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.

^{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.

*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.

*material*color constancy” (Logvinenko, 2012a; Logvinenko & Tokunaga, 2011).

^{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?

*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.

Unique hues | ||||
---|---|---|---|---|

UH_{1} | UH_{2} | UH_{3} | UH_{4} | |

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 |

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 |

*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.

*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

*i*th and

*j*th 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).

*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

*k*th 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.

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 |

*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*.

*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.

UH_{1} | UH_{2} | UH_{3} | UH_{4} | |
---|---|---|---|---|

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 |

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 |

^{2}).

Illuminant | x | y |
---|---|---|

Blue | 0.131 | 0.150 |

Red | 0.635 | 0.321 |

Yellow | 0.392 | 0.410 |

Green | 0.224 | 0.667 |

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 |

*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*).

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 |

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 |

*generalized robustness index*for the

*j*th chromaticity class (

*j*= 1, …, 4) as

*I*

_{ i }is the

*i*th illuminant (thus,

*C*

_{ j }(

*I*

_{ i }) stands for the

*j*th chromaticity class registered under the

*i*th illuminant);

*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.

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 |

*unitary color shift*.

Illuminant | UC_{1} (B) | UC_{2} (R) | UC_{3} (Y) | UC_{4} (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 |

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 |

Unitary color shift | ||||
---|---|---|---|---|

0 | ≤0.5 | ≤1 | ≤1.5 | ≤2 |

28% | 59% | 64% | 80% | 91% |

*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.

*i*th chip under the red illumination and the

*j*th chip under the neutral illumination can, generally, differ from the observer response to the

*j*th chip under the red illumination and the

*i*th 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.

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 |

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 |

*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.

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 |

*any*alteration in illuminant.

^{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

*k*th chromaticity class, and

*λ*

_{ k }

^{2}is the right end of the

*k*th chromaticity class. In other words, a paper

*μ*

_{ i }belongs to the

*k*th chromaticity class if

*μ*

_{ i }lies between

*λ*

_{ k }

^{1}and

*λ*

_{ k }

^{2}. More formally, a paper

*μ*

_{ i }belongs to the

*k*th chromaticity class if (i) when

*λ*

_{ k }

^{1}≤

*λ*

_{ k }

^{2}, one has

*λ*

_{ k }

^{1}≤

*μ*

_{ 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}.

*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

*λ*

_{ k }

^{1}≤

*λ*

_{ k }

^{2}

*λ*

_{ i }

^{1}>

*λ*

_{ i }

^{2},

*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.

*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”)

*R*(

*μ*

_{ i },

*μ*

_{ j })}.

*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*

_{ 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.

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 } ^{1} ≥ λ _{ Ak } ^{2} | λ _{ Ck } ^{1} < μ _{ i } < λ _{ Ck } ^{2} | μ _{ j } ≤ λ _{ Ak } ^{2} OR μ _{ j } ≥ λ _{ Ak } ^{1} |

λ _{ Ck } ^{1} ≥ λ _{ 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 } ^{1} ≥ λ _{ Ck } ^{2} | λ _{ Ak } ^{1} ≥ λ _{ Ak } ^{2} | μ _{ i } ≤ λ _{ Ck } ^{2} | μ _{ j } ≤ λ _{ Ak } ^{2} OR μ _{ j } ≥ λ _{ Ak } ^{1} |

λ _{ Ck } ^{1} ≥ λ _{ Ck } ^{2} | λ _{ Ak } ^{1} ≥ λ _{ Ak } ^{2} | μ _{ i } ≥ λ _{ Ck } ^{1} | μ _{ j } ≤ λ _{ Ak } ^{2} OR μ _{ j } ≥ λ _{ Ak } ^{1} |

*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: