3 Elementary step functions as optimal object stimuli
14.4 Optimal object stimuli for trichromatic human observers
14.5 Condition for all step functions of degree
to be optimal object stimuli
Notes
15 Trichromatic Regular Object‐Colour Solid
15.1 Meridians of the trichromatic regular object‐colour solid
15.2 Equator of the trichromatic object‐colour solid and strictly optimal object stimuli
Notes
16 Object‐Colour Stimulus Manifold
16.1 Object metamerism
16.2 Object atlas
16.3 Object‐colour stimulus manifold. Illuminant‐induced non‐linear object‐colour stimulus map
16.4 Trichromatic object‐colour stimulus manifold
Notes
17 Object‐Colour Perception in a Single‐Illuminant Scene
17.1 Perceptual object‐colour coordinates
17.2 Perceptual correlates of
‐coordinates
17.3 Effect of illumination on object colour in a single‐illuminant scene: Object‐colour shift induced by illumination
17.4 Object‐colour perception by dichromats in a single‐illuminant scene
Notes
18 Object Metamer Mismatching
18.1 Metamer‐mismatch regions
18.2 Numerical evaluation of metamer‐mismatch regions
18.3 Indices of object metamer mismatching
18.4 Object‐metamerism‐preserving transformations of colour mechanisms
Notes
19 Object‐Colour Perception in a Multiple‐Illuminant Scene
19.1 Object/light colour equivalence and its inseparability
19.2 Object/light atlas
19.3 Object/light colour stimulus manifold
19.4 Material colour shift induced by illumination change. Implication for the problem of ‘colour constancy’
Notes
20 Object‐Colour Indeterminacy
20.1 Trade‐off between object and light components
20.2 Trade‐off between material and lighting colours
20.3 Object‐colour indeterminacy in variegated scenes. Impact of articulation
20.4 Implication for measuring object colour
Notes
21 On Perception in General: An Outline of an Alternative Approach
21.1 What is colour for?
21.2 The need for a new approach to perception. A linguistic metaphor
Notes
22 Epilogue
Notes
References
Appendix A: Some Auxiliary Facts from Functional Analysis
A.1 Banach spaces of measures and functions, and stimulus spaces
A.2 Convex analysis
Notes
Appendix B: Proofs
Notes
Index
End User License Agreement
List of Tables
Chapter 3
Table 3.1 Parameter values used by Govardovskii et al. (2000) in Eq. (3.16)...
Chapter 7
Table 7.1 Metamer‐mismatch indices computed for various pairs of observers ...
Table 7.2 Minimal, maximal, mean, and median metamer‐mismatch indices for 2...
Table 7.3 Metamer‐mismatch indices for those pairs of the colour matching f...
Table 7.4 Indices of metamer mismatching induced by the filters shown in Fi...
Table 7.5 Metamer‐mismatch indices corresponding to the chromaticity metame...
Chapter 17
Table 17.1 Critical wavelengths (nm) specifying the optimal stimuli for pro...
List of Illustrations
Chapter 3
Figure 3.1 Colour matching functions of the CIE 1931 Standard Colorimetric O...
Figure 3.2 Colour matching functions of the CIE 1931 Standard Colorimetric O...
Figure 3.3 Colour matching functions of the CIE 1964 Complementary Standard ...
Figure 3.4 CIE 2015
2
colour matching functions.
Figure 3.5 Scaled to unity peak, photopigment absorptance spectra, calculate...
Figure 3.6 Transmittance of the macular pigment of the human eye as a functi...
Figure 3.7 Transmittance of the young human eye lens relative to that at a w...
Figure 3.8 Normalised cone spectral sensitivities (cone fundamentals) based ...
Figure 3.9 Govardovskii's cone fundamentals in semi‐logarithmic coordinates....
Figure 3.10 Smith and Pokorny's cone fundamentals normalised by their peak s...
Figure 3.11 Logarithm of the relative spectral sensitivity of the cone funda...
Figure 3.12 Colour matching functions of the CIE 1931 Standard Colorimetric ...
Figure 3.13 Stockman & Sharp 2
cone fundamentals based on the Stiles & Burc...
Figure 3.14 Spectral sensitivity of three types of human cones as evaluated ...
Chapter 4
Figure 4.1 Spectral colour‐signal curve in the
cone excitation space evalu...
Figure 4.2 Spectral colour‐signal curve in the
cone excitation space evalu...
Figure 4.3 The colour‐signal curve in the CIE 1931 colour‐signal space. Nota...
Figure 4.4 Spectral colour‐signal conical surface plotted for Govardovskii's...
Figure 4.5 Gaussian cone fundamentals. They are obtained by replacing
in E...
Figure 4.6 Spectral locus, i.e., the
‐image of
(black line), plotted for ...
Figure 4.7 Chromaticity gamut (diagram) for Govardovskii's fundamentals (Fig...
Figure 4.8
‐ and
‐cone response ratio as a function of wavelength, as calc...
Figure 4.9 Spectral locus (black line) as derived from the Stockman & Sharpe...
Figure 4.10 Spectral locus (black line) and purple interval (purple line) as...
Figure 4.11 MacLeod–Boynton chromaticity diagram. The abscissa is
, where
Figure 4.12 The CIE 1931 chromaticity diagram. The abscissa
, where
,
, a...
Figure 4.13 Logarithm of the determinant 4.17 as a function of wavelength
....
Chapter 5
Figure 5.1 Radiant spectral density function of CIE illuminant D65.
Figure 5.2 Individual colour matching functions of 49 observers employed by ...
Figure 5.3 Dominant wavelength shift induced by a change in the L‐photopigme...
Figure 5.4 Dominant wavelength shift induced by a change in the
‐photopigme...
Figure 5.5 Spectral power distribution for the blackbody radiators of variou...
Figure 5.6 The profiles of
and
. The parameters of the normal
cumulativ
...
Figure 5.7 Equivalent peak wavelength shift as a function of the blackbody c...
Chapter 7
Figure 7.1 Metamer‐mismatch region (between the blue and red lines) evaluate...
Figure 7.2 The graph of
(the blue line) and
(the red line) as a function...
Figure 7.3 Cone photopigment absorptance curves, same as in Fig. 3.5 (with p...
Figure 7.4
‐,
‐, and
‐cone photopigment excitations are along the axes. T...
Figure 7.5 Convex hull of the cluster of the metamer‐mismatch points depicte...
Figure 7.6 Chromaticity metamer‐mismatch area for an equal energy light (mar...
Figure 7.7 The same chromaticity metamer‐mismatch area as in Fig. 7.6, only ...
Figure 7.8 Volumetric metamer‐mismatch index as a function of
and
.
Figure 7.9 Chromaticity metamer‐mismatch index as a function of
and
.
Figure 7.10 Chromaticity metamer‐mismatch index as a function of
and
for...
Figure 7.11 Chromaticity metamer‐mismatch area for an equal energy light (ma...
Figure 7.12 Series of chromaticity metamer‐mismatch areas plotted in the sam...
Figure 7.13 Volumetric metamer‐mismatch index,
, computed for various value...
Figure 7.14 Chromaticity metamer‐mismatch index,
, computed for the same li...
Figure 7.15 Linear metamer‐mismatch index (Eq. 7.18) computed for the same l...
Figure 7.16 Luminance metamer‐mismatch index computed for the same light and...
Figure 7.17 Linear (squares) and luminance (diamonds) metamer‐mismatch indic...
Figure 7.18 Metamer‐mismatch volume for the equal energy light produced by s...
Figure 7.19 Cone spectral sensitivity functions based on the Govardovskii ph...
Figure 7.20 Metamer‐mismatch volume induced by taking into account of the sp...
Figure 7.21 Chromaticity metamer‐mismatch area induced by taking into accoun...
Figure 7.22 Metamer‐mismatch volume for a mixture (in equal proportions) of ...
Figure 7.23 Chromaticity metamer‐mismatch area for a mixture (in equal propo...
Figure 7.24 Metamer‐mismatch volume produced for the CIE 2006 2° LMS cone fu...
Figure 7.25 Chromaticity metamer‐mismatch area produced for the CIE 2006 2° ...
Figure 7.26 Chromaticity metamer‐mismatch area obtained for the CIE 2006 2° ...
Figure 7.27 Chromaticity metamer‐mismatch area obtained for the CIE 1964 and...
Figure 7.28 Volumetric metamer‐mismatch indices for Stiles & Burch's observe...
Figure 7.29 Chromaticity metamer‐mismatch indices for Stiles & Burch's obser...
Figure 7.30 Linear metamer‐mismatch indices for Stiles & Burch's observers. ...
Figure 7.31 Volumetric vs. chromaticity metamer‐mismatch indices for Stiles ...
Figure 7.32 Volumetric vs. linear metamer‐mismatch indices for Stiles & Burc...
Figure 7.33 Histogram of volumetric metamer‐mismatch indices for Stiles & Bu...
Figure 7.34 Histogram of chromaticity metamer‐mismatch indices for Stiles & ...
Figure 7.35 Histogram of linear metamer‐mismatch indices for Stiles & Burch'...
Figure 7.36 Volumetric metamer mismatch index was computed twice for each pa...
Figure 7.37 Average volumetric metamer‐mismatch indices for Stiles & Burch's...
Figure 7.38 Average chromaticity metamer‐mismatch indices for Stiles & Burch...
Figure 7.39 Asymmetry indices for volumetric metamer‐mismatch indices for St...
Figure 7.40 Asymmetry indices for chromaticity metamer‐mismatch indices.
Figure 7.41 Histogram of indices of asymmetry,
, for volumetric metamer‐mis...
Figure 7.42 Histogram of indices of asymmetry,
, for chromaticity metamer‐m...
Figure 7.43 Two sets of colour matching functions, making the minimal angle ...
Figure 7.44 Two sets of colour matching functions, making the maximal angle ...
Figure 7.45 The abscissa of each point is the angle between the three‐dimens...
Figure 7.46 Colour matching functions with a minimal area difference between...
Figure 7.47 Colour matching functions with a maximal area difference between...
Figure 7.48 Spectral transmittance curves of a few neutral density filters f...
Figure 7.49 Boundary contours of the chromaticity metamer‐mismatch areas ind...
Figure 7.50 Chromaticity metamer‐mismatch areas induced by the neutral densi...
Figure 7.51 Volumetric metamer‐mismatch indices produced by the neutral dens...
Figure 7.52 Chromaticity metamer‐mismatch indices produced by the neutral de...
Figure 7.53 Chromaticity metamer‐mismatch indices induced by the difference ...
Figure 7.54 Along the vertical axis is the proportion of observers lying abo...
Figure 7.55 Spectral responses of the camera sensors.
Figure 7.56 Metamer‐mismatch volume in the CIE 1931 colorimetric space induc...
Figure 7.57 Chromaticity metamer‐mismatch area for the camera sensors presen...
Chapter 8
Figure 8.1 Spectral and non‐spectral hues.
Figure 8.2 Colour patches varying mainly in saturation.
Figure 8.3 The CIE 1931 chromaticity diagram, as in Fig. 4.12, with the only...
Figure 8.4 Constant hue loci in the chromaticity diagram for two observers....
Figure 8.5 Spectral luminous efficiency functions,
and
, established by t...
Figure 8.6 Counting just noticable differences.
Figure 8.7 Simultaneous brightness contrast.
Figure 8.8 A version of simultaneous brightness contrast. Source: Logvinenko...
Figure 8.9 Rearranged version of Fig. 8.8.
Figure 8.10 HSV cylindrical representation of colour.
Figure 8.11 Trichromatic spectral colour signal curve in the cone excitation...
Figure 8.12 Protanopic spectral colour signal curve in the dichromatic cone ...
Figure 8.13 Logarithm of the absolute value of the determinant 8.33 for
an...
Figure 8.14 Logarithm of the absolute value of the determinant 8.33 for
an...
Figure 8.15 Averaged and rounded matrix of responses of Observer #1. White c...
Figure 8.16 Chromaticity classes (marked with arcs) as derived from the resp...
Figure 8.17 Averaged and rounded matrix of responses of Observer #2. Source:...
Figure 8.18 Chromaticity classes as derived from the response matrix in Fig....
Figure 8.19 Averaged and rounded matrix of responses of Observer #3. Source:...
Figure 8.20 Iso‐chromes presented in the MacLeod–Boynton chromaticity diagra...
Figure 8.21 Unique hue settings in the MacLeod–Boynton chromaticity diagram ...
Figure 8.22 The square markers represent the rescaled estimate of the amount...
Figure 8.23 The markers represent the rescaled estimate of the amount of yel...
Figure 8.24 Hue‐component values (from Figs 8.22 and 8.23) have been replott...
Chapter 11
Figure 11.1 Colour detection display. A small yellow patch of light (the tar...
Chapter 13
Figure 13.1 Naka–Rushton curves (Eq. 13.2) for different values of
(
.
Figure 13.2 Naka–Rushton curves, as in Fig. 13.1, with a logarithmic scale o...
Figure 13.3 Summary diagram of the organisation of midget pathways of the pr...
Figure 13.4 Spatial weighting functions,
(green line) and
(red line) (mo...
Figure 13.5 Spatio‐chromatic sinusoidal pattern.
Figure 13.6 Derivative–Gaussian model of the spatial sensitivity function of...
Figure 13.7 Derivative of a Gaussian with negative sign.
Figure 13.8 The spatial sensitivity function profiles of a
‐double‐opponent...
Figure 13.9 The spatial‐sensitivity‐function profiles of a
‐double‐opponent...
Figure 13.10 As viewing distance increases, the neutral horizontal diamonds ...
Chapter 15
Figure 15.1 The main meridian of the regular object‐colour solid.
Figure 15.2 CIE 1931 chromaticity coordinates of the main (red line) and opp...
Figure 15.3 Simulation of the main‐meridian colours on the monitor screen. T...
Figure 15.4 Simulation of the opposite‐meridian colours on the monitor scree...
Figure 15.5 Regular object‐colour solid as evaluated for Govardovskii's fund...
Figure 15.6 Normalised length of the main meridian for the regular object‐co...
Figure 15.7 Regular object‐colour solid, the same as in Fig. 15.5, with even...
Figure 15.8 The regular object‐colour solid from Fig. 15.1 with the equator ...
Figure 15.9 The equator (green line) and the meridian (blue line) of the reg...
Figure 15.10 CIE 1931 chromaticity coordinates of the equator colours (red l...
Chapter 16
Figure 16.1 Regular object‐colour solid with coordinate lines on its boundar...
Chapter 17
Figure 17.1
and
coordinates evaluated for Munsell papers of five hues: 1...
Figure 17.2 Reparameterised spectral bandwidth,
, and central wavelength,
Figure 17.3 Spherical representation of Munsell papers by using
‐coordinate...
Figure 17.4 Spherical representation of Munsell papers by using
‐coordinate...
Figure 17.5 Reparameterised spectral bandwidth,
, and central wavelength,
Figure 17.6 Object‐colour solid with dichromatic orbits on its boundary surf...
Figure 17.7 Log
cone response ratio curve for Govardovskii's cone fundamen...
Figure 17.8 Log
cone response ratio curve for Govardovskii's cone fundamen...
Figure 17.9 Dichromatic orbits on the
diagram, along with Munsell papers. ...
Figure 17.10 Munsell papers, which tritanopes perceive in the same way as tr...
Chapter 18
Figure 18.1 Example of a function
that does not satisfy Eq. 18.9 (see text...
Chapter 19
Figure 19.1 Although this is just a sample of computer graphics, it illustra...
Figure 19.2 Experimental display used by Logvinenko & Tokunaga (2011) to exp...
Figure 19.3 Lights reflected by Munsell papers under various illuminations, ...
Figure 19.4 Output configuration produced by the non‐metric MDS algorithm. E...
Chapter 20
Figure 20.1 Photo of the experimental display, taken under a single illumina...
Figure 20.2 Sketch of the stimulus display with a paper cone (see text). Sou...
Figure 20.3 Sketch of the display used to demonstrate the trade‐off between ...
Chapter 21
Figure 21.1 The image on the right was created so that the luminance of ever...
Part 1
Figure I.1 Bipartite field, widely used in experiments on colour matching. I...
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
Begin Reading
References
Appendix A: Some Auxiliary Facts from Functional Analysis
Appendix B: Proofs
Index
End User License Agreement
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Foundations of Colour Science
From Colorimetry to Perception
Alexander D. LogvinenkoGlasgow Caledonian UniversityUK
Vladimir L. Levin†Central Economics and Mathematics Institute of the Russian Academy of SciencesRussia
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Names: Logvinenko, Alexander D., author. | Levin, Vladimir L., author.Title: Foundations of colour science : from colorimetry to perception / Alexander D. Logvinenko, Glasgow Caledonian University, UK, Vladimir L. Levin, Central Economics and Mathematics Institute of the Russian Academy of Sciences, Russia.Description: First edition. | Chichester, UK : Wiley, 2023. | Includes bibliographical references and index.Identifiers: LCCN 2021060391 (print) | LCCN 2021060392 (ebook) | ISBN 9781119885917 (cloth) | ISBN 9781119885900 (adobe pdf) | ISBN 9781119885948 (epub)Subjects: LCSH: Color. | Colorimetry. | Color vision.Classification: LCC QC495 .L55 2022 (print) | LCC QC495 (ebook) | DDC 535.6–dc23/eng20220215LC record available at https://lccn.loc.gov/2021060391LC ebook record available at https://lccn.loc.gov/2021060392
Cover Design: Wiley
Preface
The trigger for writing this book was, for me, the realisation that Grassmann's laws – the cornerstone of colorimetry (which is the basis of colour science) – seemed not to be quite correct. True, it has been reported from time to time before that Grassmann's laws get violated under some special circumstances such as high light intensity and/or at the far periphery in the retina. However, I realised that one of the most important aspects of Grassmann's laws – the transitivity law – just could not be possibly true for colour matching even under normal condition of viewing. Intransitivity of colour matching undermines the major achievement of colorimetry: the definition of colour as a class of equivalence (i.e., a class of visually indiscriminable lights) since in this case colour matching cannot be conceptualised as an equivalence relation. The problem, as I saw it, was not only that the whole grand construction of colorimetry, it turned out, was built on sand, but that, despite this, colour television, the very idea of which would have been impossible without colorimetry, still successfully worked.
In the end, resolving this apparent paradox proved to be rather easy (Logvinenko, 2006). However, in doing this I had been becoming more and more confident in the opinion that the very foundations of colour science should be reconsidered. Furthermore, it should be done on quite different grounds, involving rather different mathematics.
It so happened that thus far the basic mathematical tool of colour science has been linear algebra. There is good reason for this. ‘Many modern ideas in linear algebra – especially the concept of linear functional and the resulting distinction between a linear space and its dual space – seem to be specially designed to clarify thinking about colour theory’ (Krantz, 1975a, p. 283).
Really, lights can be mixed together. Formally, this translates into the statement that there is an algebraic operation on the set of lights. Although colour is a mental, not physical attribute of light, a similar algebraic operation can be defined on the set of colours, namely, colour mixing. As known, the colour of the mixture of two lights depends on the colour of each light rather than their individual spectral compositions. Grassmann was, perhaps, the first to realise that this implies there being an algebraic structure of colour, which is relatively independent of that of light (Grassmann, 1853).
It is the algebraic structure of colours that enables us to represent colours as vectors in the linear space. Also it allows us to use matrix algebra to perform colour computation, that has been widely used in colour science and its applications (e.g., colorimetry, colour imaging, computer graphics). Formally, the algebraic structure of colour is defined as that of the classes of visually equivalent (i.e., subjectively indistinguishable) lights. The vector representation of colours results from mapping the lights onto the classes of visual equivalence (the colour stimulus map). The vector representation of colour, empirically based on colour matching, puts colorimetry and its methods on par with the physical sciences, which has been acknowledged by the inclusion of colorimetry by Richard Feynman in his celebrated lectures on physics (Feynman et al., 1963, Ch. 35).
It should be said, however, that there are natural limitations for the application of linear algebra even in colorimetry, not to mention colour science in general. The finite dimensionality of linear spaces which linear algebra deals with requires the discretisation of continuous functions that naturally arise in colorimetry (e.g., the light energy distributions over the visible spectrum and colour matching functions), which is a problem in itself. Being free of this limitation, functional analysis (dealing with infinite‐dimensional linear spaces) seems more appropriate for colour science purposes. We found that many problems in colorimetry are much easier to tackle from the infinite‐dimensional standpoint.
The linear structure is not all that colour inherits from light. There is a natural physical measure of proximity between different lights. The fact that close lights have colours which look subjectively close to each other testifies that the light proximity is somehow retained in colour. It means that the colour stimulus map is, in a sense, continuous. While Grassmann stated the law of continuity for colour mixing more than 150 years ago, it has not been elaborated on to the same extent as his other laws of colour mixing. Moreover, the topological concepts underlying the notion of continuity still remain alien to colour science despite the fact that, as shown in this book, they prove to be rather effective for solving a number of problems in colour science.
Topological methods are particularly pertinent to colour science because the colour manifold itself, as well as some other important objects (e.g., the object‐colour solid), is convex. All these objects inherit convexity from their physical prototypes. For example, all lights, as a whole, can be represented by a convex cone (in the infinite‐dimensional vector space). As the colour stimulus map preserves convexity, the full gamut of colours can also be represented as a convex cone in the vector space, but with finite dimensions. Using the methods of convex analysis for infinite‐dimensional topological vector spaces, we have solved some long‐standing problems in colour science.
It should be mentioned that the achievements of colorimetry in colour measurement have been made at a cost of neglecting colour perception (i.e., the structure of colour appearance). Really, colorimetry can predict whether two lights will appear of same colour or different, but not what exactly they will look like. Although a number of colour appearance models have been put forward (Wyszecki, 1986; Fairchild, 2005), they are quite far from meeting the colorimetric standards of precision and accuracy. It is hardly surprising since colorimetry has proven to be, in fact, the science about photopigments in the human eye. The participation of a human observer in a colorimetric experiment is intentionally reduced to the role of a comparator, with only two outputs: same or different. The success of a colorimetric study depends on how well a human turns into, virtually, a technical device. Ideally, the influence of all factors (including perceptual factors), with the exception of photopigments, is supposed to be eliminated. The effectiveness of colour television suggests that we are pretty close to this ideal. One has, however, to admit that colorimetry has not quite reached the level of exact sciences, like physics (as evidenced by the aforementioned violation of Grassmann's laws). The reason is that, surprisingly, a human, alas, is unable to entirely simulate such a simple device as a comparator. The fact is that human responses, even such simple ones as binary ‘yes/no’ s, are inherently fuzzy, whereas colorimetry implies crisp responses, simply ignoring this fuzziness. It was interesting for us to see what would become of colorimetry after the revision of Grassmann's laws, taking into account the fuzziness of human responses.
In order to look into colour perception, we chose to completely break with colorimetry and start from scratch. Guided by Hering's idea of component hues, we attempted a new approach to colour perception based on the same theoretical framework as that used by us to expose the main results obtained in colorimetry. It allowed us to develop techniques to measure colour appearance with an accuracy meeting, we believe, the standards adopted in colorimetric measurements.
Although, human colour vision was our main concern, we hope many results presented in the book can be applicable to animals' colour vision as well as various technical devices equipped with artificial photosensors (e.g., cameras). Our intention was to outline mathematical foundations – the most general principles – which can be used when dealing with a particular system of colour vision based on the same principles as human colour vision. For example, a notion of metamerism is applicable to any live or artificial system of photosensors with different spectral characteristics. The mathematical problem of describing metameric lights (i.e., indistinguishable for these photosensors) naturally arises. It turns out that there is a general solution to this problem (an algorithm is presented in Chapter 6). In fact, much of the book is devoted to various mathematical problems arising in colour science, and their solving.
Going beyond the usual repertoire of mathematical methods used by colour scientists would not have been possible without the participation of a professional mathematician, such as was Vladimir Levin. He made proofs of almost all theorems, lemmas, and propositions in the book. Unfortunately, his untimely death in 2012 prevented him from taking part in the final stage of work on this book. In particular, the Outline (Chapter 1), Chapters 9, 13, 17, 19–21, and the Epilogue were written after he passed away.
It is also worth mentioning the following context, which played a not insignificant part in the genesis of the premise for this book. Since colour is the subject of various disciplines, and given the rapid progress in almost every one of them in recent years, there is an urgent need for a guide that presents the subject from a unified perspective and covers it as completely as possible. At the moment, a person's ideas about colour are very much dependent on which textbook they happen to read, recalling to mind the ancient Indian parable of the blind men and the elephant. This book attempts to at least partially realise this goal, by outlining our own vision of what the science of colour could be like. We, of course, were aware that such a task was too daunting for two. We also understood that we would inevitably have to intrude into adjacent areas in which we are not experts. That being said, such intrusions concerned only colour, not mathematics, so the responsibility for all inevitable inaccuracies and errors lies entirely with me. I shall be most grateful to those readers who discover such errors, if they do not see it as too troublesome to report them to the author. In general, any comments and criticism will be gratefully received if they aid the improvement of this book.
I am also grateful to Sara Hutchinson, for her generous and attentive contribution to the grammatical proofing of the text.
And last but not least, I would never have finished this book if it had not been for the constant support of my daughter Alisa, who has shown a keen interest in my work, always enthusiastically supported me in those rare moments when I lost spirit, and, unlike me, always believed that I would have enough stamina to finish the job.
Alexander D. LogvinenkoJuly 18, 2022 Glasgow
1Outline for Readers in a Hurry
Colour is a peculiar, though fascinating, subject for a scientist to explore. A scientific study implies, in general terms, (i) outlining the subject matter; (ii) the experimental method whereby empirical data are collected; and (iii) the theoretical paradigm within which these data are interpreted.
When studying colour, difficulties begin from the very first step: it is not easy to describe colour as a subject for research, let alone give it a scientific definition.1 Whilst most of us take for granted the colourfulness of our environment, it is worth remembering that the physical objects themselves are colourless. ‘For the Rays to speak properly are not coloured’ (Newton, 1730). As such, colour is the spawn of our mind. The problem here is not that colour belongs to the perceptual world, but that there is nothing in the physical world that the colour might reflect. Apparent size and apparent shape are also the attributes of the perceptual world; however, one can readily point out their physical counterparts.
Thus, the perception of colour cannot be put in the same category as the perception of size, shape, speed, and other physical variables, which are usually given attention in textbooks. Indeed, size and shape, as they appear to us, are usually regarded as direct representations, even reflections of the corresponding physical entities in our perception. So, even though we do not know how such representations are made possible, the existence of such perceptual attributes as apparent size and apparent shape seems natural.2 On the contrary, perception of colour raises a problem as there is no colour in the world to which size and depth belong. Why does something that does not exist in the physical world nonetheless appear in our mind? To answer this question, which is in fact a philosophical one,3 is not easy. At any rate, there is no commonly accepted answer as yet. We will present our views on this matter at the very end of this book (Section 21.1).
In such a situation, there is nothing left but to try to define colour as a mental fact without referring to a particular physical entity behind it (that is, not assuming any physical fact that it reflects). For example, colour is often defined as the perceptual attribute that we mean when using such words as red, green, yellow, brown, and the like. However, these words refer only to one, albeit very important, attribute of colour, namely, hue.4 And then, how to deal with such words as white and black? On the one hand, they do not seem to point out a hue; on the other, they undoubtedly belong to the colour names. Does it follow that white is a colour?5 Also, are various shades of grey colours? While inclining to an affirmative answer, which is in line with an established tradition (Kaiser & Boynton, 1996), we have to admit that these, usually called achromatic colours, are somewhat different from so‐called chromatic colours, which are referred to with words red, green, pink, etc.
Perhaps, all these questions arise because we are trying to go from colour names to colours, and not vice versa, which, in our opinion, would be more appropriate. It should be borne in mind that our verbal colour categories are much coarser than our colour experiences. Some obvious perceptual differences, pertinent to colour, cannot be rendered by colour names. For instance, we use the same colour name, say, red, to describe physically different things: (i) solid opaque objects (e.g., red apple), (ii) liquids (e.g., red wine), and (iii) self‐luminous objects (e.g., red light). Is the redness, which we experience in all these three cases, same or different? On the one hand, they are different since one cannot find, for instance, a red light which will look absolutely identical to the red opaque object. On the other, some argue that, while appearing not quite same, they share the same quality of redness. However, we believe that the redness, that a red apple has, red wine has, and red light has, and which we can identify for all three objects, is a conceptual notion (an idea) rather than a perceptual experience. We do not perceive it. We can only think of it. Arguments in favour to this important conceptual distinction are the same as those Gibson (1979) was putting forward, arguing that space cannot be perceived, it can only be ‘conceived’ by means of abstract thinking. Specifically, Gibson claimed that it is the environment rather than the physical world that to be perceived. As a result, he considerably modified the concept of what, in his opinion, could be ‘seen’, including, for example, edibility, into it; but in his concept of ‘seeable’ no place had been left for space. In general, the issue of what we can ‘see’ turns out to be far from as trivial as it might seem to a layman and becomes decisive when investigating visual perception.
Because of certain historical reasons, understanding colour as abstract qualia (e.g., redness, greenness, orangeness, etc.), common to all the perceptual experiences, prevails in the science of colour. One reason is that when the science of colour only evolved as independent discipline within psychology the latter was dominated by the philosophical theory of sensations, colour being understood as an attribute of visual sensations (Boring, 1942). Perception was believed to be based on the sensations. Although the Gestalt psychologists (e.g., Koffka, 1935) and Gibson (1979) have convincingly showed that it is a percept (e.g., a red apple) rather than sensations that is immediately experienced, according to the major dogma of the psychology of sensations, a percept can be decomposed, by using introspection, into sensations underlying this percept, in a similar way as a molecule can be decomposed into atoms. Briefly, sensations were thought of as elements of perception, in the same way as atoms were considered as elements of matter. Sensations were believed to have some attributes, colour being one of them. Along with colour, extension and duration were also listed as unalienated attributes of visual sensations (Boring, 1942).
Within the psychology of sensations, an experimental paradigm has been formed for studying them. The way to experimentally study sensations has been paved by the belief that for each attribute of a sensation there exists the stimulus, that is, a physical agent which causes and determines this attribute.6 For example, the frequency of a sound wave is commonly accepted to be the stimulus for such an attribute of auditory sensations as pitch. Its amplitude (generally, sound pressure level) is thought of as the stimulus for loudness. So is light intensity for brightness (i.e., the subjective magnitude of a visual sensation). It is ascertaining the relationship between the stimulus intensity and the subjective magnitude of the corresponding sensation (known as the psychophysical function) that psychophysics began from7 (Fechner, 1860).
Over time, the psychophysical approach has gone beyond a separate discipline and turned into a way of thought that dominates vision science. Colour science has developed within the psychophysical approach, adopting and elaborating psychophysical experimental methods. According to this approach, the main objective of vision research is to ascertain stimuli for the attributes of visual sensation and then to establish the corresponding psychophysical relationship between them, just as the psychophysical functions have been established. However, it did not take long to realise that the search for psychophysical stimuli was a much more complicated task than it seemed at first. Admittedly, in the most general sense, the stimulus for visual sensations is believed to be light. However, it turns out that establishing what characteristics of light can be taken as the stimulus for particular attributes of visual sensations is not easy. For example, apart from the intensity of a light patch its brightness is also affected, to a certain degree, by the size of the patch, its duration, and some others dimensions (Wyszecki & Stiles, 1982). It turns out that finding all physical parameters which determine brightness of light is rather difficult, if possible at all.8 As a palliative, light intensity is considered as the core stimulus for brightness, the contribution of all other factors affecting brightness being believed to be relatively small.
To find what in the light can be considered as the stimulus for the other attributes of its colour, and especially for the hue, turned out to be even more difficult. An account of what is known on this issue will be preceded by a quick summary of the basic notions and facts concerning light which will be used throughout the book.9
Modern science holds that light is a form of electromagnetic radiation which propagates through space at a constant speed. Moreover, the physical theory states that light, as well as other forms of matter, has both a wave nature and a particle nature (wave‐particle duality of light).
In the wave model of electromagnetic radiation, light is viewed as an electromagnetic wave, that is, oscillation of electric and magnetic fields. The electric and magnetic field components oscillate at right angles to each other, and to the direction of propagation. An important characteristic of an electromagnetic wave is frequency, that is, its rate of oscillation measured in hertz. The frequency of the electromagnetic wave, , is related to its wavelength, (measured in metres), by the equation: , where is the speed of the wave. The electromagnetic waves of all possible wavelengths (in practice, from thousands of metres to m) are referred to as the electromagnetic spectrum. The range of the electromagnetic spectrum within which the sun radiation hits a maximum (wavelengths between approximately and m) is called the visible spectrum. Light is said to be electromagnetic radiation in this range of wavelengths. The wavelength of light is commonly expressed in terms of nanometres ( m).
The particle model of electromagnetic radiation states that the energy of electromagnetic waves (i.e., radiant energy) is emitted by a source (and absorbed by an object) in discrete packets of energy, or quanta. A quantum of light is often referred to as a photon. Within this point of view, light can be considered as a stream of photons. The energy of a photon, , is proportional to its frequency, , and inversely proportional to its wavelength, :
where is Planck's constant and is the speed of light in a vacuum.
Electromagnetic waves of a single wavelength are referred to as monochromatic. No real electromagnetic radiation can be, strictly speaking, monochromatic. Even such sources as lasers, which are designed to produce monochromatic radiation, do, in fact, emit radiant energy across a whole (though, quite narrow) band of the spectrum. Hence, real light is always a complex wave, that is, a superposition of electromagnetic waves of different wavelengths. An important characteristic of light is its radiant spectrum which describes how the radiant energy10 is distributed with wavelength (i.e., across the spectrum).
Electromagnetic waves reveal themselves only when interacting with other objects. When any electromagnetic waves interact with (i.e., are absorbed by) an object, the energy of the waves is converted to heat. As to the eye, apart from warming, light can result in another, specific effect – isomerisation of the light‐sensitive molecules in the retina. This triggers a complex (and not yet fully understood) process of converting a photon into electrical signals in the eye, called visual phototransduction (Schnapf & Baylor, 1987; M. E. Burns & Pugh, 2014).
Light absorption causes a change to the photopigment (light‐sensitive molecules) contained in the outer segment of the photoreceptors.11 The probability that a photon is caught by the photopigment is different for different wavelengths. Such a spectral selectivity (or spectral tuning) of the light absorption in the photopigment is characterised by the photopigment absorption spectrum, which indicates the amount of light absorbed depending on the wavelength.12 However, having been captured, each photon contributes into the photoreceptor output signal regardless of the wavelength (the univariance principle) (Rushton, 1972). In other words, the process of transforming light absorption into electrical response is assumed to be wavelength‐independent, the output electrical signal of a photoreceptor being proportional to the number of absorbed photons (the quantum catch).13
The probability of a photon capture by a photopigment is maximal for a particular wavelength (different for different photopigments), the photopigment absorption spectrum having an inverted U‐shaped form.14 Actually, the end points of the visible spectrum wavelength interval are defined as the wavelengths where the photon capture probability is negligibly small for all the photoreceptors involved. Monochromatic electromagnetic waves falling out of this range are not considered as stimuli for vision.
The relationship between the radiant spectrum of a light and its colour has been noticed a long time ago (for a historical review see e.g., Mollon, 2003). For example, it has been known since Newton's time that monochromatic lights differ in hue, each monochromatic light having its own hue. The rainbow allows everyone to see that literally with their own eyes. However, not every hue can be found in the rainbow. For instance, purple, lilac, violet are not there. Such hues are called non‐spectral. However, what determines the hue of a light with the broadband spectrum had remained unclear until the nascence of colorimetry (Schanda, 2007a).
The major idea behind colorimetry is rather simple. Since the light that enters the eye is the only reality the visual system encounters, it seems a good idea to first examine how the colour of light depends on its radiant spectrum in the laboratory, where you can control all the characteristics of light with the desired accuracy. Over time, certain standards for conducting such experiments were formed (Wyszecki & Stiles, 1982), that will be referred to, for brevity, as the colorimetric experimental paradigm. In experiments of this sort, either a single spotlight is presented in isolation (in complete darkness) or light is projected directly to the eye using special optics (with no objects or medium between the light and the observer).15 The rationale is to study disembodied light, that is, light as such. Behind this is an implicit belief that studying light as such will result in the knowledge of colour as such. Indeed, in this situation, the colour is usually perceived as being attributed to neither object nor medium. A term unrelated colour is used to distinguish this type of colour experience from a more natural one (referred to as related colour) when colours are perceived as being attributed to objects or media (Kaiser & Boynton, 1996, pp. 30–41). Alternatively, the terms light colour, and object (or medium) colour will be used in this book.
Significant advances have been made in studying light colours under the colorimetric experimental paradigm. First of all, it has been found that lights which are physically very different may be indistinguishable for humans. Such lights obviously match each other in colour. Furthermore, it has been discovered that lights evoking an equal quantum catch in the cone photoreceptors of the same type appear to make a match.16 Thus, being indistinguishable for the brain, all the lights producing the same quantum catch match each other, and they have same colour.
Then, it has been established that for each light either it matches a mixture of some amount of daylight and a monochromatic light, or its mixture with some monochromatic light matches daylight. The wavelengths of these monochromatic lights are called dominant or complementary wavelengths for the light, respectively (see Section 5.1). Dominant (respectively, complementary) wavelength is often called the stimulus correlate of hue.17
It must be emphasised, however, that colorimetry holds its explanatory capacity only within the colorimetric experimental paradigm. However, the latter implies so impoverished an environment (e.g., a dark laboratory room, presentation of a homogeneous light patch of fixed size and duration, etc.) that one cannot help questioning its relation to natural vision (i.e., vision in the natural environment). The justification of the colorimetric paradigm is rooted in the psychology of sensations the major dogma of which is that sensation as an element of perception is identical to elemental perception, the latter being understood as perception of an isolated stimulus. However, that this dogma does not withstand any criticism was clear already for Gestalt psychologists (e.g., Koffka, 1935).
Furthermore, colorimetry remains vulnerable even on its own territory. No doubt, it has been a great achievement for colour science that a theory of light colour has been created that can predict rather successfully the colour of lights viewed under some standardised conditions (i.e., within the colorimetric experimental paradigm) (Wyszecki & Stiles, 1982). However, apart from being very restrictive, these conditions are practically impossible to implement. Indeed, these conditions ideally imply studying light which is structured neither in space nor in time so as to prevent confounding these variables with the spectral variable (i.e., wavelength). However, it is very difficult to present absolutely homogeneous light. Furthermore, it turns out that light that is absolutely homogeneous in space and time always tends to be perceived as neutral (i.e., hueless, such as daylight) irrespective of its radiant spectrum. Whilst this goes against our everyday life experience, it has been confirmed in experiments with optical stabilisation of the image on the retina. The human visual system was found to rapidly cease responding to unchanging visual stimulation. Specifically, a stabilised image was found to fade within 3–5 seconds (Krauskopf, 1963; Yarbus, 1965/1967).
Therefore, to study how the colour of a patch of light depends on its radiant spectrum, the patch should be presented not in isolation but be accompanied by other lights. However, the colour of the patch may change dramatically in the presence of other patches of light (despite that all the patches are perceived unrelated to any objects). Colour induction in its two varieties – chromatic simultaneous contrast and chromatic assimilation – can serve as an example (Kaiser & Boynton, 1996, pp. 30–41). Colorimetry loses its predictive power in this situation. Although a number of models have been put forward to explain the effect of the spatial and/or temporal context on colour (e.g., Wyszecki, 1986; Chichilnisky & Wandell, 1995; Mausfeld, 1998), there is no general account as yet.
True, the effect of the spatial and/or temporal context on colour in principle does not undermine colorimetry but only shows its limitations. Really, the colorimetric experimental paradigm implies varying the radiant energy only across the spectrum, keeping the spatial and temporal parameters of the stimulus display constant. As a result, one always has to remember that the explanatory power is limited by this particular spatial and temporal configuration of the stimulus display. For the stimulus display with a different spatio‐temporal configuration one will need, strictly speaking, a new colorimetry.18
This state of affairs is by no means satisfactory. It seems reasonable to expect that the visual environment for which the theory holds its explanatory power (let us call it the admissible environment for the theory) is considerably wider than the test environment (i.e., which was used to conduct the experiments on which the theory is based). Ideally, the admissible environment for the colour science should encompass the natural scenes. Colour induction indicates that the admissible environment for colorimetry is way far from that objective. An obvious reason for this is that the colorimetric description of light as a radiant spectrum is too restrictive and should be reconsidered. Really, since (i) spatial and temporal modulations are necessary for light to be perceived as coloured, and (ii) the spatial and/or temporal context affect the colour, one should describe light as the radiant energy distribution not only across the spectrum but also across the space and time. An attempt at such generalisation will be given at the end of Part I (Section 13.3.2).
Such a generalisation is a recognition of the fact that a theory of colour as such (i.e., regardless of all other dimensions, such as space and time) cannot be created; that it can only be created as part of the general theory of perception.19 Besides, this generalisation is made in an attempt to broaden the admissible environment so as to make the theory accountable for more realistic conditions of observation (up to real scenes). However, in the natural conditions, the spatio‐temporal modulation of light is produced by reflecting objects, transparent media, and other elements of the environment. Furthermore, under the natural viewing conditions, we see the colour of these objects and/or media rather than that of light. Therefore, we have to include in consideration object colours and transparent medium colours. After all, the ability to see the colour of objects (i.e., the related colours) is much more important from an ecological point of view than the colour of lights.20
At this point, we have to return to the issue already raised above. How do unrelated and related colours ‘relate’ to each other? Is the colour of an object the same as the colour of the light reflected from that object? As mentioned above, the major dogma of the psychology of sensations entails the positive answer. Although there is abundant evidence against this dogma (provided by Gestalt Psychologists and Gibson21) these questions in colour science remain without a clear answer.
