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Article  |   April 2012
Color constancy investigated via partial hue-matching
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Journal of Vision April 2012, Vol.12, 17. doi:10.1167/12.4.17
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      Alexander D. Logvinenko, Anja Beer; Color constancy investigated via partial hue-matching. Journal of Vision 2012;12(4):17. doi: 10.1167/12.4.17.

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Abstract

Each hue is believed to be made up of the four component hues (yellow, blue, red, and green). A hue consisting of just one component hue is called unitary (or unique). A new technique—partial hue-matching—has been used to reveal the component and unitary hues for a sample of 32 Munsell papers, which were illuminated by neutral, yellow, blue, green, and red lights and assessed by four normal trichromatic observers. The same set of four component hues has been found under both the neutral and the chromatic illuminations for all of the observers. On average, more than 87% of the papers containing a particular component hue under the neutral illumination also have this component hue when lit by the chromatic lights. However, only a quarter of the papers perceived as unitary under the neutral illumination continues being perceived as unitary under all of the chromatic illuminations. In other words, most unitary colors shift along the hue circle due to change in an illuminant's chromaticity. Still, this shift of unitary colors is relatively small: On average, it does not exceed one Munsell hue step.

Introduction
Although color constancy has been studied for more than a century (Foster, 2011; Jameson & Hurvich, 1989; Katz, 1911/1999; Pokorny, Shevell, & Smith, 1991; Shevell & Kingdom, 2008; Smithson, 2005), there is no clear understanding of the nature of the very phenomenon, not to mention its mechanisms. According to the textbook definition, one refers to the constancy of the color of an object with respect to illumination 1 (Kaiser & Boynton, 1996; von Helmholtz, 1867); this is to say that color constancy endures in spite of variations in illumination. However, this definition cannot be taken literally, because the color appearance of an object definitely does change with its illumination. 2 In order to reconcile the definition with this obvious fact, color constancy is usually said to be imperfect, the following terms being used—“relative constancy” (Brainard, 2009), “relational constancy” (Foster & Nascimento, 1994), “qualitative constancy” (van Trigt, 2007), and the like. As it is not always clear what these terms mean, we will use a generic term “approximate color constancy” (e.g., Brainard & Wandell, 1986; Hering, 1964) that can be understood in at least three different ways. 
First, approximate color constancy can be thought of as, literally, imperfect color constancy. In other words, while there are some deviations from perfect color constancy, they are believed to be insignificant, especially for natural illuminants (Brainard, 2009; Hurlbert, 2007). For example, while looking pure yellow under daylight, a banana may get a slight tinge of green when illuminated by a tungsten filament light source, remaining very close to the previous yellow. An important implicit assumption behind such an observation is that the object color palette as a whole does not change with respect to illumination change. In other words, although the color of a particular object might change with illumination, the nomenclature of object colors remains the same under all illuminations. It is this assumption that justifies asymmetric (i.e., across two adjacent illuminants) color matching—a technique widely used to measure color constancy (for a review, see, e.g., Foster, 2011; Smithson, 2005). Indeed, asymmetric color matching would not make any sense if the object color palette itself were different under different lights. 
As stated by Krantz, the possibility of the asymmetric color matching is “so much taken for granted that it seems never to have been stated explicitly in textbooks or in the technical literature on color” (Suppes, Krantz, Luce, & Tversky, 1989). Nevertheless, it seems as if an exact asymmetric color match can hardly be achieved, that is, the color of an object lit by one light cannot be exactly matched by the color of another object lit by the other light (Brainard, Brunt, & Speigle, 1997; Logvinenko & Maloney, 2006). An exact asymmetric color match is impossible because there is an unavoidable difference in color appearance of the same object lit by different lights. For example, two white surfaces illuminated by, say, blue and yellow lights, on the one hand, look white, but on the other hand, they appear two different “whites.” Multidimensional scaling of colored papers lit by different lights showed that the dissimilarity between such papers never became zero, even when the papers reflected metameric lights (Tokunaga & Logvinenko, 2010a, 2010b, 2010c; Tokunaga, Logvinenko, & Maloney, 2008). In other words, manipulating surface reflectance can only minimize (but not nullify) the color difference produced by a difference in illumination. This difference has been recently conceptualized as a difference in the lighting dimensions of object color. Distinguishing between the material and lighting dimensions of object color appears to be crucial for understanding perception of colored objects under variegated illumination (Logvinenko, 2012a; Logvinenko & Tokunaga, 2011; Tokunaga & Logvinenko, 2010b). 
It should be stressed that the very existence of the lighting dimensions of object color is understood as the existence of the whole variety of object color palettes (Logvinenko, 2012a; Tokunaga & Logvinenko, 2010c). Strictly speaking, all these object color palettes are entirely different. This leads to the view that color constancy—as color appearance constancy—does not exist at all (Foster, 2003). The observer's ability to establish an “approximate asymmetric color match”, then, stands as a purely behavioral phenomenon without any foundation in the color's precise appearance. 
A compromising stance is that while the object color palettes differ under different illuminations, there is a correspondence between them in the observer's perception. For example, although a white surface under red light looks differently as compared to the same white surface lit by blue light (differing along the lighting dimensions), they are readily recognized as the same “white”—that is, they have the same (white) material color. From this point of view, approximate color constancy means that an illumination change mainly affects the object color in terms of the lighting dimensions, whereas in terms of the material dimensions the object color remains very much the same (Logvinenko & Tokunaga, 2011). For example, two yellow bananas under different illuminations differ in the lighting dimensions, being the same in terms of the material dimensions. Such a view can be tentatively named “material color constancy” (Logvinenko, 2012a; Logvinenko & Tokunaga, 2011). 
Admittedly, one should not expect material color constancy to be perfect. Moreover, there is a physical reason (referred to as the mismatch of metamers; Wyszecki & Stiles, 1982) for material color constancy to be somewhat imperfect (Logvinenko, 2009b, 2012a). How can, then, material color constancy be measured? Clearly, asymmetric color matching will not do. At first glance, an alternative might be color naming and/or color categorizing that have been used in color constancy studies (Granzier, Brenner, & Smeets, 2009; Hansen, Walter, & Gegenfurtner, 2007; Olkkonen, Hansen, & Gegenfurtner, 2009; Olkkonen, Witzel, Hansen, & Gegenfurtner, 2011; Smithson & Zaidi, 2004; Troost & de Weert, 1991; Uchikawa, Uchikawa, & Boynton, 1989). The rationale is that although illumination change alters the color appearance of, say, a red apple, it keeps belonging to the same color category—“red.” The problem with this method is that, first, it is not clear how to measure the color categories (more or less) precisely; second, there are large interindividual differences between subjects; and third, the illumination may affect the color categories themselves. Indeed, will what the observer means by “red” actually remain the same under daylight and under red light? 
All this criticism is also applicable to hue scaling—the method that has also been employed in color constancy studies (Foster, 2008, 2011; Smithson, 2005). This method implies the observer's ability to rate by number the amount of a particular hue (usually, red, green, yellow, and blue) in the color of an object. Apart from some controversy concerning such an ability (Logvinenko & Geithner, 2012), a similar question arises. Will the redness of an illumination interfere with evaluated redness of the object? 
Among other things, all these techniques (i.e., color naming, color categorizing, and hue scaling) operate with predetermined sets of hues, color names, or color categories in terms of which observers are instructed to assess colors. However, as noticed by many previous researchers (for a review, see Mausfeld, 1998), objects take on a rather unusual appearance under chromatic illumination. If chromatic illumination alters the object color palette so dramatically that the classical four component hues (i.e., red, green, yellow, and blue) are not enough to describe this alteration, the hue scaling technique (as well as color naming) will not allow for the object color palette alteration, as induced by the illumination change, to be revealed. As an alternative, a new method of color assessment—partial hue-matching (Logvinenko, 2012b)—can be used instead. 
The partial hue-matching technique has been designed to assess object colors in terms of hue components without presupposing their number nor defining them verbally beforehand. The observer is asked whether two colors have any hue in common. The complete set of hues in which the observer is to discern all of the sample colors (these hues referred to as the component hues) is derived from an observer response matrix. Moreover, one can also derive from this matrix which of these component hues each sample color contains. Therefore, the colors can be characterized in terms of component hues, even though the nomenclature of the component hues is not known in advance. 
In this article, we present the results of an experiment in which the partial hue-matching technique was used to assess the color of 32 Munsell papers under five different illuminations 3 (neutral, red, green, yellow, and blue) to address the following issues. Does the change in the illumination chromaticity affect the object color palette, that is, the nomenclature of the component hues? Can the change in the illumination chromaticity induce a change in the chromatic content of the color of a particular paper, that is, in the set of its component hues? In particular, will the papers whose color is perceived as “unique” or unitary (i.e., possessing only one component hue) under neutral illumination remain as such after the illumination chromaticity has been changed? 
Experiment 1: Partial hue-matching under symmetric achromatic illumination
The purpose of the experiment was (i) to make observers familiar with the partial hue-matching technique, (ii) to reveal the component hues under neutral illumination using the partial hue-matching technique, and (iii) to establish which Munsell hues are perceived as unique by using the partial hue-matching technique (Logvinenko, 2012b) and the conventional unique hue selection technique (Shamely, Sedito, & Kuehni, 2010). 
Within the theoretical framework of partial hue-matching, the notion of a component hue is operationalized in terms of the so-called chromaticity class (Logvinenko, 2012b). Given a sample of colors, the chromaticity class is defined as the maximal subset of colors that contain the same component hue. If the component hues are known in advance, then the chromaticity classes are readily determined. For example, adopting the classical conception of four component hues, yellow, blue, red, and green (Hering, 1964; Hurvich & Jameson, 1957), one can easy recognize the four chromaticity classes corresponding to these component hues in a sample of Munsell hues displayed in Figure 1. Specifically, the chromaticity class made up, say, by the blue component hue is the sector in the hue circle in Figure 1 containing all the chips tinged with blue. 
Figure 1
 
Munsell chips used in the preliminary experiment on unique hue selection. On the circumference is the Munsell notation of the page from which the chip was selected.
Figure 1
 
Munsell chips used in the preliminary experiment on unique hue selection. On the circumference is the Munsell notation of the page from which the chip was selected.
As mentioned above, the chromaticity classes can be derived if we know for each pair of colors whether these colors contain any common hue (Logvinenko, 2012b). Furthermore, chromaticity classes derived by the partial hue-matching technique for a Munsell sample ordered in a hue circle as in Figure 1 prove to have no gaps, that is, they always take the form of a sector. Thus, they can be specified by the ends of these sectors (Logvinenko & Beattie, 2011; Logvinenko & Geithner, 2012). 
Observers
Four normal trichromatic observers participated in the experiment. Their color vision was tested with Ishihara pseudo-isochromatic plates. They had normal visual acuity and were aged between 23 and 32 years. Two of them (Observers DW and MT) were naive to the purpose of the experiment and untrained in color vision experiments. Observers AB and CG were aware of the purpose but not trained either. 
Preliminary experiment: Unique hue selecting
The unique hue selecting was made from 40 Munsell hues. The chip of maximum Munsell chroma was taken from each page of the Munsell book of colors (Munsell, 1915). The stimulus sample was placed on a table covered with a white cloth either in random order or arranged in a hue circle as Figure 1. Observers made four unique hue selections under both spatial arrangements. The results were rather robust. The median hue selections are presented in Table 1
Table 1
 
Munsell chips chosen as having unique hue by the four observers. Each entry represents the median choice across the eight trials. Although selection was made from the 40 Munsell chips, the median chips are given according to the numeration of the 32 Munsell chips engaged in the subsequent experiments (Table 2).
Table 1
 
Munsell chips chosen as having unique hue by the four observers. Each entry represents the median choice across the eight trials. Although selection was made from the 40 Munsell chips, the median chips are given according to the numeration of the 32 Munsell chips engaged in the subsequent experiments (Table 2).
Unique hues
UH1 UH2 UH3 UH4
Blue Red Yellow Green
AB 5 17 21 29
CG 6 17 21.5 30
DW 5 17 21 30
MT 5 17 21 28
Methods
Apparatus
A sample of 32 from the 40 Munsell chips depicted in Figure 1 was used as a stimulus set (Table 2). The Munsell chips were mounted in a random order on a rectangle (38 × 65 cm) vertical stand covered by white paper with a random dot pattern. The stimulus display consisted of two identical stands (32 Munsell chips in each; Figure 2). The stimulus display was mounted on a wall covered with black cloth. Red light-emitting diodes (3 mm in diameter) were set up next to each chip indicating which pair of chips was to be assessed (one chip on each half of the experimental display). They were driven by a PC equipped with a special program to run this experiment. 
Table 2
 
Munsell notations of the chips used in the partial hue-matching experiments.
Table 2
 
Munsell notations of the chips used in the partial hue-matching experiments.
1 2 3 4 5 6 7 8
2.5BG5/10 2.5B5/10 5B5/10 7.5B5/10 10B5/12 2.5PB5/12 5PB5/12 7.5PB5/10
9 10 11 12 13 14 15 16
10PB4/12 10P4/12 2.5RP4/12 5RP5/12 7.5RP5/14 10RP5/14 2.5R5/14 5R4/14
17 18 19 20 21 22 23 24
7.5R4/16 10R5/16 2.5YR5/14 7.5YR7/14 2.5Y8.5/12 5Y8/14 7.5Y8.5/12 10Y8.5/12
25 26 27 28 29 30 31 32
2.5GY8/10 5GY7/12 7.5GY6/12 10GY6/12 2.5G5/10 5G5/10 7.5G5/10 10G5/10
Figure 2
 
Stimulus display.
Figure 2
 
Stimulus display.
The stimulus display was illuminated by a neutral light source (52 lux; the CIE 1931 chromaticity coordinates x = 0.323 and y = 0.360). The experimental room was semi-darken. (Apart from the main light source, there was a little desk lamp illuminating the rear wall.) Observers sat at a distance of 1.5 m from the stimulus display. Viewing was binocular with no restriction concerning observation time and eye and head movements. 
Procedure
In one experimental trial, a pair of Munsell chips (one in each stand) was randomly indicated by the light-emitting diodes. The observer's task was to decide whether these chips shared any hue. More specifically, they were asked to enter an integer (0 to 10) indicating their confidence in the affirmative answer using a keypad. Observers were suggested to use rate “0” when the chips definitely had no hue in common and rate “10” when they definitely did have a common hue. Rates between 1 and 9 were to be chosen in the case of an uncertainty concerning the availability of a common hue. 
The experiment was preceded by a period of adaptation to the given illumination conditions that lasted at least 10 min. Each of the 32 × 32 pairs had been evaluated once during an experimental session. Five sessions were conducted with each observer. 
Results and analysis
For each observer, their responses to a single stimulus pair obtained in all the five sessions were summed up and divided by 50 so as to reduce the response range to the interval [0; 1]. The observer's responses to all the 32 × 32 stimulus pairs were recorded as a response matrix, {r(i, j)}, where indices i, j vary from 1 to 32, and the entry r(i, j) stands for the cumulative rate for the pair comprising the ith and jth Munsell chips. The Munsell chips are numbered as in Table 2. Figure 3 displays the response matrix produced by one observer (AB). Figure 4 presents a binary response matrix with sharp borders such that the least-squared difference between it and the response matrix (Figure 3) is minimal (see 9). 
Figure 3
 
Response matrix produced by Observer AB. Various shades of gray encode the response rate, r (from white representing r = 1 to black representing r = 0).
Figure 3
 
Response matrix produced by Observer AB. Various shades of gray encode the response rate, r (from white representing r = 1 to black representing r = 0).
Figure 4
 
Least-square approximation with a binary matrix to the response matrix in Figure 3.
Figure 4
 
Least-square approximation with a binary matrix to the response matrix in Figure 3.
It was found that the best approximation was achieved when the number of chromaticity classes n = 4, and the ends of the chromaticity classes were as follows: i 1 1 = 32, i 1 2 = 15; i 2 1 = 7, i 2 2 = 22; i 3 1 = 18, i 3 2 = 29; i 4 1 = 25, i 4 2 = 5 (here the ends of the kth chromaticity class are designated as i k 1 and i k 2). The rest of the observers also yielded four chromaticity classes. The ends of the chromaticity classes obtained for them can be found in Table 3
Table 3
 
Chromaticity class ends as evaluated by the least-square technique for the four observers. Each entry is the Munsell chip number according to the numeration given in Table 2. C i stands for the ith chromaticity class. The letters in brackets (e.g., B) indicates the likely verbal name of the corresponding component hue (e.g., “blue”).
Table 3
 
Chromaticity class ends as evaluated by the least-square technique for the four observers. Each entry is the Munsell chip number according to the numeration given in Table 2. C i stands for the ith chromaticity class. The letters in brackets (e.g., B) indicates the likely verbal name of the corresponding component hue (e.g., “blue”).
Chromaticity classes
C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
AB 32 15 7 22 18 29 25 4
CG 30 16 7 22 18 28 24 5
DW 30 15 7 20 18 28 25 3
MT 30 15 7 22 18 27 24 4
In Figure 5, the chromaticity classes of Observer AB are represented by arcs in the hue wheel. Specifically, each arc covers the Munsell chips belonging to the same chromaticity class. The four chromaticity classes fall into the two opponent pairs: C 1 (B) and C 3 (Y) and C 2 (R) and C 4 (G). The members of each pair are opponent to each other in the sense that, first, they cover the opposite parts of the hue wheel, and second, they do not overlap, that is, there are gaps between their ends. As a result, some chips are covered by one arc, whereas the others by two arcs. Those that are marked by only one arc (that is, which fall in a gap between the opponent chromaticity classes) belong to just one chromaticity class. This means that these chips contain just one component hue. According to the commonly accepted terminology (e.g., Wyszecki & Stiles, 1982), the color of such chips will be referred to as unitary. Belonging to two chromaticity classes, all the other chips in the sample have two component hues. Their colors will be called binary
Figure 5
 
Chromaticity classes revealed by Observer AB.
Figure 5
 
Chromaticity classes revealed by Observer AB.
A Munsell chip of unitary color gives an idea of the color appearance of the component hue constituting the chromaticity class it belongs to. For example, being of unitary color for Observer AB, Munsell chips 5 and 6 (10B5/12 and 2.5PB5/12) belong to the chromaticity class C 1 (with the ends λ 2 1 = 32 and λ 2 2 = 15). Therefore, the component hue constituting the chromaticity class C 1 can apparently be named as “blue.” Likewise, Munsell chips 16 and 17 (5R4/14 and 7.5R4/16) indicate the hue constituting the chromaticity class C 2 that can be named as “red.” The other two pairs of chips of unitary color—23 and 24 (10Y8.5/12 and 7.5Y8.5/12) and 30 and 31 (7.5G5/10 and 10G5/10)—represent the hues constituting the chromaticity classes C 3 (“yellow”) and C 4 (“green”), respectively. Thus, Observer AB yielded the response matrix such that it is best approximated by the binary matrix induced by the two pairs of opponent component hues 4 : yellow and blue and red and green. 
Figures 68 display the chromaticity classes for the rest of the observers. As one can see, all the observers yielded similar results that are in line with the classical notion of color opponency based on the four (yellow, blue, red, and green) hues (Hering, 1964; Hurvich & Jameson, 1957). 
Figure 6
 
Chromaticity classes revealed by Observer CG.
Figure 6
 
Chromaticity classes revealed by Observer CG.
Figure 7
 
Chromaticity classes revealed by Observer DW.
Figure 7
 
Chromaticity classes revealed by Observer DW.
Figure 8
 
Chromaticity classes revealed by Observer MT.
Figure 8
 
Chromaticity classes revealed by Observer MT.
As more than one chip were found to have unitary color, we evaluated the midpoint of the sector of chips in the hue circle that have the same unitary color and used this midpoint as an index specifying the location of the unitary colors in the hue circle for every observer (Table 4). By and large, these are in agreement with the unique hue-selecting data in Table 1
Table 4
 
Unitary colors as revealed in Experiment 1. Each entry is the Munsell chip number according to the numeration given in Table 2. When an entry is not an integer (e.g., 5.5), it means that the center of the gap between the chromaticity classes happens to be in between the adjacent chips (e.g., the chips with numbers 5 and 6).
Table 4
 
Unitary colors as revealed in Experiment 1. Each entry is the Munsell chip number according to the numeration given in Table 2. When an entry is not an integer (e.g., 5.5), it means that the center of the gap between the chromaticity classes happens to be in between the adjacent chips (e.g., the chips with numbers 5 and 6).
UH1 UH2 UH3 UH4
Blue Red Yellow Green
AB 5.5 16.5 23.5 30.5
CG 6 17 23 29
DW 5 16.5 22.5 29
MT 5.5 16.5 23 28.5
In order to characterize the location of a chromaticity class in the hue circle by a single index, we also evaluated the chromaticity class center as the midpoint between its ends. Table 5 displays the chromaticity class centers in the hue circle for all the four observers. 
Table 5
 
Centers of the chromaticity classes obtained for 4 observers in Experiment 1. The center of a chromaticity class is defined as the midpoint between its ends. For instance, C k denotes the center of the chromaticity class with the endpoints i k 1 and i k 2 (see Table 3).
Table 5
 
Centers of the chromaticity classes obtained for 4 observers in Experiment 1. The center of a chromaticity class is defined as the midpoint between its ends. For instance, C k denotes the center of the chromaticity class with the endpoints i k 1 and i k 2 (see Table 3).
Chromaticity classes
C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
AB 7.5 14.5 23.5 30.5
CG 7 14.5 23 30.5
DW 6.5 13.5 23 30
MT 6.5 14.5 22.5 30
Experiment 2: Partial hue-matching under symmetric chromatic illumination
If material color constancy exists, then the chips of unitary color should remain such despite a change in illumination, even if the illumination change alters the overall color appearance of these chips. One objective of the present experiment is to find out which Munsell chips in Table 2, if any, will look as being of unitary color under chromatic illumination. 
As one can see from Experiment 1, both the traditional unique hue selecting and partial hue-matching yield similar sets of unitary (unique) colors. However, using the unique hue selection when the illumination is chromatic becomes problematic. Indeed, this technique is based on an introspective criterion of “uniqueness” or “irreducibility.” Under neutral illumination, a chip is of unitary color when it looks as having just one hue, say, yellow. Imagine now that the chip is illuminated by a green light. In this case, the presence of at least two hues will be noticed: the yellow (material) hue of the chip and the green (lighting) hue of the illumination. It is not quite clear, even for experienced observers, how to judge such a dual color experience under the traditional unique hue selection task. Indeed, because of the material vs. lighting duality of the color percept in this condition, there is no chip satisfying the criterion of “uniqueness.” 
To its advantage, the partial hue-matching technique can, in principle, be used under chromatic illumination as well. The material vs. lighting duality seems to make the task of detecting a common hue neither ambiguous nor more difficult. Admittedly, one might get a trivial outcome: All the pairs of the chips illuminated by, say, a green light will get an affirmative response because of the presence of a common (green) lighting hue. This is also what should occur if the observers will rest their judgments on the chromaticity of the reflected light. Another objective of Experiment 2 is to find out whether this happens in the reality. 
An intriguing issue is also whether the material and lighting hues will interact. In other words, will observers give an affirmative answer to, say, a red chip lit by the achromatic light when compared with an achromatic chip lit by a red light? As we will see below, the answer to both these questions proves negative. 
Methods
Observers, apparatus, and procedure remained the same as in Experiment 1. The only difference between Experiments 1 and 2 was that, instead of the neutral illumination, four chromatic (red, green, yellow, and blue) lights were used in succession. The spectral power distributions of these lights are depicted in Figure 9. Their CIE chromaticity coordinates can be found in Table 6. The luminance of the lights was approximately equal (10–11 cd/m2). 
Figure 9
 
Spectral power distribution of the illuminants employed in the experiment. The line color matches the color of the light each line represents.
Figure 9
 
Spectral power distribution of the illuminants employed in the experiment. The line color matches the color of the light each line represents.
Table 6
 
CIE 1931 chromaticity coordinates of the illuminants used in Experiment 2.
Table 6
 
CIE 1931 chromaticity coordinates of the illuminants used in Experiment 2.
Illuminant x y
Blue 0.131 0.150
Red 0.635 0.321
Yellow 0.392 0.410
Green 0.224 0.667
Results
All the observers under all the four chromatic illuminations (except for Observer MT under the red illumination) yielded four chromaticity classes. Observer MT brought about five chromaticity classes under the red illumination (Figure 10). The ends of the chromaticity classes produced by each observer under all the chromatic illuminations are presented in Table 7. Comparing these with the data obtained under the neutral light (Table 3), one can see that although the ends of the chromaticity classes revealed under the chromatic illuminations are somewhat different, the difference is relatively small and not systematic, if any. This conclusion is supported by Figures 1114 where the chromaticity classes of all observers for all the five illuminants are depicted. The chromaticity classes for the neutral illumination are marked with the black arcs as in Figure 5. The arcs of blue, red, yellow, and green colors are used to mark the chromaticity classes for the blue, red, yellow, and green illuminations, respectively. As one can see, it is safe to conclude that the four chromaticity classes revealed under the chromatic illuminations are the same yellow, blue, red, and green chromaticity classes found under the neutral light, the end of which are somewhat altered due to the change of the illumination chromaticity. 
Figure 10
 
Chromaticity classes obtained for Observer MT under the red illumination.
Figure 10
 
Chromaticity classes obtained for Observer MT under the red illumination.
Table 7
 
Chromaticity class ends derived for the four chromatic illuminations (denoted by B, R, Y, and G in the first column) under the symmetric illumination condition. Each entry is the Munsell chip number according to the numeration given in Table 2.
Table 7
 
Chromaticity class ends derived for the four chromatic illuminations (denoted by B, R, Y, and G in the first column) under the symmetric illumination condition. Each entry is the Munsell chip number according to the numeration given in Table 2.
Observer AB Observer CG
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 31 15 6 22 19 29 25 4 30 17 6 22 18 28 24 5
R 29 12 7 20 17 27 22 3 29 17 7 20 18 26 22 5
Y 30 15 7 22 18 28 24 4 30 16 7 22 18 28 24 5
G 30 15 6 22 19 28 24 4 30 17 6 22 18 28 24 3
Observer DW Observer MT
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 2 15 6 22 18 28 24 2 1 15 4 22 19 28 23 1
R 29 14 7 19 15 28 22 1 2 11 4 20 12 26 22 1
Y 1 15 7 20 18 28 24 2 1 15 6 21 18 27 23 2
G 29 19 7 20 20 28 25 2 29 13 6 20 20 28 23 2
Figure 11
 
Chromaticity classes produced by Observer AB for the five illuminants under the symmetric illumination condition. The stimulus colors are arranged in a hue circle. Specifically, Chip 8 (7.5PB) is located at 12 o'clock, Chip 16 (5R) at 9 o'clock, Chip 24 (10Y8) at 6 o'clock, and Chip 32 (10G) at 3 o'clock. Each chromaticity class is marked with an arc (as in Figure 5), the color of which indicates the color of the illumination.
Figure 11
 
Chromaticity classes produced by Observer AB for the five illuminants under the symmetric illumination condition. The stimulus colors are arranged in a hue circle. Specifically, Chip 8 (7.5PB) is located at 12 o'clock, Chip 16 (5R) at 9 o'clock, Chip 24 (10Y8) at 6 o'clock, and Chip 32 (10G) at 3 o'clock. Each chromaticity class is marked with an arc (as in Figure 5), the color of which indicates the color of the illumination.
Figure 12
 
Chromaticity classes produced by Observer CG for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 12
 
Chromaticity classes produced by Observer CG for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 13
 
Chromaticity classes produced by Observer DW for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 13
 
Chromaticity classes produced by Observer DW for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 14
 
Chromaticity classes produced by Observer MT for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 14
 
Chromaticity classes produced by Observer MT for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Note that not every alteration of the chromaticity class ends results in a shift of the chromaticity class center. For instance, a symmetrical contraction (or expansion) of a chromaticity class leaves the location of its center unchanged. An important particular case of such an alteration takes place when the observers change their own criterion. It has been found in the previous studies that some observers exhibited broader gaps between the opponent chromaticity classes than the other (Logvinenko & Beattie, 2011; Logvinenko & Geithner, 2012). This can be a result of that different observers set different criterion of the absence of a common hue between two colors. Those with a high criterion produce wider gaps as compared to those with lower criteria. Such criterion fluctuation is unlikely to affect the chromaticity class center location. So, it seems reasonable to take into consideration only those chromaticity class alterations that are accompanied by the chromaticity class center shifts. In this case, the chromaticity class center shift can be taken as a natural indicator of deviation from constancy. Admittedly, it does not reflect symmetrical alterations of the chromaticity classes that might, in principle, be caused by the illumination change. Still, it is a simple and informative summary of the illumination effect on the chromaticity classes. Figure 15 displays the chromaticity class centers for all the five illuminations. 
Figure 15
 
Chromaticity class centers evaluated for four observers and five illuminations. On the circumference are the 32 stimulus colors arranged counterclockwise according to the numeration in Table 2 as in Figures 1114. A marker represents a chromaticity class center for a particular observer under a particular illuminant. The marker color encodes the illuminant color (the gray standing for the neutral illuminant). The marker shape encodes the observer (circle: AB; star: CG, diamond: DW, square: MT). Note that some markers do overlap.
Figure 15
 
Chromaticity class centers evaluated for four observers and five illuminations. On the circumference are the 32 stimulus colors arranged counterclockwise according to the numeration in Table 2 as in Figures 1114. A marker represents a chromaticity class center for a particular observer under a particular illuminant. The marker color encodes the illuminant color (the gray standing for the neutral illuminant). The marker shape encodes the observer (circle: AB; star: CG, diamond: DW, square: MT). Note that some markers do overlap.
As one might expect, the largest shift of the chromaticity class centers has been produced by the red light (the spectral power distribution of which differs most from that of the neutral light). However, even under the red illumination the chromaticity class centers do not shift by more than three Munsell hue steps. Note that for all observers the blue chromaticity class center moves toward green (i.e., from red) whereas the yellow chromaticity class center moves toward red (i.e., from green). Therefore, there is no overall shift in object color chromaticity due to illumination that can be described as either “red” or “green” one. 
In order to express the chromaticity class shift induced by illumination in the quantitative terms, we evaluated the proportion of chips that remained in the same chromaticity class under both the neutral and chromatic lights. For example, consider the red chromaticity class, C 2 (R), obtained for Observer AB under the neutral illumination (chips from 7 to 22; Table 3 and Figure 5) and the blue illumination (chips from 6 to 22; Table 7 and Figure 11). As we can see, 16 chips from 7 to 22 belong to C 2 (R) under both illuminations. That is, they remain in the same (C 2 (R)) chromaticity class despite the change of the illumination color (from neutral to blue). The red chromaticity classes under the neutral and blue illuminations consist of 16 and 17 chips, respectively. Dividing the number of chips common to both the chromaticity classes (under the neutral and blue illuminations), i.e., 16, over the number of chips in the union of these classes, i.e., 17, we get the proportion of 0.94 that quantifies the robustness of C 2 (R) to the switch of illumination from neutral to blue (referred to as the chromaticity class robustness index). 
The chromaticity class robustness index averaged across observers has been evaluated for each chromaticity class and each illumination (Table 8). As could be expected, the most robust chromaticity classes are found to be under the yellow light, the least under the red light. The total mean is found to be 0.87. 
Table 8
 
Chromaticity class robustness index averaged across observers. Rows correspond to chromaticity classes and columns to illuminations.
Table 8
 
Chromaticity class robustness index averaged across observers. Rows correspond to chromaticity classes and columns to illuminations.
B R Y G Mean
C 1 (B) 0.88 0.76 0.89 0.87 0.85
C 2 (R) 0.89 0.85 0.97 0.93 0.91
C 3 (Y) 0.93 0.73 0.98 0.84 0.87
C 4 (G) 0.89 0.73 0.89 0.87 0.84
Mean 0.90 0.77 0.93 0.88 0.87
Table 9 shows the chromaticity class robustness index averaged across illuminations for individual observers. Observer CG yielded the most robust results. Interestingly, she also produced the narrowest gaps between the opponent chromaticity classes (Figure 12). As a matter of fact, she has got some experience in painting. Perhaps, practicing in painting has made an impact on her ability to discern various shades of object color. 
Table 9
 
Chromaticity class robustness index averaged across illuminations. Rows correspond to chromaticity classes and columns to observers.
Table 9
 
Chromaticity class robustness index averaged across illuminations. Rows correspond to chromaticity classes and columns to observers.
AB CG DW MT Mean
C 1 (B) 0.85 0.95 0.82 0.77 0.85
C 2 (R) 0.94 0.94 0.94 0.82 0.91
C 3 (Y) 0.86 0.95 0.90 0.78 0.87
C 4 (G) 0.89 0.93 0.80 0.74 0.84
Mean 0.87 0.94 0.87 0.78 0.87
The chromaticity class robustness index as determined above reflects the stability of a chromaticity class evaluated under two illuminations, namely, the neutral and one of the chromatic illuminations. It does not show how the chromaticity class alters due to switching over between the whole set of chromatic illuminants. So, we generalize chromaticity class robustness index as follows. We define the so-called generalized robustness index for the jth chromaticity class (j = 1, …, 4) as 
N ( i = 1 i = 5 ( C j ( I i ) ) ) N ( i = 1 i = 5 ( C j ( I i ) ) ) ,
(1)
where I i is the ith illuminant (thus, C j (I i ) stands for the jth chromaticity class registered under the ith illuminant);
and
denote the set intersection and union, respectively; and N(x) stands for the number of elements in set x. Hence, the numerator in Equation 1 is the number of common chips in the chromaticity classes C j obtained under all the illuminations, and the denominator is the number of chips that appear in the chromaticity class C j at least for one illumination. The generalized robustness indices for all the chromaticity classes and for all the observers are given in Table 10
Table 10
 
Generalized chromaticity class robustness index.
Table 10
 
Generalized chromaticity class robustness index.
AB CG DW MT Mean
C 1 (B) 0.68 0.90 0.57 0.53 0.67
C 2 (R) 0.82 0.82 0.76 0.74 0.79
C 3 (Y) 0.69 0.82 0.64 0.41 0.64
C 4 (G) 0.73 0.75 0.64 0.67 0.70
Mean 0.73 0.82 0.65 0.59 0.70
Likewise under neutral illumination, no overlapping was found between opponent chromaticity classes under the chromatic illuminations, there being gaps between them. Approximately the same four classical unitary hues have been revealed under all the chromatic illuminations (Figure 16). Note that for some illuminations (such as yellow and blue) some unitary colors, such as yellow and blue, remain almost constant, whereas for the other illuminations these unitary colors undergo some chromaticity shift. For instance, under the red illumination the papers looking unitary blue under the neutral illumination become tinged with red. To the contrary, the papers looking unitary yellow under the neutral illumination become tinged with green under the red illumination. Hence, as mentioned above, there is no overall shift that all the papers are subject to due to the change of the illuminant. 
Figure 16
 
Unitary colors evaluated for four observers and five illuminations. Each marker represents the median unitary color chip for a particular observer under a particular illuminant. Notations are the same as in Figure 15.
Figure 16
 
Unitary colors evaluated for four observers and five illuminations. Each marker represents the median unitary color chip for a particular observer under a particular illuminant. Notations are the same as in Figure 15.
Observing how the gaps between the opponent chromaticity classes vary with the illumination chromaticity, one can get an idea of how stable the unitary colors of different kinds under the illumination alteration. In order to estimate the robustness of unitary colors to a change of the illumination from neutral to chromatic, we evaluated for the unitary colors an index similar to the chromaticity class robustness index. It shows that only a quarter of Munsell chips perceived as having unitary colors under the neutral illumination remain being perceived so under all the chromatic illuminations. In other words, the colour of 3 of 4 chips perceived unitary under the neutral light, is perceived binary under chromatic illuminations. We will refer to this as the unitary color shift
The unitary color shift has been quantified in Munsell hue steps as follows. For each observer and each chromatic illumination, the difference between the positions of the unitary color of a particular kind in the hue wheel under the neutral and the chromatic illuminant was evaluated. In other words, we evaluated the difference (in Munsell hue step) between the position of a gray marker in Figure 16 and the marker of the same shape but different color lying in the same quadrant. For example, for Observer AB the median blue unitary color lies between Munsell chips 5 and 6 under the neutral illumination. Under the red illumination, the median blue unitary color is Munsell chip 5 for this observer. Thus, the blue unitary color shift induced by a switch from the neutral to the red light is measured as 0.5 for Observer AB. 
The unitary color shifts averaged across observer are presented in Table 11. Averaging also over illumination (see the last row in Table 11), one can see that for all the four unitary colors the shift is practically one Munsell hue step, the total average being 0.95. The rightmost column in Table 11 presents the averaged (over observer and unitary color type) unitary color shifts for different illuminations. Not surprisingly, the smallest shift (nearly half a Munsell hue step) is induced by the yellow illuminant and the largest (almost two Munsell hue steps) by the red illuminant. 
Table 11
 
Unitary color shift averaged across observer.
Table 11
 
Unitary color shift averaged across observer.
Illuminant UC1 (B) UC2 (R) UC3 (Y) UC4 (G) Mean
B 0.25 1.13 1.25 0.38 0.75
R 2.13 1.50 1.13 2.38 1.78
Y 0.50 1.13 0.50 0.00 0.53
G 0.50 0.50 1.00 1.00 0.75
Mean 0.84 1.06 0.97 0.94 0.95
The unitary color shifts averaged across unitary color type are presented in Table 12. In the last row of this table, one can find the averaged (over illumination and unitary color type) unitary color shifts for different observers. Once again, Observer CG has the most robust unitary colors. The averaged unitary color shift for this observer is less than half a Munsell hue step. The maximal averaged unitary color shift is registered for Observer MT—more than one and a half Munsell hue step. 
Table 12
 
Unitary color shifts averaged across unitary color type.
Table 12
 
Unitary color shifts averaged across unitary color type.
Illuminant AB CG DW MT Mean
B 0.38 0.25 0.88 1.50 0.75
R 1.88 1.00 1.38 2.88 1.78
Y 0.50 0.00 0.63 1.00 0.53
G 0.75 0.50 1.00 0.75 0.75
Mean 0.88 0.44 0.97 1.53 0.95
Notably, more than a quarter of unitary colors (for all observers and chromatic illuminations) did not change at all as compared to those under the neutral light (see the first column in Table 13). The unitary color shift not more than half a Munsell hue step was registered in 59% of the cases, 64% of the unitary colors having shifted by not more than a Munsell hue step. 
Table 13
 
Proportions of unitary colors that undergone the unitary color shift not more than zero, half, one, one and a half, and two Munsell hue steps.
Table 13
 
Proportions of unitary colors that undergone the unitary color shift not more than zero, half, one, one and a half, and two Munsell hue steps.
Unitary color shift
0 ≤0.5 ≤1 ≤1.5 ≤2
28% 59% 64% 80% 91%
Experiment 3: Partial hue-matching under asymmetric—achromatic vs. chromatic—illumination
One might argue that the resemblance of the Experiments 1 and 2 results can be accounted for by “discounting the illuminant” (von Helmholtz, 1867). Indeed, because of chromatic adaptation at the receptor level, e.g., the adaptation of von Kries' (1970) type, the sensory inputs into the post-receptor neural structures might be approximately equal under the neutral and chromatic lights. 
It should be noted, however, that the notion of discounting the illuminant is usually applied to a single-illuminant scene, the illumination of which might change over time. Color constancy observed in this situation is sometimes referred to as successive color constancy, in contrast to simultaneous color constancy that occurs in a multiple-illuminant scene where the illumination changes over space (Brainard, 2004). It is clear that if the discounting of the illumination of multiple-illuminant scenes is possible at all, it can happen only locally, that is, it should occur independently in each area (frame) of homogeneous illumination. In other words, in a scene comprising a few areas homogeneously lit by different lights, each illuminant is discounted independently within the corresponding area. Such local discounting of the illuminants requires minimizing the eye movements because of the local nature of the cone adaptation (MacLeod, Williams, & Makous, 1992). Therefore, under the condition of free eye and head movements in our experiments, the local simultaneous discounting of the illuminants is rather unlikely. At any rate, Logvinenko and Tokunaga (2011) showed that the local simultaneous discounting of the illuminants failed to predict the observers' asymmetric color matches made across the illumination border (i.e., under the experimental conditions typical for simultaneous color constancy). In the following experiment, we used the experimental conditions very similar to those used by Logvinenko and Tokunaga so as to explore simultaneous color constancy via partial hue-matching, the same way as we did successive color constancy in Experiment 2
Specifically, in Experiment 3 partial hue-matching was carried out under asymmetric—achromatic vs. chromatic—illumination condition, that is, when two different illuminants were employed. If the chromatic adaptation is responsible for the similarity of the results of Experiments 1 and 2, then Experiment 3 (in which there was no restriction on eye and head movements) should bring about different results. Yet, as shown below, the results of Experiment 3 are in remarkable correspondence to those obtained in Experiments 1 and 2
Methods
Experiment 3 differed from the previous experiments (Experiments 1 and 2) only in that the illumination condition was asymmetric, that is, one half of the experimental display (Figure 2) was lit with the neutral light whereas the other half with a chromatic light. Observers were instructed to move their gaze between the stands so as to minimize local chromatic adaptation. 
Results
Figure 17 displays the response matrix produced by Observer MT under the neutral vs. red illumination condition. It is clearly asymmetric. The asymmetry means that the observer response to the ith chip under the red illumination and the jth chip under the neutral illumination can, generally, differ from the observer response to the jth chip under the red illumination and the ith chip under the neutral illumination. The pattern in Figure 17 seems to comprise four overlapping rectangles, some of which are definitely not regular (i.e., they are not squares). This is hardly surprising since, as established in Experiment 2, the chromaticity classes under the neutral and chromatic (especially, red) illuminations are not identical. If there were just one chromaticity class with different ends for the neutral and chromatic illuminations, then the response matrix under the asymmetric illumination conditions would be a rectangle with the sides determined by the chromaticity class ends for the neutral and chromatic illuminations. If one assumes that under asymmetric illumination conditions the number of chromaticity classes, say n, remains the same for both illuminations but the ends of the chromaticity classes alter with illumination, then the response matrix under the asymmetric illumination conditions should be an overlap of n rectangles. Each rectangle is determined by two chromaticity classes: one for the neutral and the other for the chromatic illuminations. Specifically, the vertical coordinates of the rectangle are the ends of the chromaticity class under the neutral illumination, the horizontal coordinates being those under the chromatic illumination. 
Figure 17
 
Response matrix (Observer MT) obtained for the neural vs. red illumination condition. An entry (i, j) stands for the observer response to the ith chip under the red illumination and the jth chip under the neutral illumination.
Figure 17
 
Response matrix (Observer MT) obtained for the neural vs. red illumination condition. An entry (i, j) stands for the observer response to the ith chip under the red illumination and the jth chip under the neutral illumination.
Using a similar least-square-based algorithm as that used for the symmetric matrices (see 10), the response matrix in Figure 17 was approximated by an asymmetric matrix consisting of the four overlapping rectangles. The horizontal coordinates of these rectangles brought about the ends of the chromaticity classes under the chromatic illumination (Table 14) and the vertical coordinates brought about those under the neutral illumination (Table 15). Note that all the data in Table 15 are obtained for the neutral illumination. As the neutral illuminant was present in each of the four asymmetric illumination conditions, the chromaticity class ends for the neutral illumination were replicated four times in Experiment 3
Table 14
 
Chromaticity class ends derived for the four chromatic illuminations (B, R, Y, and G) under the asymmetric illumination conditions. Each entry is the Munsell chip number according to the numeration given in Table 2.
Table 14
 
Chromaticity class ends derived for the four chromatic illuminations (B, R, Y, and G) under the asymmetric illumination conditions. Each entry is the Munsell chip number according to the numeration given in Table 2.
Observer AB Observer CG
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 31 15 7 23 19 29 24 4 30 18 7 23 18 28 24 5
R 30 12 6 21 17 28 24 3 30 16 4 21 18 28 24 3
Y 30 15 7 22 18 28 24 4 30 16 7 22 18 28 24 5
G 30 15 6 22 18 28 24 4 30 17 7 22 18 28 24 6
Observer DW Observer MT
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 2 15 6 20 19 28 24 2 1 15 4 22 20 28 24 1
R 29 15 7 19 16 28 24 1 2 14 4 20 12 28 24 1
Y 2 15 7 20 18 28 24 4 2 15 6 20 18 27 24 1
G 2 14 6 20 19 28 24 4 2 12 4 20 19 28 21 2
Table 15
 
Chromaticity class ends obtained for the achromatic illuminant under the asymmetric illumination conditions. Each entry is the Munsell chip number according to the numeration given in Table 2. Note that the difference between the rows is produced by the chromaticity (B, R, Y, or G) of the illumination of the adjacent stand. The illumination of the chips for which the chromaticity classes are obtained was neutral.
Table 15
 
Chromaticity class ends obtained for the achromatic illuminant under the asymmetric illumination conditions. Each entry is the Munsell chip number according to the numeration given in Table 2. Note that the difference between the rows is produced by the chromaticity (B, R, Y, or G) of the illumination of the adjacent stand. The illumination of the chips for which the chromaticity classes are obtained was neutral.
Observer AB Observer CG
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 30 15 8 22 18 28 25 4 30 16 7 22 18 28 24 5
R 30 15 8 22 18 28 23 4 30 16 7 22 18 28 23 5
Y 30 15 8 22 18 28 24 4 30 16 7 22 18 28 24 5
G 30 15 8 22 18 28 25 4 30 16 7 22 18 28 24 5
Observer DW Observer MT
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 2 15 7 20 18 28 25 2 1 15 6 20 18 27 25 2
R 1 15 8 20 18 28 23 2 1 15 7 20 18 26 23 1
Y 2 15 7 20 18 28 25 3 1 15 6 20 18 27 25 2
G 1 15 8 22 18 28 24 5 31 15 7 22 18 28 23 2
Comparing Tables 3 and 15 shows that the chromaticity class ends for the neutral illumination obtained under the symmetric and asymmetric illumination conditions are rather close. Figure 18 supports this conclusion. This graph was plotted as follows. One entry to Table 3, say i 1 1 = 32 for Observer AB, is taken as an abscissa for the four points in the graph, the ordinates of which are all the four values of i 1 1 for Observer AB in Table 15 (as a matter of fact, all those happen to equal 30). Though, because of overlapping we see in Figure 18 only one yellow circle with the coordinates (32, 30) instead of four circles of various colors. When the chromaticity class ends for the neutral illumination obtained under the symmetric and asymmetric illumination conditions are equal, the corresponding point lies on the black diagonal. Therefore, the deviation from the diagonal indicates the difference between the symmetric and asymmetric illumination conditions. 5 In other words, the deviation from the diagonal manifests the effect of illumination condition on the location of the chromaticity class ends for the neutral illumination. As can be seen in the first column of Table 16, 64% of the points in Figure 18 are located on the diagonal and 83% of the points depart from the diagonal by less than two Munsell hue steps. 
Figure 18
 
Chromaticity class ends for the achromatic illuminant obtained under the symmetric versus asymmetric conditions. The entries to Table 3 are plotted along the horizontal axis. The corresponding entries to Table 15 are plotted along the vertical axis. Different markers represent different observers: AB ( Image not available ), CG ( Image not available ), DW (♦), and MT ( Image not available ). The color of the markers indicates that of the chromatic illumination (of the neighboring stand) under the asymmetric condition.
Figure 18
 
Chromaticity class ends for the achromatic illuminant obtained under the symmetric versus asymmetric conditions. The entries to Table 3 are plotted along the horizontal axis. The corresponding entries to Table 15 are plotted along the vertical axis. Different markers represent different observers: AB ( Image not available ), CG ( Image not available ), DW (♦), and MT ( Image not available ). The color of the markers indicates that of the chromatic illumination (of the neighboring stand) under the asymmetric condition.
Table 16
 
Chromaticity class shift in percentage from symmetric to asymmetric condition (see text).
Table 16
 
Chromaticity class shift in percentage from symmetric to asymmetric condition (see text).
Munsell hue step rate Achromatic illumination Chromatic illumination
Zero-step rate 64.06 62.50
One-step rate 18.75 23.44
Two-step rate 10.94 9.38
More-than-two-step rate 6.25 4.69
Figure 19 shows how the chromaticity class ends for the chromatic illumination obtained under the symmetric illumination conditions correlate with those obtained under the asymmetric illumination conditions. The entries to Table 7 are taken as abscissas, the corresponding entries to Table 14 being the ordinates. The second column in Table 16 testifies that the correlation is nearly as good as in Figure 18
Figure 19
 
Chromaticity class ends for the chromatic illuminants obtained under the symmetric versus asymmetric conditions. The entries to Table 7 are plotted along the horizontal axis. The corresponding entries to Table 14 are plotted along the vertical axis. Different markers represent different observers: AB ( Image not available ), CG ( Image not available ), DW (♦), and MT ( Image not available ). The color of the markers indicates that of the chromatic illuminant.
Figure 19
 
Chromaticity class ends for the chromatic illuminants obtained under the symmetric versus asymmetric conditions. The entries to Table 7 are plotted along the horizontal axis. The corresponding entries to Table 14 are plotted along the vertical axis. Different markers represent different observers: AB ( Image not available ), CG ( Image not available ), DW (♦), and MT ( Image not available ). The color of the markers indicates that of the chromatic illuminant.
General discussion
In this study, we looked into color categories of a particular type—chromaticity classes—under the neutral and the four chromatic illuminations. Recall that a chromaticity class is a subset of all the colors containing one particular component hue (Logvinenko, 2012b). Being well defined, these color categories (i.e., chromaticity classes) are not as burdened with cognitive components as those used before in color constancy studies (Granzier et al., 2009; Hansen et al., 2007; Olkkonen et al., 2009, 2011; Troost & de Weert, 1991; Uchikawa et al., 1989). Indeed, when an observer is asked whether a particular color is, say, green, there are many plausible interpretations of this instruction. For example, the affirmative answer can be expected only when the color is perceived as pure (unique) green being tinged with no other hues. Alternatively, one can expect the affirmative answer when any, even the slightest, tinge of green is present in the color. Only subject to two conditions could such a strategy (that is, the latter) bring about the color category amounting to the chromaticity class. 
One condition is that by “green” the observer understands the green component hue. This is a rather unrealistic assumption, because while it is safe to assume the existence of the component hues, there is no easy way to explain to naive observers what component hues are. The advantage of the partial hue-matching technique is that it brings out the component hues as an outcome without having to specify them in advance. The other condition is that the observer is capable of discerning the presence of the green component hue irrespective of its amount. One can hardly expect observers to be so thorough. When using the partial hue-matching technique, we instruct observers to make a judgment concerning the presence of some common hue no matter how pronounced it is. 
When the color categories are not well specified, as is the case in the studies devoted to so-called categorical color constancy, the task is, actually, reduced to verbal naming. In this case, it is left to observers to specify what the word “green” means. In such a situation, one might expect the whole continuum of the strategies intermediate between the two extremes outlined above. Green might mean the component green slightly contaminated with the other hues. Otherwise, it might be any color containing the component green in some quantity. Depending on the threshold criterion for this quantity, one might expect the whole series of the color categories of green. There is no guarantee that this criterion will not change from trial to trial, not to mention from experiment to experiment, because, as is typical of a threshold criterion, one cannot control it. 
Such proneness of color categories to variation seriously undermines color categorizing as a tool for studying color constancy (e.g., Foster, 2003), except for one situation—perfect categorical color constancy. Indeed, if a particular color category does not change with illumination, it means that both perceptual and cognitive components are constant. However, although some authors summarized their findings as demonstrating rather good categorical color constancy, close inspection of their results shows that the variability of the color categories they used is, in fact, larger than that revealed in our study (see, e.g., Olkkonen et al., 2011). However, even if it were reliably established that some color categories, that is, verbal naming of colored objects, remained constant, how much would this tell us about color perception and how much about word perception? 
Let us discuss now what we can learn about color perception from our study of perceptual categories based on the component hues. Experiment 1 shows that under the neutral illumination, the partial hue-matching technique yields the same chromatic structure of object color as that revealed by the hue-selecting technique based on pure introspective analysis—that is, the classical four component hues (yellow, blue, red, and green) that have been noticed by artists and psychologists for a long time (for a review, see, e.g., Kuehni & Schwarz, 2008; Valberg, 2001). Furthermore, the same four-component-hue structure has also been found under chromatic illumination (Experiment 2). It follows that, despite the unusual appearance that objects have under chromatic illumination—which many previous authors have described as “strange” and even “weird” (for a review, see Mausfeld, 2003)—the color of objects lit by a chromatic light does not contain any additional component hues. Moreover, the four component hues revealed under the chromatic illuminations are in fact the same yellow, blue, red, and green component hues that are known to subsist under neutral illumination. 
We believe that this apparent weirdness of the color appearance induced by a chromatic illumination reflects the material-lighting dualism of object color (Logvinenko & Tokunaga, 2011; Tokunaga & Logvinenko, 2010a, 2010b, 2010c). Indeed, as noticed by many scientists (e.g., Mausfeld, 2003), under chromatic illumination the whole scene appears as if it were being viewed through a veil of prevailing hue. In other words, if the illuminant is, say, red, the reddish tint is present everywhere in the scene. Experiment 2 proves that our observers abstracted their evaluations out of such a prevailing tint (as produced by illuminant) altogether. Otherwise, their response matrices would have been full of white entries. Specifically, the very existence of negative answers to some pairs of Munsell chips (i.e., no, these chips do not have a common hue) proves that our observers did not take into account this prevailing tint of illumination. According to our terminology, they based their responses on the material rather than lighting dimensions of object color. It should be borne in mind that our observers were not instructed to ignore the illumination color. No additional instructions were given in Experiment 2 as compared to Experiment 1 at all. 
Experiment 3, conducted under asymmetric illumination conditions (i.e., under the simultaneous presence of two different illuminants), shows that this result cannot be accounted for by chromatic adaptation. In reality, the same four-component-hue structure has been found under asymmetric illumination conditions as well. However, it is unlikely that chromatic adaptation could have taken place simultaneously to two different fields without any restrictions on eye movements. Moreover, if this were to have happened, then the apparent difference in the color appearance of the two fields would essentially reduce, if not disappear. However, this was not the case. Apart from the immediate introspection, this is corroborated by the fact that the chromaticity classes derived in the asymmetric illumination conditions (Experiment 3) are very similar to those established in the symmetric illumination conditions (Experiments 1 and 2). 
This remarkable resemblance between the chromaticity classes in the symmetric and asymmetric illumination conditions testifies in favor of the material color existence. Really, the affirmative answer to a pair of Munsell chips lit by different lights means that the observer sees a common hue in the color of them. Therefore, the corresponding chromaticity classes (say, green ones) established under the neutral and chromatic illuminations are underlain by the same (green) material hue. This is in line with Logvinenko and Tokunaga's (2011) finding that the asymmetric color matching based on the least dissimilarity is transitive. If, as they argue, such least dissimilarity matching is based on material color, then it must be transitive. 
One might argue that if all the corresponding chromaticity classes are constituted by the same material component hue, why do they differ for different illuminations? Admittedly, the chromaticity classes of the same type are found to shift relative to each other in the hue circle (Figure 15). Note, however, that, first, this illuminant-induced shift is relatively small. In other words, the robustness of chromaticity classes to illumination change is high. Indeed, the average chromaticity-class-robustness index (respectively, generalised chromaticity-class-robustness index) was found to be 0.87 (respectively, 0.70). These are higher values than for other quantitative indices registered in the other studies of categorical color constancy (Granzier et al., 2009; Olkkonen et al., 2011; Troost & de Weert, 1991). Second, this is exactly what could be expected to result from illuminant-induced mismatching of metamers. The fact is that, because of the illuminant-induced mismatching of metamers, a particular reflectance becomes metameric to different reflectances under different illuminations. Since material hue is assigned to a whole class of metamerism, a particular reflectance is assigned different material hues, depending on illumination, because of this illuminant-induced mismatching of metamers. Therefore, as argued elsewhere (Logvinenko, 2009b; Logvinenko & Tokunaga, 2011), whatever the material hue's initial assignment to a class of metameric reflectances, a particular reflectance will undergo a hue shift with any alteration in illuminant. 
It follows that if mismatching of metamers did not exist, the chromaticity class robustness indices would be even higher. Interestingly, there are spectral reflectance functions that are not subject to the mismatching of metamers. These are so-called optimal object color stimuli, that is, those that map to the object color solid boundary (Wyszecki & Stiles, 1982). Indeed, as shown elsewhere (Logvinenko, 2009b; Logvinenko & Levin, 2012), there are no metamers for the optimal object color stimuli; thus, there is no mismatching of metamers. Therefore, ideally, one should employ optimal object color stimuli to ascertain the robustness of chromaticity classes to illumination change. However, the optimal object color stimuli happen to be ideal rectangular spectral reflectance functions (Logvinenko, 2009b; Schrodinger, 1970). Hence, it is not possible to create them in practice. Yet, they can be approximated with physically realizable reflectances to a limited extent, so that the mismatching of metamers is made negligibly small. Replicating our experiments with such reflectances would allow us to find out how constant the chromaticity classes really are with illumination. 
Thus, we come up with the following understanding of what is invariant (constant) and what is variant in color perception of multi-illuminant scenes. First, there is the four-component-hue chromatic structure of material color, which is the same (i.e., constant) for any illumination. Second, there is the assignment of a combination (of a certain proportion) of the material component hues to every class of metamerism for each illuminant (this is referred to as the material color map; Logvinenko, 2012a). Third, there is a map that determines for each pair of reflectance and light the class of metamerism to which this reflectance belongs, provided it is illuminated by this light (Logvinenko, 2009b). Because of light–object interaction, the same reflectance can be mapped to different classes of metamerism under different illuminations (the illuminant-induced mismatching of metamers). As a result, the same reflectance will be assigned different material hues under different illuminations (the illuminant-induced material hue shift). Hence, the material color of an object is not (and cannot be) invariant (constant) with respect the object's illumination. Yet, the illuminant-induced material hue shift can be, in principle, predicted from the illuminant-induced mismatching of metamers—provided that the material color map is established. 
Appendix A
Deriving chromaticity classes: Symmetric illumination conditions
In what follows, the stimulus chips will be assumed to be ordered in terms of Munsell hue (as in Table 2). Although it is not necessary (Logvinenko, 2012b), this assumption makes the subsequent analysis technically much simpler. As pointed out above, in this case the chromaticity classes can be fully specified by their endpoints. Therefore, the experimental identification of chromaticity classes reduces to establishing the chromaticity class ends. 
Designating each stimulus paper by a number 6 μ i (i = 1, …, 32), the stimulus array can be represented as a matrix {(μ i , μ j )} where indices i, j vary from 1 to 32. Let us designate r(μ i , μ j ) as the observer response to the stimulus pair (μ i , μ j ). After normalizing the observer response, r(μ i , μ j ) varies from 0 to 1. We assume that if there were no uncertainty, the observer response would be binary, taking either 1 or 0. In this case, the response function is fully determined by the ends of the chromaticity classes: λ 1 1, λ 1 2,…, λ n 1, λ n 2, where n is the number of chromaticity classes (we considered only n < 6), λ k 1 is the left end of the kth chromaticity class, and λ k 2 is the right end of the kth chromaticity class. In other words, a paper μ i belongs to the kth chromaticity class if μ i lies between λ k 1 and λ k 2. More formally, a paper μ i belongs to the kth chromaticity class if (i) when λ k 1λ k 2, one has λ k 1μ i λ k 2 and (ii) when λ k 1 > λ k 2, one has μ i λ k 2 and μ i λ k 1
Now let us define the observer response to a stimulus pair (μ i , μ j ) (written r(μ i , μ j ; λ 1 1, λ 1 2, …, λ n 1, λ n 2) where μ i and μ j are variables, and λ 1 1, λ 1 2,…, λ n 1, λ n 2 are parameters) as 
r ( μ i , μ j ; λ 1 1 , λ 1 2 , , λ n 1 , λ n 2 ) = max ( r ( μ i , μ j ; λ 1 1 , λ 1 2 ) , , r ( μ i , μ j ; λ n 1 , λ n 2 ) ,
(A1)
where if λ k 1λ k 2  
r ( μ i , μ j ; λ k 1 , λ k 2 ) = { 1 , i f λ k 1 μ i , μ j λ k 2 ; 0 , i f o t h e r w i s e .
(A2)
If λ i 1 > λ i 2, 
r ( μ i , μ j ; λ k 1 , λ k 2 ) = { 1 , i f μ i , μ j λ k 2 ; 1 , i f μ i , μ j λ k 1 ; 1 , i f μ i λ k 2 a n d μ j λ k 1 ; 1 , i f μ j λ k 2 a n d μ i λ k 1 ; 0 , i f o t h e r w i s e .
(A3)
 
The graph of r(μ i , μ j ; λ 1 1, λ 1 2,…, λ n 1, λ n 2) is an intersection of the n squares, the sides of which are determined by parameters λ 1 1, λ 1 2,…, λ n 1, λ n 2. For instance, Figure 4 is the graph of r(μ i , μ j ; λ 1 1, λ 1 2,…, λ n 1, λ n 2), where n = 4, and λ 1 1 = 32, λ 1 2 = 15, λ 2 1 = 7, λ 2 2 = 22, λ 3 1 = 18, λ 3 2 = 29, λ 4 1 = 25, and λ 4 2 = 5. 
Now let us designate R(μ i , μ j ) as the real observer response to the stimulus pair (μ i , μ j ) normalized so that it varies between 0 and 1. For each real response matrix {R(μ i , μ j )}, we have found such numbers n, λ 1 1, λ 1 2,…, λ n 1, λ n 2 that the sum of the squared differences (“residuals”) 
i = 1 , j = 1 32 ( R ( μ i , μ j ) r ( μ i , μ j ; λ 1 1 , λ 1 2 , , λ n 1 , λ n 2 ) ) 2 ,
(A4)
reaches its minimum. In other words, we have found the binary matrix of a similar form as in Figure 4 (with the sharp borders) that approximates (in the sense of the least squares) the real response matrix {R(μ i , μ j )}. 
Deriving chromaticity classes: Asymmetric illumination conditions
For the asymmetric illumination conditions, we denote the observer response to a chip μ i lit by the chromatic illumination and a chip μ j lit by the achromatic illumination as r(μ i , μ j ; λ C1 1, λ C1 2,…, λ Cn 1, λ Cn 2, λ A1 1, λ A1 2,…, λ An 1, λ An 2), where n is the number of chromaticity classes; λ C1 1, λ C1 2,…, λ Cn 1, λ Cn 2 are the chromaticity class ends under the chromatic illumination; and λ A1 1, λ A1 2,…, λ An 1, λ An 2 are the chromaticity class ends under the achromatic illumination. Likewise for the symmetric illumination conditions (Equation 1), the model of the observer response to a stimulus pair (μ i , μ j ) is 
r ( μ i , μ j ; λ C 1 1 , λ C 1 2 , , λ C n 1 , λ C n 2 , λ A 1 1 , λ A 1 2 , , λ A n 1 , λ A n 2 ) = max ( r ( μ i , μ j ; λ C 1 1 , λ C 1 2 , λ A 1 1 , λ A 1 2 ) , , r ( μ i , μ j ; λ C n 1 , λ C n 2 , λ A n 1 , λ A n 2 ) ,
(A5)
where r(μ i , μ j ; λ Ck 1, λ Ck 2, λ Ak 1, λ Ak 2) (k = 1, …, n) takes 1 when the logical statement in at least one row in Table A1 is true. Otherwise, it takes 0. 
Table A1
 
Matrix of conditions resulting in the positive response to the stimulus pair (μ i , μ j ). Each row should be taken as a logical statement. The four inequalities in each row are supposed to be combined by the logical AND operator.
Table A1
 
Matrix of conditions resulting in the positive response to the stimulus pair (μ i , μ j ). Each row should be taken as a logical statement. The four inequalities in each row are supposed to be combined by the logical AND operator.
λ Ck 1 < λ Ck 2 λ Ak 1 < λ Ak 2 λ Ck 1 < μ i < λ Ck 2 λ Ak 1 < μ j < λ Ak 2
λ Ck 1 < λ Ck 2 λ Ak 1λ Ak 2 λ Ck 1 < μ i < λ Ck 2 μ j λ Ak 2 OR μ j λ Ak 1
λ Ck 1λ Ck 2 λ Ak 1 < λ Ak 2 μ i λ Ck 2 OR μ i λ Ck 1 λ Ak 1 < μ j < λ Ak 2
λ Ck 1λ Ck 2 λ Ak 1λ Ak 2 μ i λ Ck 2 μ j λ Ak 2 OR μ j λ Ak 1
λ Ck 1λ Ck 2 λ Ak 1λ Ak 2 μ i λ Ck 1 μ j λ Ak 2 OR μ j λ Ak 1
Then, as for the symmetric illumination conditions, for each rectangular response matrix {R(μ i , μ j )} recorded in the experiment, we sought for such numbers n, λ C1 1, λ C1 2,…, λ Cn 1, λ Cn 2, λ A1 1, λ A1 2,…, λ An 1, λ An 2 that minimized the residuals: 
i = 1 , j = 1 32 ( R ( μ i , μ j ) r ( μ i , μ j ; λ C 1 1 , λ C 1 2 , , λ C n 1 , λ C n 2 , λ A 1 1 , λ A 1 2 , , λ A n 1 , λ A n 2 ) ) 2 .
(A6)
 
Acknowledgments
Commercial relationships: none. 
Corresponding author: Alexander D. Logvinenko. 
Email: a.logvinenko@gcu.ac.uk. 
Address: Glasgow Caledonian University, Cowcaddens Road UK, Glasgow, G4 0BA, UK. 
Footnotes
Footnotes
1  There are other types of color constancy (see, e.g., Foster et al., 1997) that are left out of scope of the present article.
Footnotes
2  The effect of illumination on object's color is a serious problem for illumination engineers who try to minimize it (Logvinenko, 2009a; Schanda, 2007; Wyszecki & Stiles, 1982).
Footnotes
3  It is important to emphasize that real surfaces lit by real light sources (i.e., not computer-generated images) were employed as stimuli in the present study. As argued elsewhere (Logvinenko & Tokunaga, 2011), there are serious questions concerning the adequacy of using computer-generated images instead of real objects for color constancy studies because the perception of real objects differs from the perception of pictures of objects. In particular, the material vs. lighting duality established for perception of real objects disappears for their pictorial representations (Logvinenko, Petrini, & Maloney, 2008).
Footnotes
4  Component hues are understood to be opponent in the sense that they constitute the opponent chromaticity classes, as defined above.
Footnotes
5  Note that the deviation from the diagonal of the blue triangle in the right bottom corner in Figure 18 is effectively 3 Munsell hue steps because of the circular nature of the hue wheel. For example, in the hue wheel both Chip 31 and Chip 1 depart from Chip 32 by one step.
Footnotes
6  This number can be either the number from Table 2 or some other ordinal index, for instance, central wavelength (Logvinenko, 2009b).
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Figure 1
 
Munsell chips used in the preliminary experiment on unique hue selection. On the circumference is the Munsell notation of the page from which the chip was selected.
Figure 1
 
Munsell chips used in the preliminary experiment on unique hue selection. On the circumference is the Munsell notation of the page from which the chip was selected.
Figure 2
 
Stimulus display.
Figure 2
 
Stimulus display.
Figure 3
 
Response matrix produced by Observer AB. Various shades of gray encode the response rate, r (from white representing r = 1 to black representing r = 0).
Figure 3
 
Response matrix produced by Observer AB. Various shades of gray encode the response rate, r (from white representing r = 1 to black representing r = 0).
Figure 4
 
Least-square approximation with a binary matrix to the response matrix in Figure 3.
Figure 4
 
Least-square approximation with a binary matrix to the response matrix in Figure 3.
Figure 5
 
Chromaticity classes revealed by Observer AB.
Figure 5
 
Chromaticity classes revealed by Observer AB.
Figure 6
 
Chromaticity classes revealed by Observer CG.
Figure 6
 
Chromaticity classes revealed by Observer CG.
Figure 7
 
Chromaticity classes revealed by Observer DW.
Figure 7
 
Chromaticity classes revealed by Observer DW.
Figure 8
 
Chromaticity classes revealed by Observer MT.
Figure 8
 
Chromaticity classes revealed by Observer MT.
Figure 9
 
Spectral power distribution of the illuminants employed in the experiment. The line color matches the color of the light each line represents.
Figure 9
 
Spectral power distribution of the illuminants employed in the experiment. The line color matches the color of the light each line represents.
Figure 10
 
Chromaticity classes obtained for Observer MT under the red illumination.
Figure 10
 
Chromaticity classes obtained for Observer MT under the red illumination.
Figure 11
 
Chromaticity classes produced by Observer AB for the five illuminants under the symmetric illumination condition. The stimulus colors are arranged in a hue circle. Specifically, Chip 8 (7.5PB) is located at 12 o'clock, Chip 16 (5R) at 9 o'clock, Chip 24 (10Y8) at 6 o'clock, and Chip 32 (10G) at 3 o'clock. Each chromaticity class is marked with an arc (as in Figure 5), the color of which indicates the color of the illumination.
Figure 11
 
Chromaticity classes produced by Observer AB for the five illuminants under the symmetric illumination condition. The stimulus colors are arranged in a hue circle. Specifically, Chip 8 (7.5PB) is located at 12 o'clock, Chip 16 (5R) at 9 o'clock, Chip 24 (10Y8) at 6 o'clock, and Chip 32 (10G) at 3 o'clock. Each chromaticity class is marked with an arc (as in Figure 5), the color of which indicates the color of the illumination.
Figure 12
 
Chromaticity classes produced by Observer CG for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 12
 
Chromaticity classes produced by Observer CG for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 13
 
Chromaticity classes produced by Observer DW for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 13
 
Chromaticity classes produced by Observer DW for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 14
 
Chromaticity classes produced by Observer MT for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 14
 
Chromaticity classes produced by Observer MT for the five illuminants under the symmetric illumination condition. Notations are the same as in Figure 11.
Figure 15
 
Chromaticity class centers evaluated for four observers and five illuminations. On the circumference are the 32 stimulus colors arranged counterclockwise according to the numeration in Table 2 as in Figures 1114. A marker represents a chromaticity class center for a particular observer under a particular illuminant. The marker color encodes the illuminant color (the gray standing for the neutral illuminant). The marker shape encodes the observer (circle: AB; star: CG, diamond: DW, square: MT). Note that some markers do overlap.
Figure 15
 
Chromaticity class centers evaluated for four observers and five illuminations. On the circumference are the 32 stimulus colors arranged counterclockwise according to the numeration in Table 2 as in Figures 1114. A marker represents a chromaticity class center for a particular observer under a particular illuminant. The marker color encodes the illuminant color (the gray standing for the neutral illuminant). The marker shape encodes the observer (circle: AB; star: CG, diamond: DW, square: MT). Note that some markers do overlap.
Figure 16
 
Unitary colors evaluated for four observers and five illuminations. Each marker represents the median unitary color chip for a particular observer under a particular illuminant. Notations are the same as in Figure 15.
Figure 16
 
Unitary colors evaluated for four observers and five illuminations. Each marker represents the median unitary color chip for a particular observer under a particular illuminant. Notations are the same as in Figure 15.
Figure 17
 
Response matrix (Observer MT) obtained for the neural vs. red illumination condition. An entry (i, j) stands for the observer response to the ith chip under the red illumination and the jth chip under the neutral illumination.
Figure 17
 
Response matrix (Observer MT) obtained for the neural vs. red illumination condition. An entry (i, j) stands for the observer response to the ith chip under the red illumination and the jth chip under the neutral illumination.
Figure 18
 
Chromaticity class ends for the achromatic illuminant obtained under the symmetric versus asymmetric conditions. The entries to Table 3 are plotted along the horizontal axis. The corresponding entries to Table 15 are plotted along the vertical axis. Different markers represent different observers: AB ( Image not available ), CG ( Image not available ), DW (♦), and MT ( Image not available ). The color of the markers indicates that of the chromatic illumination (of the neighboring stand) under the asymmetric condition.
Figure 18
 
Chromaticity class ends for the achromatic illuminant obtained under the symmetric versus asymmetric conditions. The entries to Table 3 are plotted along the horizontal axis. The corresponding entries to Table 15 are plotted along the vertical axis. Different markers represent different observers: AB ( Image not available ), CG ( Image not available ), DW (♦), and MT ( Image not available ). The color of the markers indicates that of the chromatic illumination (of the neighboring stand) under the asymmetric condition.
Figure 19
 
Chromaticity class ends for the chromatic illuminants obtained under the symmetric versus asymmetric conditions. The entries to Table 7 are plotted along the horizontal axis. The corresponding entries to Table 14 are plotted along the vertical axis. Different markers represent different observers: AB ( Image not available ), CG ( Image not available ), DW (♦), and MT ( Image not available ). The color of the markers indicates that of the chromatic illuminant.
Figure 19
 
Chromaticity class ends for the chromatic illuminants obtained under the symmetric versus asymmetric conditions. The entries to Table 7 are plotted along the horizontal axis. The corresponding entries to Table 14 are plotted along the vertical axis. Different markers represent different observers: AB ( Image not available ), CG ( Image not available ), DW (♦), and MT ( Image not available ). The color of the markers indicates that of the chromatic illuminant.
Table 1
 
Munsell chips chosen as having unique hue by the four observers. Each entry represents the median choice across the eight trials. Although selection was made from the 40 Munsell chips, the median chips are given according to the numeration of the 32 Munsell chips engaged in the subsequent experiments (Table 2).
Table 1
 
Munsell chips chosen as having unique hue by the four observers. Each entry represents the median choice across the eight trials. Although selection was made from the 40 Munsell chips, the median chips are given according to the numeration of the 32 Munsell chips engaged in the subsequent experiments (Table 2).
Unique hues
UH1 UH2 UH3 UH4
Blue Red Yellow Green
AB 5 17 21 29
CG 6 17 21.5 30
DW 5 17 21 30
MT 5 17 21 28
Table 2
 
Munsell notations of the chips used in the partial hue-matching experiments.
Table 2
 
Munsell notations of the chips used in the partial hue-matching experiments.
1 2 3 4 5 6 7 8
2.5BG5/10 2.5B5/10 5B5/10 7.5B5/10 10B5/12 2.5PB5/12 5PB5/12 7.5PB5/10
9 10 11 12 13 14 15 16
10PB4/12 10P4/12 2.5RP4/12 5RP5/12 7.5RP5/14 10RP5/14 2.5R5/14 5R4/14
17 18 19 20 21 22 23 24
7.5R4/16 10R5/16 2.5YR5/14 7.5YR7/14 2.5Y8.5/12 5Y8/14 7.5Y8.5/12 10Y8.5/12
25 26 27 28 29 30 31 32
2.5GY8/10 5GY7/12 7.5GY6/12 10GY6/12 2.5G5/10 5G5/10 7.5G5/10 10G5/10
Table 3
 
Chromaticity class ends as evaluated by the least-square technique for the four observers. Each entry is the Munsell chip number according to the numeration given in Table 2. C i stands for the ith chromaticity class. The letters in brackets (e.g., B) indicates the likely verbal name of the corresponding component hue (e.g., “blue”).
Table 3
 
Chromaticity class ends as evaluated by the least-square technique for the four observers. Each entry is the Munsell chip number according to the numeration given in Table 2. C i stands for the ith chromaticity class. The letters in brackets (e.g., B) indicates the likely verbal name of the corresponding component hue (e.g., “blue”).
Chromaticity classes
C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
AB 32 15 7 22 18 29 25 4
CG 30 16 7 22 18 28 24 5
DW 30 15 7 20 18 28 25 3
MT 30 15 7 22 18 27 24 4
Table 4
 
Unitary colors as revealed in Experiment 1. Each entry is the Munsell chip number according to the numeration given in Table 2. When an entry is not an integer (e.g., 5.5), it means that the center of the gap between the chromaticity classes happens to be in between the adjacent chips (e.g., the chips with numbers 5 and 6).
Table 4
 
Unitary colors as revealed in Experiment 1. Each entry is the Munsell chip number according to the numeration given in Table 2. When an entry is not an integer (e.g., 5.5), it means that the center of the gap between the chromaticity classes happens to be in between the adjacent chips (e.g., the chips with numbers 5 and 6).
UH1 UH2 UH3 UH4
Blue Red Yellow Green
AB 5.5 16.5 23.5 30.5
CG 6 17 23 29
DW 5 16.5 22.5 29
MT 5.5 16.5 23 28.5
Table 5
 
Centers of the chromaticity classes obtained for 4 observers in Experiment 1. The center of a chromaticity class is defined as the midpoint between its ends. For instance, C k denotes the center of the chromaticity class with the endpoints i k 1 and i k 2 (see Table 3).
Table 5
 
Centers of the chromaticity classes obtained for 4 observers in Experiment 1. The center of a chromaticity class is defined as the midpoint between its ends. For instance, C k denotes the center of the chromaticity class with the endpoints i k 1 and i k 2 (see Table 3).
Chromaticity classes
C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
AB 7.5 14.5 23.5 30.5
CG 7 14.5 23 30.5
DW 6.5 13.5 23 30
MT 6.5 14.5 22.5 30
Table 6
 
CIE 1931 chromaticity coordinates of the illuminants used in Experiment 2.
Table 6
 
CIE 1931 chromaticity coordinates of the illuminants used in Experiment 2.
Illuminant x y
Blue 0.131 0.150
Red 0.635 0.321
Yellow 0.392 0.410
Green 0.224 0.667
Table 7
 
Chromaticity class ends derived for the four chromatic illuminations (denoted by B, R, Y, and G in the first column) under the symmetric illumination condition. Each entry is the Munsell chip number according to the numeration given in Table 2.
Table 7
 
Chromaticity class ends derived for the four chromatic illuminations (denoted by B, R, Y, and G in the first column) under the symmetric illumination condition. Each entry is the Munsell chip number according to the numeration given in Table 2.
Observer AB Observer CG
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 31 15 6 22 19 29 25 4 30 17 6 22 18 28 24 5
R 29 12 7 20 17 27 22 3 29 17 7 20 18 26 22 5
Y 30 15 7 22 18 28 24 4 30 16 7 22 18 28 24 5
G 30 15 6 22 19 28 24 4 30 17 6 22 18 28 24 3
Observer DW Observer MT
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 2 15 6 22 18 28 24 2 1 15 4 22 19 28 23 1
R 29 14 7 19 15 28 22 1 2 11 4 20 12 26 22 1
Y 1 15 7 20 18 28 24 2 1 15 6 21 18 27 23 2
G 29 19 7 20 20 28 25 2 29 13 6 20 20 28 23 2
Table 8
 
Chromaticity class robustness index averaged across observers. Rows correspond to chromaticity classes and columns to illuminations.
Table 8
 
Chromaticity class robustness index averaged across observers. Rows correspond to chromaticity classes and columns to illuminations.
B R Y G Mean
C 1 (B) 0.88 0.76 0.89 0.87 0.85
C 2 (R) 0.89 0.85 0.97 0.93 0.91
C 3 (Y) 0.93 0.73 0.98 0.84 0.87
C 4 (G) 0.89 0.73 0.89 0.87 0.84
Mean 0.90 0.77 0.93 0.88 0.87
Table 9
 
Chromaticity class robustness index averaged across illuminations. Rows correspond to chromaticity classes and columns to observers.
Table 9
 
Chromaticity class robustness index averaged across illuminations. Rows correspond to chromaticity classes and columns to observers.
AB CG DW MT Mean
C 1 (B) 0.85 0.95 0.82 0.77 0.85
C 2 (R) 0.94 0.94 0.94 0.82 0.91
C 3 (Y) 0.86 0.95 0.90 0.78 0.87
C 4 (G) 0.89 0.93 0.80 0.74 0.84
Mean 0.87 0.94 0.87 0.78 0.87
Table 10
 
Generalized chromaticity class robustness index.
Table 10
 
Generalized chromaticity class robustness index.
AB CG DW MT Mean
C 1 (B) 0.68 0.90 0.57 0.53 0.67
C 2 (R) 0.82 0.82 0.76 0.74 0.79
C 3 (Y) 0.69 0.82 0.64 0.41 0.64
C 4 (G) 0.73 0.75 0.64 0.67 0.70
Mean 0.73 0.82 0.65 0.59 0.70
Table 11
 
Unitary color shift averaged across observer.
Table 11
 
Unitary color shift averaged across observer.
Illuminant UC1 (B) UC2 (R) UC3 (Y) UC4 (G) Mean
B 0.25 1.13 1.25 0.38 0.75
R 2.13 1.50 1.13 2.38 1.78
Y 0.50 1.13 0.50 0.00 0.53
G 0.50 0.50 1.00 1.00 0.75
Mean 0.84 1.06 0.97 0.94 0.95
Table 12
 
Unitary color shifts averaged across unitary color type.
Table 12
 
Unitary color shifts averaged across unitary color type.
Illuminant AB CG DW MT Mean
B 0.38 0.25 0.88 1.50 0.75
R 1.88 1.00 1.38 2.88 1.78
Y 0.50 0.00 0.63 1.00 0.53
G 0.75 0.50 1.00 0.75 0.75
Mean 0.88 0.44 0.97 1.53 0.95
Table 13
 
Proportions of unitary colors that undergone the unitary color shift not more than zero, half, one, one and a half, and two Munsell hue steps.
Table 13
 
Proportions of unitary colors that undergone the unitary color shift not more than zero, half, one, one and a half, and two Munsell hue steps.
Unitary color shift
0 ≤0.5 ≤1 ≤1.5 ≤2
28% 59% 64% 80% 91%
Table 14
 
Chromaticity class ends derived for the four chromatic illuminations (B, R, Y, and G) under the asymmetric illumination conditions. Each entry is the Munsell chip number according to the numeration given in Table 2.
Table 14
 
Chromaticity class ends derived for the four chromatic illuminations (B, R, Y, and G) under the asymmetric illumination conditions. Each entry is the Munsell chip number according to the numeration given in Table 2.
Observer AB Observer CG
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 31 15 7 23 19 29 24 4 30 18 7 23 18 28 24 5
R 30 12 6 21 17 28 24 3 30 16 4 21 18 28 24 3
Y 30 15 7 22 18 28 24 4 30 16 7 22 18 28 24 5
G 30 15 6 22 18 28 24 4 30 17 7 22 18 28 24 6
Observer DW Observer MT
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 2 15 6 20 19 28 24 2 1 15 4 22 20 28 24 1
R 29 15 7 19 16 28 24 1 2 14 4 20 12 28 24 1
Y 2 15 7 20 18 28 24 4 2 15 6 20 18 27 24 1
G 2 14 6 20 19 28 24 4 2 12 4 20 19 28 21 2
Table 15
 
Chromaticity class ends obtained for the achromatic illuminant under the asymmetric illumination conditions. Each entry is the Munsell chip number according to the numeration given in Table 2. Note that the difference between the rows is produced by the chromaticity (B, R, Y, or G) of the illumination of the adjacent stand. The illumination of the chips for which the chromaticity classes are obtained was neutral.
Table 15
 
Chromaticity class ends obtained for the achromatic illuminant under the asymmetric illumination conditions. Each entry is the Munsell chip number according to the numeration given in Table 2. Note that the difference between the rows is produced by the chromaticity (B, R, Y, or G) of the illumination of the adjacent stand. The illumination of the chips for which the chromaticity classes are obtained was neutral.
Observer AB Observer CG
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 30 15 8 22 18 28 25 4 30 16 7 22 18 28 24 5
R 30 15 8 22 18 28 23 4 30 16 7 22 18 28 23 5
Y 30 15 8 22 18 28 24 4 30 16 7 22 18 28 24 5
G 30 15 8 22 18 28 25 4 30 16 7 22 18 28 24 5
Observer DW Observer MT
C 1 (B) C 2 (R) C 3 (Y) C 4 (G) C 1 (B) C 2 (R) C 3 (Y) C 4 (G)
i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2 i 1 1 i 1 2 i 2 1 i 2 2 i 3 1 i 3 2 i 4 1 i 4 2
B 2 15 7 20 18 28 25 2 1 15 6 20 18 27 25 2
R 1 15 8 20 18 28 23 2 1 15 7 20 18 26 23 1
Y 2 15 7 20 18 28 25 3 1 15 6 20 18 27 25 2
G 1 15 8 22 18 28 24 5 31 15 7 22 18 28 23 2
Table 16
 
Chromaticity class shift in percentage from symmetric to asymmetric condition (see text).
Table 16
 
Chromaticity class shift in percentage from symmetric to asymmetric condition (see text).
Munsell hue step rate Achromatic illumination Chromatic illumination
Zero-step rate 64.06 62.50
One-step rate 18.75 23.44
Two-step rate 10.94 9.38
More-than-two-step rate 6.25 4.69
Table A1
 
Matrix of conditions resulting in the positive response to the stimulus pair (μ i , μ j ). Each row should be taken as a logical statement. The four inequalities in each row are supposed to be combined by the logical AND operator.
Table A1
 
Matrix of conditions resulting in the positive response to the stimulus pair (μ i , μ j ). Each row should be taken as a logical statement. The four inequalities in each row are supposed to be combined by the logical AND operator.
λ Ck 1 < λ Ck 2 λ Ak 1 < λ Ak 2 λ Ck 1 < μ i < λ Ck 2 λ Ak 1 < μ j < λ Ak 2
λ Ck 1 < λ Ck 2 λ Ak 1λ Ak 2 λ Ck 1 < μ i < λ Ck 2 μ j λ Ak 2 OR μ j λ Ak 1
λ Ck 1λ Ck 2 λ Ak 1 < λ Ak 2 μ i λ Ck 2 OR μ i λ Ck 1 λ Ak 1 < μ j < λ Ak 2
λ Ck 1λ Ck 2 λ Ak 1λ Ak 2 μ i λ Ck 2 μ j λ Ak 2 OR μ j λ Ak 1
λ Ck 1λ Ck 2 λ Ak 1λ Ak 2 μ i λ Ck 1 μ j λ Ak 2 OR μ j λ Ak 1
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