Color constancy of color-deficient observers under illuminations defined by individual color discrimination ellipsoids

Ruiqing Ma; Ken-ichiro Kawamoto; Keizo Shinomori

doi:10.1364/JOSAA.33.00A283

1. INTRODUCTION

Color constancy is an important mechanism in the human color vision system. It helps us to perceive the color of a surface consistently under different illumination conditions. For contiguous color constancy, there are a number of possible mechanisms that humans might adopt to reduce the influence of illumination color on objects’ surfaces. Numerous studies have investigated color constancy (for a review, see Foster [1]).

It is commonly accepted that the adaptation of photoreceptors in the retina plays a critical role in the color constancy of color-normal observers. The mechanism underlying the von Kries model is the prominent cone adaptation model, in which the sensitivities of three classes of photoreceptors can be suppressed independently by constant values describing their adaptation state [2]. This von Kries principle was supported by the results of a variety of color constancy experiments [3–10]. It has been reported in previous literature that the simple von Kries model can describe matched data reasonably, and very well in most cases [3–10]. In fact, the long duration and full-field adaptation of photoreceptors to illuminations enabled the achievement of almost complete color constancy [8,9], although L- and M-cones tended to adapt so as to support color constancy, whereas S-cones were strongly influenced by illumination changes [10]. This simple application of the von Kries model has been recognized as a mathematical expression of the independent adaptation and/or gain control of photoreceptors describing the cone responses in the color constancy task, although such adaptation and gain control do not necessarily occur only at the receptor level (they most likely also occur at post-receptoral levels, as do other adaptations [11]).

The opponent color mechanisms in color constancy were also investigated: the simple von Kries principle performed worse for the opponent color signals than for the original photoreceptoral cones [7]. Red-green and blue-yellow dimensions respond differently to illuminant changes [12]: red-green dimensions undergo an additive change, while blue-yellow dimensions undergo a multiplicative change. Illumination changes affect opponent color mechanisms, both red-green and S-cone related mechanisms, in different ways [10]. For the above reasons, we treat the von Kries adaptation principle as a concept in which the adaptation stage is not necessarily limited to the photoreceptoral stage, but can be in the post-receptoral opponent color stage.

Color constancy must also be considered in the case of red-green color-deficient people who are characterized as having a loss of the red-green color discrimination ability due to the lack of or mutation of the L- and M-cone photopigments. Several studies have investigated color constancy in red-green color-deficient observers [13–18]. As for the underlying mechanism mediating color constancy in red-green color deficiency, it has been suggested that the adaptation of the cone photoreceptors could be a significant factor [13], that the relatively intact blue-yellow subsystem of color vision might be the basis for color constancy [14], and that the utilization of cone-excitation ratios in the remaining cone class could be a contributing factor [19].

The main purpose of this study was to investigate to what extent color constancy depends on the mechanism underlying the statistical operation of the scene [20,21] and on the illuminant-by-illuminant estimation strategy [22,23], which depends on the appearance of the scene. The contribution of these complex mechanisms will be compared to that of the von Kries adaptation mechanism, which uses simple adaptation and/or gain control. We expected that if red-green color-deficient observers had difficulty recognizing the red and green illuminations prepared in this study, the complex mechanism would have little effect on color constancy and there would be some significant differences in color constancy performance (as measured using the paper match task) between the performance of color-deficient observers and that of color-normal observers. For that purpose, we needed to determine whether or not the simple von Kries adaptation model could provide a satisfactory description of the response of the remaining photoreceptoral cones and blue-yellow opponent color signals in the color constancy of red-green color-deficient observers. The question was explored using the simple assumption that color-deficient observers have the same blue-yellow color opponent mechanism as color-normal observers, except for the absence of a certain type of cones: L- and S-cones in deuteranopes, M- and S-cones in protanopes, and L-, S-, and varied (hybrid) M-cones ( $M^{'}$ -cone) in deuteranomalous observers. The investigation was performed under two different illumination change conditions: first, under red and green illuminations, with the illumination changes along the longer axis of the individually obtained discrimination ellipsoid, which cannot be discriminated by color-deficient observers; second, with illumination changes from D65 illumination to blue or yellow illumination, which can be discriminated by color-deficient observers.

2. METHODS

A. Observers

Six color-deficient observers (three protanopes, one deuteranope, and two deuteranomalous observers, all male) from 19 to 25 years old (mean, 21.2) and five color-normal observers (3 female, 2 male) from 19 to 25 years old (mean, 23.4) participated in the experiments. Although three of the color-deficient observers were students in the Faculty of Health Science and Technology and had basic knowledge about color deficiency, all observers were naïve regarding color constancy and the purposes of the experiments; the authors did not serve as observers. All observers had normal or corrected-to-normal acuity, with the best-corrected visual acuity better than 0.6 (1.67 min. of visual angle). The color vision of observers was tested by a set of color vision tests: Ishihara color test plates (International 38 plates edition), the Farnsworth D-15 test, Standard Pseudo-Isochromatic Plates, the Farnsworth-Munsell 100 hue test, the Cambridge Colour Test (CCT) [24,25]. We used the Neitz OT anomaloscope (Model: OT (one lamp model), Neitz co. Ltd.) to classify color-deficient observers. Three protanopes (KMY, OTK, and PB) and one deuteranope (KBY) could make anomaloscope matches over the full range of red/green (R/G) settings, indicating that they were dichromats; one deuteranomalous observer (OK) had R/G settings ranging from 8.5 to 16 when $Y$ settings was in the range from 14.5 to 15 (classified as deuteranomaly). The other deuteranomalous observer (LX) had R/G settings ranging from 0 to 45 when $Y$ settings was in the range from 13.5 to 14 (classified as severe deuteranomaly), although both observers were classified as severe deuteranopes by other tests.

In the calculations for the test stimulus and data analysis of this study, we assumed that the protanopes had M- and S-cones, which are the same in spectral sensitivity as those of color-normal observers, and that the deuteranope had L- and S-cones. We used Smith–Pokorny’s cone fundamentals [26] and the CIE 1931 color-matching function [27] with the cone matrix by Kaiser and Boynton [28]. In addition, we assumed that the deuteranomalous observers had the hybrid cone of the fusion gene $M^{'}$ [29], rather than the M gene found in color-normal observers.

The procedures and experiments described here conform to the principles expressed in the Declaration of Helsinki and were approved by the Kochi University of Technology Research Ethics Committee. Written informed consent was obtained from each observer prior to testing.

B. Apparatus and Calibration

The stimuli were presented on a 19-inch CRT color monitor (CPD-G220, Sony Inc.) at a $1024 \times 768$ resolution at 120 Hz frame rates using a special graphics card for vision experiments (ViSaGe, Cambridge Research Systems (CRS), Inc.), which provides a 14-bit resolution for each red, green, and blue (RGB) phosphor in a Dell PC. Gamma correction of the monitor was carried out using the dedicated light measurement instrument (ColorCAL, CRS) of the graphics card controlled by calibration software (VSG desktop, CRS). The six-button response box (CB6, CRS) of the card was used to input the observer responses. A black paper board ( $90 cm \times 60 cm$ ) was placed in front of the observer to create a haploscopic presentation in which the left half of the screen was viewed by the left eye and the right half was viewed by the right eye. The viewing distance was 90 cm. The CCT was performed with the same apparatus and using the same calibration data as in the main experiments, except the black paper board was not used and the observers used both eyes; here, a four-button response box (CT6, CRS) was used. All experiments were performed in a darkened room where the only illumination was from the CRT monitor used for the main experiments and the CCT.

The chromaticity coordinates and luminance of all colors in the stimuli, measured by a colorimeter (CS-200, Konica-Minolta, Inc.) and a spectral radiometer (CS-1000, Konica-Minolta, Inc.), confirmed that the error of the screen presentation was less than 3% in the CIE 1931 $x y$ chromaticity coordinates and less than 5% in the luminance.

C. Stimulus

A 5 deg square standard pattern virtually illuminated by standard illumination, D65 (6500 K daylight), and a 5 deg square test-pattern virtually illuminated by red, green, blue, and yellow test illuminations were presented side by side, separated by a 1 deg, completely black space made with a black paper board covering the entire monitor screen except for those two patterns. Paired patterns had an identical spatial arrangement: a 1 deg rectangular color patch at the center was surrounded by background ellipses. The background of the patterns was composed of 230 superimposed ellipses painted one of eight colors (see below). Each ellipse had a random position and orientation with a random size change between 0.8 and 1.2 deg along the shorter axis and between 1.6 and 2.0 deg along the longer axis. Figure 1 shows an example of the test stimulus for the red test illumination condition.

Fig. 1. Example of test stimulus for red illumination condition. The standard pattern under D65 illumination (left) and the test pattern under colored illumination (right) were presented haploscopically in each trial. The left and right locations of patterns were changed from session to session.

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All colors used in the stimuli were simulations of Munsell matte color surfaces. The colors were calculated using spectral reflectance from the Munsell Book of Color [30,31] and the spectral radiance of the illumination. The central colored patches were selected so as to have the middle Value and Chroma, that is 5 and 6, and the hue covering the color region. At the same time, the simulated color under different illuminations was not allowed to exceed the available color gamut of the monitor. Consequently, the spectral reflectance of the central colored patch was selected from among 12 Munsell surfaces: Munsell 5R5/6, 2.5YR5/6, 10YR5/6, 7.5Y5/6, 5GY5/6, 2.5G5/6, 10B5/6, 7.5PB5/6, 5P5/6, 2.5RP5/6, 10RP5/6, and 20% flat-reflectance surface (neutral color) as an achromatic reference. Twelve colored patches were presented in a pseudo-random sequence. The CIE 1976 $u^{'} v^{'}$ chromaticity coordinates of the 12 Munsell color surfaces under D65 illumination are shown in Fig. 2.

Fig. 2. CIE 1976 $u^{'} v^{'}$ chromaticity coordinates of the twelve central colored patches, illuminated by D65 illumination. The label denotes the code in the Munsell color system. The Value and Chroma were 5 and 6.

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The reflectance spectra in the background ellipses were composed of eight Munsell surfaces, taken from the Munsell Book of Color with Value 5 and Chroma 6, yielding an angle distance of approximately 45 deg in the hue circle of the Munsell Color System. It is expected that the background consisting of these eight surfaces caused no or little bias in the color constancy, because it has been known that a chromatic bias of the background surfaces has a small effect on the color constancy performance, while the illuminant change has a large effect [6,32,33]. Considering that the color of the background surfaces cannot be the same as that of the central patches, in case observers might refer to the background color in the adjustment of the central patch, the Chroma of the background color was changed at random to either 4 or 8 if the randomly selected hue coincided with one of the twelve surfaces used in the central colored patch. The eight surfaces forming one background pattern, different for each of the six sessions, were obtained by rotating the Munsell hue circle clockwise.

Red and green illuminations were arranged so as to have a special effect on dichromats and anomalous trichromats; chromatic changes from the standard illumination (D65) to red and green illuminations were set on the longer axis of the color discrimination ellipsoid, measured individually by means of the CCT at $u^{'} v^{'} = 0.198, 0.464$ with 20 test vectors used for the ellipse fit [25]. For the protanopic observers, the red and green test illuminations were obtained by increments (expressed as red illumination, because the illumination appears reddish for color-normal observers) or decrements (as green illumination) of a L-cone stimulation of 5% from D65. For the deuteranopic and deuteranomalous observers, the red and green test illuminations were obtained by increments (green) or decrements (red) of a M-cone stimulation of 10% from D65. For the color-normal observers, red and green illuminations were obtained from the standard deutan confusion line ( $x_{d} = 1.4000$ , $y_{d} = - 0.4000$ ) passing through D65. All observers used the same blue and yellow illuminations, which are close to the short axis of color discrimination ellipsoids of the observers. We did not use the individual shorter axis because the change of illumination color from white (D65) to blue or yellow was apparent to all observers and the usage of the common illuminations was accurate enough in this study, in which a red-green color control was not used in the matching of color-deficient observers (see the next section for details). The locations of all test illuminations relative to the individual discrimination ellipsoids in the CIE 1976 $u^{'} v^{'}$ diagram are shown in Fig. 3. The corresponding $u^{'} v^{'}$ chromaticity coordinates and cone stimulations of all test illuminations are summarized in Table 1. Table 2 shows the mean contrast of L-, M-, and S-cones between one of chromatic illumination and D65 illumination on the 20% flat-reflectance surface. The contrasts in Table 2 were the mean between individually adjusted illuminations in Table 1. Regardless of the direction difference between the longer axes of the color-deficient observers and the standard confusion lines used for the color-normal observers, the cone contrasts calculated from the same cone sensitivities were close between observers’ color vision type, suggesting that the assumption of the cone sensitivities in color-deficient observers is reasonable.

Fig. 3. Location of test illuminations compared to individual discrimination ellipsoids (denoted by red lines for protans and green lines for deutan and deuteranomalous observers) in the CIE 1976 $u^{'} v^{'}$ chromaticity diagram. Open squares (protanopes) and circles (deuteranope and deuteranomalous observers) denote the corresponding illuminations obtained individually. Red and green filled circles denote the illuminations for color-normal observers. The black circle denotes D65 illumination. Blue and yellow illuminations are denoted by open triangles. Symbol colors denote the color of illuminations. Black lines denote the longer axis of discrimination ellipsoids for color-deficient observers and the standard deutan line for color-normal observers.

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Table 1. CIE1976 u $^{'}$ v $^{'}$ Chromaticity Coordinates and Cone Stimulations of All Illuminations^a^,^b

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Table 2. Mean Contrast of Cones and Luminance between Colored and D65 Illuminations on 20% Flat-Reflectance Surface^a^,^b^,^c^,^d^,^e

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The intensity of each illumination was adjusted to achieve the same luminance on the 20% flat-reflectance surface illuminated to be $25 cd / m^{2}$ . Table 3 shows the maximum, minimum, and average luminance ( $cd / m^{2}$ ) of the 48 Munsell surfaces of all background ellipsoids for six sessions (8 Munsell surfaces for each session) rendered under all illuminations for color-normal observers. The luminance varied over a relatively narrow range. It was necessary to assume, however, that the luminance for color-deficient observers should depend on the assumed spectral radiance and the spectral radiance of the stimulus. In this study, for the luminance of the color-deficient observers, we assumed that the luminous efficiency functions of the protanopes and deuteranope are the same as those for color-normal observers’ M- and L-cone sensitivities; the function of the deuteranomalous observer is the sum of the L- and $M^{'}$ -cones [29]. The contrasts of luminance between one of chromatic illumination and D65 illumination are in Table 2.

Table 3. Luminance ( $cd / m^{2}$ ) of Background under All Illuminations for Color-Normal Observers

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The spectral radiances of the test illuminations were constructed as linear combinations of daylight spectral basis functions [34]. The chromaticity coordinates of the Munsell surfaces under D65 and the test illuminations were calculated using CIE 1931 standard color matching functions [27]. Spectra were sampled at 5 nm intervals and integrated from 380 to 780 nm.

D. Procedure

In the haploscopic matching of the central colored patches, the observer used the six-button response box to adjust the chromaticity and intensity of the test patch in the test pattern under the virtual colored illumination. The functions of the six buttons for all sessions for the color-normal observers were as follows: two buttons to control blue-yellow colors, two buttons to control red-green colors, and two buttons to control intensity (bright-dark). Chromaticity was controlled in the CIE 1931 chromaticity diagram base and the intensity was controlled in the luminance base. The red-green and blue-yellow color changes were set in the monitor’s phosphor base; red-green control was accomplished by increasing and decreasing the intensity of R and G phosphors to maintain the luminance set by the bright-dark buttons. Blue-yellow control was achieved by varying the intensity of yellow (equal digits of R and G phosphors) and the B phosphor. This setting enabled the setting of the possible chromatic change to the maximum and made color control by observers as easy as possible, since colors were changed to the most saturated RGYB colors on the monitor. The relative intensities of the RGB phosphors were set so as to make the monitor white $x, y = 0.333, 0.333$ .

The color-deficient observers were all asked to complete one session with red-green color control and to decide whether the method with red-green color control was better or worse than that without it. All of the color-deficient observers reported that the usage of red-green color control buttons made adjustments more complicated, and that they could get any color without the red-green buttons. Thus, color-deficient observers here completed sessions using only the blue-yellow buttons and intensity buttons to adjust the color and luminance. This indicates that for the color-deficient observers, the matched point was always on the blue-yellow adjustment line connecting yellow-white (the monitor white) to blue (B phosphor) in the CIE 1931 chromaticity diagram. Possible effects of this binding condition will be discussed in section 4.C.

Before the first session, the observers were given 3 min to practice controlling the chromaticity and intensity with the six-button response box. At the beginning of the session, the observers adapted to the D65, $25 cd / m^{2}$ white-screen for 5 min with both eyes. Before starting the trials under each illumination condition, the observers adapted to the backgrounds for 5 min, with no central patch under the colored illumination for one eye and under the D65 illumination in the other eye. The observers viewed the stimuli haploscopically and were instructed to adjust the color and brightness of the central colored patch in the test pattern so as to make the test patch look “as if it were cut from the same piece of paper as the corresponding patch in the standard pattern” [35,36]. In each trial, the starting chromaticity of the test patch was set as $x, y = 0.333, 0.333$ . Each session contained five illumination conditions (D65; the red and green illuminations (or blue and yellow illuminations) as shown in Fig. 3, and desaturated red-green illuminations (or desaturated blue and yellow illuminations), which are not shown in this paper). All trials of one illumination condition were performed in pseudo-random order and the illumination would be changed for the adaptation; there were 12 trials for each illumination condition, corresponding to the 12 different central colored patches. Each matching datum was averaged over 6 sessions. In 3 sessions, the left side was a standard pattern illuminated by D65, and the right side was a test pattern illuminated by a test illumination; the reverse arrangement was used in the other 3 sessions. Each session took about 90 min. On average, the stimulus presentation time for each trial was about 1.5 min.

3. RESULTS

A. Color Constancy Performance

For color-normal observers, the color constancy index proposed by Arend et al. [36] was used to quantitatively evaluate the degree of color constancy. The index, $I$ , is defined as

I = 1 - b / a,

where

b

denotes the Euclidean distance from the matched point to the theoretical point (the chromaticity coordinates of the patch under test illumination) at which the color constancy would be perfect in the sense of predicting the color appearance of the patch under the test illumination.

a

denotes the Euclidean distance from the standard point (the chromaticity coordinates of the patch under D65 illumination) to the theoretical point. In this study, the distance was defined in two-dimensional CIE 1976

u^{'} v^{'}

color space. An index value of 1 indicates perfect color constancy (the matched point would coincide with the theoretical point); an index value of 0 indicates no color constancy, meaning observers made the appearance match. In the appearance match, the observers neglected the background surrounding the test patch and made the hue, saturation, and brightness of the test patch exactly the same as those of the standard patch [35,36].

Figure 4 shows the constancy indices of the color-normal observers on twelve color patches under four test illuminations. Under yellow illumination, the color-normal observers even showed negative indices on patches 7.5PB 5/6 and 10B 5/6, suggesting that the color-normal observers could not perform the paper match well under yellow illumination. The averaged indices over twelve color patches were 0.55, 0.34, 0.35, and 0.12 for the red, green, blue, and yellow illuminations, respectively.

Fig. 4. Constancy indices of color-normal observers on twelve color patches under the red, green, blue, and yellow illumination conditions. The value for each color patch was averaged over five color-normal observers (N). The error bars represent the standard error of the mean.

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As described in the procedure section, since the color-deficient observers controlled the color only on the blue-yellow adjustment line without using the red-green buttons, the matched point should be on the blue-yellow line. We modified Arend et al.’s index calculation by projecting the theoretical points and standard points onto the blue-yellow line, so the constancy indices were calculated in the blue-yellow dimension to evaluate the color constancy performance of the color-deficient observers. The projection of the theoretical points and standard points onto the blue-yellow line was performed in the additional appearance matching experiment in which the standard pattern and test pattern were under the same illumination.

Before calculating the indices on the blue-yellow line, however, it was necessary to address the question of whether or not the illumination change from D65 illumination to a test illumination induced enough color change for each color patch. If the distance between the standard and matched points was less than the color discrimination range of an observer, it would be expected that these two colors would be perceived as the same and that the observer would attempt appearance matching instead of the expected paper match. Thus, the distance between the standard point and the matched point on the blue-yellow line for each patch was compared with the color discrimination range shown in Fig. 5. The color discrimination range of the color-deficient observers was determined using the CCT with three color discrimination ellipsoids obtained from three starting points, $x, y = 0.313$ ,0.329; 0.346,0.407; and 0.280,0.253, in the CIE 1931 chromaticity coordinates. The color discrimination range of each observer was set as the mean of the lengths, which was half the distance between the intersections of the blue-yellow line and the color discrimination ellipsoids. The overall color discrimination range (denoted by the black bars in Fig. 5) of 6 color-deficient observers was averaged. The means of half the length of the short and long axes in three color discrimination ellipsoids, averaged over all color-deficient observers, are also shown for reference (denoted by two gray bars in each panel).

Fig. 5. Distance on blue-yellow adjustment line between the standard point illuminated by D65 and the average of the matched points illuminated by test illumination (red, green, blue, and yellow illuminations from top to bottom) for color-deficient observers. Black and gray bars denote the color discrimination range as the mean of the ranges of all color-deficient observers (see text for details).

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It could be assumed that the distance on the blue-yellow line within the color discrimination range has the same color appearance obtained by the appearance match. It can be seen that the distance on a considerable number of color patches was smaller than the color discrimination range, meaning that the illumination changes brought out small color changes for color-deficient observers, and that it was adequate to perform an appearance match. Although the adjustment of the luminance was not reflected in the data, as shown in Fig. 5, the color-deficient observers mainly performed brightness matching to achieve the appearance match, whereas the color-normal observers showed a typical color constancy performance, as shown in Fig. 4.

Considering that, we calculated Arend et al.’s color constancy index. In some colors, the index values were unexpectedly high or less than zero (minus value). In almost all cases, the unexpected values were obtained from color patches in which the distance was within the color discrimination range. After we removed the data of such color patches, however, the index data of the remaining color patches did not indicate an apparent tendency relating to illuminations and/or type of color deficiency. It can be explained by the lack of accuracy in the additional matching data for the calculation, which were obtained by the additional appearance matching experiment for the projection performed between colors that had such small color differences. Overall, although the Arend et al.’s color constancy index is useful to express strength of the color constancy in color-normal observers, the indices could not be compared between the color-normal and color-deficient observers because of the difficulty to obtain them in the color-deficient observers. The data analysis in the color-deficient observers has required using cone responses.

B. Von Kries Model Prediction at the Photoreceptoral Stage

The simple application of the von Kries model [2] assumes that the sensitivity of each cone type would be reduced by adaptation to an illumination. Reduction under the model is treated as a linear and an independent effect expressed by changes in the coefficient values, which can be used separately to multiply the sensitivity of each cone type, meaning that the influence of each object’s surface is not included, in principle. For one patch, the post-adapted L-, M-, and S-cone signals should be the same under the D65 and test illumination; thus, the following equation should hold:

(\begin{matrix} L_{post-adapted} \\ M_{post-adapted} \\ S_{post-adapted} \end{matrix}) = (\begin{matrix} k_{L, T} & 0.0 & 0.0 \\ 0.0 & k_{M, T} & 0.0 \\ 0.0 & 0.0 & k_{S, T} \end{matrix}) \cdot (\begin{matrix} L_{T} \\ M_{T} \\ S_{T} \end{matrix}) = (\begin{matrix} k_{L, D 65} & 0.0 & 0.0 \\ 0.0 & k_{M, D 65} & 0.0 \\ 0.0 & 0.0 & k_{S, D 65} \end{matrix}) \cdot (\begin{matrix} L_{D 65} \\ M_{D 65} \\ S_{D 65} \end{matrix}) .

The responses of the post-adapted cones,

L_{post-adapted}

,

M_{post-adapted}

, and

S_{post-adapted}

, which are the same regardless of the illumination in the von Kries model, can be obtained independently by the constants

k_{L}

,

k_{M}

, and

k_{S}

.

L_{D 65}

,

M_{D 65}

, and

S_{D 65}

are the cone responses of the surface rendered under D65 illumination;

L_{T}

,

M_{T}

, and

S_{T}

are the cone responses for that surface rendered under the test illumination. The constants can be defined as the inverse of the L-, M-, and S-cone responses for a perfect white patch under the D65 illuminant and test illumination, meaning the illumination determines the constants multiplied in each of the three kinds of cones. Thus,

{\begin{cases} k_{L, T} = 1 / L_{W, T} \\ k_{M, T} = 1 / M_{W, T} \\ k_{S, T} = 1 / S_{W, T} \end{cases},

{\begin{cases} k_{L, D 65} = 1 / L_{W, D 65} \\ k_{M, D 65} = 1 / M_{W, D 65} \\ k_{S, D 65} = 1 / S_{W, D 65} \end{cases} .

L_{W, D 65}

and

L_{W, T}

denote the L-cone responses excited by a perfect white patch (100% reflectance in all wavelengths) illuminated by the D65 and test illumination, respectively.

M_{W, D 65}

,

M_{W, T}

,

S_{W, D 65}

, and

S_{W, T}

denote the same except the M- and S-cone responses, respectively.

From Eqs. (2)–(4), the cone responses of the theoretical colors under the von Kries model can be predicted from the cone responses under the standard D65 illumination, as follows:

(\begin{matrix} L_{T} \\ M_{T} \\ S_{T} \end{matrix}) = (\begin{matrix} k_{L, D 65} / k_{L, T} & 0.0 & 0.0 \\ 0.0 & k_{M, D 65} / k_{M, T} & 0.0 \\ 0.0 & 0.0 & k_{S, D 65} / k_{S, T} \end{matrix}) \cdot (\begin{matrix} L_{D 65} \\ M_{D 65} \\ S_{D 65} \end{matrix}),

{\begin{cases} k_{L, D 65} / k_{L, T} = L_{W, T} / L_{W, D 65} (= k_{L, trans}), \\ k_{M, D 65} / k_{M, T} = M_{W, T} / M_{W, D 65} (= k_{M, trans}), \\ k_{S, D 65} / k_{S, T} = S_{W, T} / S_{W, D 65} (= k_{S, trans}) . \end{cases}

Coefficients

k_{L, trans}

,

k_{M, trans}

, and

k_{S, trans}

for all test illuminations are shown in Table 4. Since the chromaticity coordinates of the three protanopes and the three deuteranopic and deuteranomalous observers were almost identical in each observer group, as shown in Table 1, the coefficients were based on the average value of the illuminants for each group. In the model calculation, we used Smith–Pokorny’s cone fundamentals [26] and the CIE 1931 color-matching function [27], meaning these coefficients were obtained for the CIE standard observer but not for the color-normal observers in this study. For deuteranomalous observers, the

M^{'}

-cone sensitivity was used instead of the standard M-cone [29]. As expected, the red and green illuminations produced relatively large cone stimulation changes in the L- and M-cones. The cone stimulation changes in the S-cone were not negligible; in fact, the changes were relatively large between observer groups even for the same color of illumination. On the other hand, the blue and yellow illuminations produced cone stimulation changes almost entirely in the S-cone. It is obvious that the

k

values reflect the amount of cone excitation change from the standard D65 illumination to the test illuminations, as can be seen in Table 2.

Table 4. Coefficients in the Von Kries Model^a^,^b

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We applied this von Kries model with a simple adaption and/or gain control to the cone responses to describe the color constancy performance by the color-deficient observers. In this study, the dichromatic color vision system is treated as a reduced form of the trichromatic color vision system: neural information from two kinds of photopigments in the first stage (L- and S-cones for deutan, M- and S-cones for protan) was combined to yield a blue-yellow opponent chromatic pathway mediated by S-L or S-M and a luminance pathway mediated by the L- or M-cone. Deuteranomalous observers have a trichromatic color vision system, but with the varied M-cone. Because the adaptation (gain control) with the von Kries model initially has no interaction between different types of cones, to calculate the cone responses predicted by the von Kries model, we simply applied the constants in Table 4: $k_{M, trans}$ and $k_{S, trans}$ for the protanopes, $k_{L, trans}$ and $k_{S, trans}$ for the deuteranope, and $k_{L, trans}$ , $k_{M, trans}$ , and $k_{S, trans}$ for the deuteranomalous observers.

Figure 6 shows the comparison of L- and M-cone responses on twelve color patches matched by the color-normal observers and those predicted by the von Kries model under four test illumination conditions. If the matched cone responses in the color constancy task could be perfectly predicted by the von Kries model, the data points would be on a diagonal line; if the observers made the appearance match, indicating no adaptation, the data points would be located exactly on the black line representing the cone responses of the color patches under D65 illumination. In some conditions, the cone responses were influenced little by changes of illumination from D65 to test illuminations, as shown in the cases where the slopes of the black lines were close to one (perfect adaptation). The response predicted by the von Kries model and those under D65 illumination overlapped, and the diagonal line indicates both perfect adaptation and no adaptation. This did occur for blue and yellow illuminations, which resulted in mainly S-cone stimulation changes; the L- and M-cone stimulations were almost the same as those for D65, as shown in Table 4. The fitted lines (denoted by the red lines) to the data points were obtained by multiplying the von Kries model prediction by a constant to minimize the sum of the squared error between the prediction and the match. The slope $k$ and coefficient of determination $R^{2}$ for the fitted line in each panel are shown in Table 5. Note that in this study, the coefficient of determination was calculated using the common definition with the linear line of zero intercept but not by the square of the correlation coefficient. It is well know that the von Kries-type adaptation is often modest under certain experimental conditions for a variety of reasons, meaning the adaptation effect is not necessarily of perfect strength (100%). Thus, the slope coefficient $k$ of the fitted line, which can reflect the strength of the von Kries-type adaptation, can vary between the slope of the no adaptation line (black line) and that of the perfect adaptation line (diagonal line).

Fig. 6. Comparison of the L-cone (top) and M-cone (bottom) matched by color-normal observers (ordinate) with those predicted by the von Kries model (abscissa) for twelve color patches. N denotes color-normal observers. The four columns of panels correspond to the red, green, blue, and yellow illuminations. Diagonal (dotted) lines indicate perfect von Kries-type adaptation. The black lines indicate no adaptation. The red lines denote the best fits, defined by the least sum of squared error between the prediction and the match, to the data points. Each data point was averaged over five observers and six sessions.

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Table 5. Slope Coefficient $k$ and Coefficient of Determination $R^{2}$ for Fitted Lines in Fig. 6

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In Table 5, the coefficients of determination suggest that the matched L-cone and M-cone responses under each illumination condition are in accordance with the principle of the von Kries model, but the slope $k$ was not simply in the range between the slope coefficient of no adaptation and one (perfect adaptation) in the L- and M-cone responses, even considering the reasonable amount of difference, which was about 4% to 5%. This indicates that they could not be described well quantitatively by the von Kries model, in which the constants indicating the changing amount of the cone sensitivity due to the adaptation were defined as the inverse of the L-, M-, and S-cone responses of a perfect white patch under the D65 illuminant and the test illumination. As can be seen in Fig. 6, the matched L-cone responses under the red illumination are substantially higher than those predicted by the von Kries model; the matched M-cone responses are close to the response under D65 illumination and even exceed it. These results suggest that the observers did not use any amount of simple L- and M-cone stimulations in the matching under the red illumination. Under the green illumination, the matched L-cone responses showed almost no adaptation; the matched M-cone responses showed a considerable amount of adaptation under the von Kries model. In the blue and yellow illuminations, the matched L- and M-cone responses approach the diagonal lines, which means both no adaptation and perfect adaptation, but with a slight upward deviation.

We also compared the M-cone responses matched by three protanopes, the L-cone response matched by one deuteranope, and the L-, M-, and $M^{'}$ -cone responses matched by two deuteranomalous observers with those predicted by the von Kries model, as shown in Fig. 7. The slope coefficients and coefficients of determination for the fitted lines are shown in Table 6. The slope coefficients $k$ and the coefficients of determination in Table 6 indicate that the M-cone responses matched by the protanopes can be explained well by the von Kries model. The model prediction in the M-cone response was even more accurate in the protanopes than that in the color-normal observers under the red and green illuminations in terms of the coefficients of determination and the sum of the square residuals. The model prediction of the L-cone response by the deuteranope was worse than that by the color-normal observers. However, that observation cannot be confirmed because there was only one deuteranope observer in this study. As can be seen in Fig. 7, under red illumination, the matched M-cone responses of protanopes (first row) are close to the no adaptation line, indicating less adaptation; under green illumination, the matched M-cone responses are exactly on the diagonal line, indicating perfect adaptation. The difference can be explained by the difference of the M-cone stimulation strength between the red and green illuminations. Under blue illumination, the matched M-cone responses of the protanopes are exactly on the diagonal line; under yellow illumination, the performance of the M-cone responses is similar to that of the color-normal observers. The L-cone responses matched by the deuteranopic observer deviated systematically from the von Kries model prediction and under red illumination were substantially larger than predicted. Unexpectedly, the tendencies of the data point distributions both in the protanopes and the deuteranope are similar to those in the color-normal observers, even though the color-deficient observers are expected not to have a red-green color opponent mechanism coded by the L-M response. The L-cone responses matched by the deuteranomalous observers showed the same tendency as those matched by the deuteranope. Apparently, the M-cone responses matched by the deuteranomalous observers (fourth row) cannot be explained by the von Kries model, which is calculated using the coefficient $k_{M, trans}$ of the color-normal observers. One possible reason for this is that deuteranomalous observers may have one of the hybrid M-cones, whose spectral sensitivity is different from that of the common type of M-cone. For that reason, the $M^{'}$ -cone response [29] (fifth row) was calculated. Since the $M^{'}$ -cone spectral sensitivity is rather closer to that of L-cone than that of M-cone, the overall tendency was more similar to the L-cone response (third row), although the points were not on a straight line. The performance of the $M^{'}$ -cone responses of the two deuteranomalous observers will be discussed in the discussion section later.

Fig. 7. Comparison of cone responses matched by color-deficient observers (ordinate) with those predicted by the von Kries model (abscissa) for twelve color patches. First row: M-cone responses matched by protanopes (denoted by P); each data point was averaged over 3 protanopes. Second row: L-cone responses matched by 1 deuteranope (D). Third row: L-cone responses matched by deuteranomalous observers (DA); each data point was averaged over 2 deuteranomalous observers. Fourth row: M-cone responses by deuteranomalous observers (DA). Fifth row: $M^{'}$ -cone responses by deuteranomalous observers (DA). The four columns from left to right correspond to red, green, blue, and yellow test illuminations. Notations of the symbol and lines are the same with Fig. 6.

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Table 6. Slope Coefficient $k$ and Coefficient of Determination $R^{2}$ for Fitted Lines in Fig. 7^a

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Figure 8 shows a comparison of the S-cone responses for twelve color patches, obtained from the color constancy task with the von Kries-model predicted responses for each observer group. The slope coefficients and the coefficients of determination for the fitted lines are shown in Table 7. Overall, the S-cone responses matched by both the color-normal and the color-deficient observers are better explained by the von Kries model than by the L- and M-cone responses. However, the fit for the color-deficient observers under the red and green illuminations was better than that for the color-normal observers. The deviation of the S-cone matching in the color-normal observers was mainly caused by the strong S-cone stimulation by the color patches (5P5/6, 7.5PB5/6, and 10B5/6). This result is in agreement with those of previous studies [6,8,10,37,38]. Romero et al. [39] suggested two possible reasons for this: the color discrimination threshold by the S-cone increases with the increment of the S-cone stimulation; additionally, color-normal observers tend to use a red-green color control to obtain color constancy and thus lose a certain amount of S-cone sensitivity.

Fig. 8. Comparison of matched S-cone responses with the von Kries model predictions for each observer group under four illumination conditions. Notations are the same as in Fig. 7, except N denotes color-normal observers (first row) and there is only one set of data (fourth row) for deuteranomalous observers (DA).

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Table 7. Slope Coefficient $k$ and Coefficient of Determination $R^{2}$ for Fitted Lines in Fig. 8^a

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In the case of the red and green illuminations, the von Kries model cannot explain some of the matched S-cone response, because the fitted lines are not always located between the no adaptation lines (black lines) and the perfect adaptation lines (diagonal lines). In these illuminations, the S-cone stimulation was little changed from the D65 illumination, which may have resulted in less accurate matching. Conversely, in the blue and yellow illuminations, the S-cone stimulation changes were relatively large and may have resulted in a better prediction of the S-cone response by the model because of the stronger adaptation effect. In the blue illumination, S-cone matching shifts toward the perfect von Kries adaptation line, indicating a certain amount of S-cone adaptation; in the yellow illumination, the matched S-cone is rather close to the no adaptation lines, indicating little adaptation.

C. Prediction at Postreceptoral Stages

It has been argued that the von Kries gain control may also occur at postreceptoral stages [11]. In a multiplicative gain control model, the postreceptoral response is additionally suppressed by the von Kries postreceptoral gain control after the von Kries cone adaptation. Considering that, we investigated whether or not the opponent color signal changes could be described only by the von Kries-type simple adaptation to cones. The postreceptoral response obtained from the linear summation of the cone responses under the von Kries model adaptation was used to fit the red-green and blue-yellow opponent color signal changes from the D65 illumination to the observers’ matches under the test illumination.

We introduced the L-2M response in the analysis, which can be considered as a coding of the red-green chromatic opponent pathway for color-normal observers. Figure 9 shows the model fit of L-2M matches of the color-normal observers. If the changing of opponent color signals can be described in terms of a simple adaptation (and gain control) on each cone type separately, the data points would fall on the diagonal line. Apparently, under red and green illuminations, the data points deviate systematically from the diagonal line, meaning that the red-green opponent signal matched by the color-normal observers does not follow the von Kries model adaptation and gain control. This result is consistent with the result reported in the literature [7], which shows that the fit of the red-green opponent color signal with the von Kries model was unsatisfactory. The fit, however, could not be improved by the additional von Kries model postreceptoral gain control (data not shown). On the contrary, under the blue and yellow illuminations, the von Kries model fits the data well. Under the blue and yellow illuminations, however, the color-normal observers generally made an appearance match in redness and greenness because the L-2M response would be adapted little in the maximum; it is more rational that other possible mechanisms, related to S-cone response and/or the blue-yellow chromatic opponent response, would be the main contributors to matching for the color constancy mostly in blueness and yellowness. The L-2M response of the deuteranomalous observers was much worse than that of the color-normal observer (data not shown). The L- $2 M^{'}$ response had almost the same value for all color patches because the $M^{'}$ -cone spectral sensitivity is close to that of the L-cone, as shown in Fig. 9. This suggests that deuteranomalous observers do not use the L-M cone response for matching either.

Fig. 9. Comparison between the matched L-2M (top) and L- $2 M^{'}$ (bottom) responses of color-normal (N) and deuteranomalous (DA) observers and the von Kries model predictions for the red, green, blue, and yellow illuminations (from left to right). Black lines denote the value under D65 illumination. If the matches perfectly comply with a principle of the separated adaptation and gain control on each cone type, the data points would fall onto the diagonal line.

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We introduced the blue-yellow chromatically opponent pathway response (chromatic response), $T_{blue-yellow}$ , that is, $S - (L + M)$ , for the color-normal observers, S-M for the protanopes, S-L for the deuteranope, and $S - (L + M)$ and $S - (L + M^{'})$ for the deuteranomalous observers. To derive the balance of the L- and M-cones with the S-cone on the blue-yellow chromatic response, the balance coefficients $u_{n}$ , $u_{p}$ , $u_{d}$ , $u_{d a}$ , and $u_{d a^{'}}$ were additionally introduced, as in Eq. (7):

{\begin{cases} T_{blue-yellow} = S - u_{n} \cdot (L + M) & (normal) \\ T_{blue-yellow} = S - u_{p} \cdot M & (protan) \\ T_{blue-yellow} = S - u_{d} \cdot L & (deutan) \\ T_{blue-yellow} = S - u_{d a} \cdot (L + M) & (deuteranomaly) \\ T_{blue-yellow} = S - u_{d a^{'}} \cdot (L + M^{'}) & (deuteranomaly) \end{cases} .

Balance coefficients were defined to make these blue-yellow responses zero on the 20% spectral reflectance patch (neutral patch), which was used as one of the colored patches under D65 illumination. The values of

u_{n}

,

u_{p}

,

u_{d}

,

u_{d a}

, and

u_{d a^{'}}

were 0.0175, 0.0507, 0.0267, 0.0175, and 0.0153, respectively, and were used for all illumination conditions.

Figure 10 shows the mean matched blue-yellow opponent color responses of each observer group for twelve color patches. Except for the deviation in the color-normal observers for color patches stimulating the S-cone strongly (5P5/6, 7.5PB5/6, and 10B5/6), as shown in Fig. 10, the matched data for all observers under all illumination conditions could be fitted by a single line, meaning that the blue-yellow opponent color responses made by the color-normal and color-deficient observers in color constancy can always be described well by the von Kries model.

Fig. 10. Comparison of the matched blue-yellow response [as defined in Eq. (7)] of color-normal (denoted by N), protan (P), deutan (D), and deuteranomalous (DA) observers (from top to bottom), and the von Kries model predictions for red, green, blue, and yellow illuminations (from left to right). Deuteranomalous observers’ data were analyzed for both M-cone (fourth row) and $M^{'}$ -cone (fifth row). The other denotations are the same as those used in Fig. 9.

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4. DISCUSSION

In this study, the von Kries-type of adaptation/gain control model was applied to predict the color constancy of color-deficient observers. In the red and green illuminations, which produced enough strong L- and M-cone stimulations but could hardly be detected in the red-green chromatic change from D65 illumination, the L- and M-cone responses matched by the color-deficient observers were unexpectedly similar to those matched by the color-normal observers. The L- and M-cone responses of all the observers could not be explained even qualitatively by the simply application of the von Kries model alone, although the red and green illuminations invoked mainly brightness change for the color-deficient observers, and the possible luminance pathway of color-deficient observers was expected to use only the L- or M-cone signal. In the blue and yellow illuminations, which mainly alter the S-cone responses, the S-cone response matched by the color-deficient observers could be described partly by the von Kries model, but a substantial S-cone adaptation was only observed under blue illumination, which produces a strong S-cone stimulation. It seems that the S-cone adaptation may make some contribution to the color constancy, although only for illuminations that can produce a strong S-cone stimulation.

A. Blue–Yellow Mechanism in the Color Constancy

The von Kries model can also explain blue-yellow chromatic responses reasonably well for both red and green illuminations, where the chromatic change was difficult to distinguish from the standard D65 illumination, and the blue and yellow illuminations, where the change was easily detected. In the red and green illuminations, it is natural for color-deficient observers to make the appearance match in which the adaptation effect should be minimal, since red-green illumination mainly causes brightness changes. In the blue and yellow illuminations, the amount of change in the chromatic appearance of the blue-yellow color was relatively large, but the blue-yellow response of the color-deficient observers still agrees with the simple adaptation model. In addition, because the illumination changes along the daylight locus are primarily mediated in the S-cone responses, the human color system may have developed a higher sensitivity along the blue/yellow dimension [28]. Before the experiment in this study was conducted, it was expected that both the color-normal and color-deficient observers would make matches under color illuminations that would mainly reflect possible mechanisms of the statistical operation of the scene [20,21] or the illuminant-by-illuminant estimation strategy [22,23] in color constancy, rather than the simple von Kries adaptation mechanism. It must be mentioned, however, that the stimuli used in this study are not necessarily suitable for deriving these possible mechanisms using the illuminant estimation strategy [40] because there was no actual color illumination in the experimental room. The kinds of observation factors that concern the real perception of surfaces is supposed to influence the color constancy, although a treatment of the precise perception is beyond the scope of this study because we used the matching method. Considering that the L- and M-cone adaptations mainly contribute to the luminance matches by the color-deficient observers and the S-cone adaptation only exists under blue illumination, the compliance of the blue-yellow opponent responses in the color matching with the von Kries model still suggests that color constancy in color-deficient observers can be mediated by the blue-yellow color system. However, considering that the amount of adaptation in the S-cone is small and that the gain control for the blue-yellow opponent color system was not used (not required) for model prediction, the color-deficient observers may have achieved color constancy by possible usage of other cues, such as the luminance, shape, and texture of the objects.

B. Color Constancy of Anomalous Observers

In this study, we used the von Kries model with the coefficient $k_{M, trans}$ calculated for the M-cone on the CIE standard observer and for $M^{'}$ -cone [29] in the prediction of the M- and $M^{'}$ -cone responses matched by the two deuteranomalous observers. As shown in Fig. 9, the von Kries model apparently could not explain the M- and $M^{'}$ -cone responses. We tried to estimate the characteristics of their M-cone responses by means of an additional matching experiment, in which both the standard and test patterns were virtually illuminated by D65. In the D65 and D65 illumination conditions, observers had to make the appearances match. Figure 11 shows the comparison of the M- and $M^{'}$ -cone responses of the matched points by the deuteranomalous observers with the L-, M-, and $M^{'}$ -cone responses by the standard normal and deuteranomalous observers under D65 and D65 illumination. If the observer were color normal (the standard observer), he or she would control the test color in the aspects of red-green and blue-yellow with brightness in the appearance match, which would naturally make the L-, M-, and S-cone responses identical for the two color patches. If the deuteranomalous observers had an M-cone close to the M-cone of the standard observer, the M-cone responses for 12 color patches, calculated using the matching data of deuteranomalous observers with the standard sensitivity of the M-cone, should have been similarly distributed to the M-cone response of the standard observers (top panel in Fig. 11). The same is true for the $M^{'}$ -cone response with the standard deuteranomalous sensitivity of the $M^{'}$ -cone (bottom panel). Conversely, if the deuteranomalous observers had the hybrid type of M-cone, whose sensitivity has shifted to a longer wavelength [41], it would be expected that the distribution of their M-cone responses would be more similar to the L-cone distribution of the standard observers (middle panel). The M-cone responses matched by two deuteranomalous observers are closer to the L-cone responses of the standard observer. This suggests that the spectral sensitivity of the M-cones of the two deuteranomalous observers is closer to that of the L-cone of the standard observer than to that of the M- and $M^{'}$ -cones. This result is consistent with the subjects’ performance on the CCT: the color discrimination ellipsoids of the two deuteranomalous observers were highly similar to that of the deuteranope in this study.

Fig. 11. Comparison of M-cone (top and middle panels) and $M^{'}$ -cone (bottom panel) responses matched by the two deuteranomalous observers (denoted separately by open and filled squares) under D65 illumination with the standard normal observer’s M- (top panel) and L-cone (middle panel) responses and the standard deuteranomalous observer’s $M^{'}$ -cone response (bottom panel) under D65 illumination. The L- and M-cone responses in the middle panel were normalized from 0 to 1 separately for comparison and the point at 1.0,1.0 denotes the white patch of 20% reflectance. The red lines denote the best fit by the least-squares method.

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C. Effect of the Adjustment Line in the Paper Match Task

The color distance and the index calculated on the blue-yellow line can be affected by the adjustment line used by color-deficient observers to make the blue-yellow color adjustment. However, it is certain that color-deficient observers cannot detect apparent blue-yellow color changes induced by red-green illumination changes. Under both red-green and blue-yellow illuminations, the color-deficient observers performed similarly in the L-, M-, and S-cone matches to color-normal observers. As shown in the results (Fig. 10), their very stable gain control adjustment of blue-yellow opponent color signals indicates that the blue-yellow opponent color system plays a very important role in the color constancy of color-deficient observers.

It should be noted that in this study, deuteranomalous observers also adjusted only blue-yellow color, as did dichromatic observers. The two deuteranomalous observers, classified by the limited R/G range in the anomaloscope matches, showed a similar color discrimination ability to that of the deuteranope in the CCT, with the longer axis of ellipsoids extended beyond the color gamut. They were also asked to complete the task in the manner of the normal observers, but the matching result was anomalous and the data points were located randomly along the red-green direction. More precisely, the comparison between the matched data obtained using the red-green buttons and not using them (using only blue-yellow buttons) revealed that the L- and S-cone responses and $S - u (L + M)$ response were almost the same, and only the M-cone response differed noticeably (correspondingly, the L-2M response also differed). However, the performance by the protanomalous observer who participated in the pilot study (data not shown) ranked in between the color discrimination ability of the protanopes and the color-normal observers. He needed to use the red-green color adjustment and achieved the same good color constancy as the color-normal observers. In previous literature, anomalous observers showed similar color constancy with color-normal observers [14,18]. These anomalous observers may be close to this protanomalous observer. The variability of color constancy performance in anomalous observers may furthermore demonstrate the speculation of Baraas et al. [18]: besides the number of cone pigments, the spectral location of the cone pigment or the interaction of the cone signals may also play an important role in the color constancy.

Overall, for color-deficient observers, the direction of the blue-yellow adjustment line should have had little influence in the paper match task, since possible changes of redness and greenness caused by a reasonably small, arbitrary rotation in the direction of the blue-yellow adjustment line could not be detected. Although accidental correlations and/or relationships of the L- and M-cone responses resulting from a one-dimensional change on the line had to be checked, because the L- and M-cone responses varied only slightly along this line, no systematic relations was found among the L- and M-cone responses.

D. Comparison of Results with Those of Previous Studies and Limitations of This Study

The following is an evaluation of the agreement between the results of this study and those of the previous studies listed in the introduction. In this study, the dichromatic observers showed almost no color constancy under red and green illuminations. This result is inconsistent with the previous literature [14], in which dichromats displayed considerable color constancy. The red-green illumination in this study was close to the dichromatic confusion line and almost caused no chromatic changes, but there was a brightness change because of the strong L- or M-cone stimulations. It is expected that the adjustment made by observers to compensate for the red-green illumination change was mostly the luminance adjustment by red-green color control. This might be the reason that there is greater variability along the red-green direction. It has been found that the contour plot representing the response of the illumination changes in the task, in which the observers were asked to discriminate illuminant changes from surface-reflectance changes, were elongated along the red-green direction [16]. In addition, in the previous literature, dichromats showed relatively good color constancy under daylight illuminant changes [14,18], especially with natural surface reflectance [17]. However, in this study, under blue-yellow illumination, which mainly alters S-cone responses, the color constancy is not so strong as that of these studies. This can support that dichromats are better at color constancy under natural reflectance and daylight illuminants, which are frequently experienced in daily life, than under Mondrian-like patterns and unfamiliar illuminants.

Three key limitations of this study must be mentioned: the current results are based on the assumption that the color vision system of dichromats is a reduction of the normal trichromatic system and the spectral sensitivity of the normal cone of anomalous observers is the same as that in a normal trichromatic system. The second limitation is that the cone contributions and luminous efficiency were not controlled individually, and heterochromatic flicker photometry and artificial pupils [42] were not employed. It can be expected that the narrow age range of the observers minimized the effect of age-related changes on luminous detection, which is caused by age-related change in the ocular lens density [43,44] and age-related reduction of the pupil size (senile miosis) [45–47]. However, there was still observer variation in luminous efficiency, even within the same age group [42]. Individual differences in tritan (S-cone) confusion lines were also not controlled, although those differences were not negligible, even without the complex aging effects related to the S-cone [48]. However, individual tritan confusion lines for dichromats cannot be measured using the S-cone adaptation method [48], because adaptation to the S-cone only makes dichromats temporary monochromats. These limitations caused the color mechanisms involving the S-cone to be stimulated along the L- and M-cones; this could cause chromatic responses on both the red-green and blue-yellow opponent pathways for the color normals, making the analysis of the results of the paper match task and model prediction more complex [49]. On the other hand, the dichromats were expected not to have a red-green opponent pathway [50]; this made the matching task simpler and model prediction better. This could be further investigated with the precise control of the luminous efficiency and blue-yellow test illuminations on the individual S-cone confusion line. Future work is expected to clarify the debate over model prediction in color normals.

Another limitation of this study is related to the experimental conditions. We found little contribution of the possible luminous pathway to color constancy through separated M- and L-cone responses for the protanopes and the deuteranopic observers. In this study, however, the stimuli were not optimized to estimate the possible contribution of the luminous (or L+M) pathway. We tried to set the luminance variation between colored patches to small (by using the same value) and to control the test illumination (by setting the luminance of the neutral patch to $25 cd / m^{2}$ ), but knowledge about the structure of the luminous pathways in color-deficient observers was insufficient. The pseudo-equal luminance setting did not result in blackness perception in either the central colored patches or the background [51,52], and it was expected that for all observers, certainly for the color-normal observers and most likely for the color-deficient observers, luminance adjustment as a brightness matching task may have been necessary for the paper match task in this study. This means that in this experiment, the luminance adjustment had little influence on the results. However, that does not necessarily imply that the contribution of the luminous pathway to the color constancy is small. This point also requires further investigation.

Funding

Japan Society for the Promotion of Science (JSPS) KAKENHI (24300085, 25245065).

Acknowledgment

We gratefully acknowledge the anonymous reviewers for their important suggestions and thoughtful comments.

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Illumination	Obs.	$u^{'}$	$v^{'}$	L	M	S
D65		0.198	0.468	16.37	8.63	0.44
Red (P)	KMY	0.239	0.478	17.19	7.81	0.37
	OTK	0.238	0.477	17.19	7.81	0.37
	PB	0.238	0.477	17.19	7.81	0.37
Red (D)	KBY	0.241	0.469	17.23	7.77	0.41
Red (DA)	LX	0.241	0.468	17.23	7.77	0.41
	OK	0.241	0.469	17.23	7.77	0.40
Red (N)		0.242	0.459	17.23	7.77	0.46
Green (P)	KMY	0.159	0.459	15.55	9.45	0.51
	OTK	0.159	0.459	15.55	9.45	0.51
	PB	0.159	0.459	15.55	9.45	0.51
Green (D)	KBY	0.155	0.468	15.51	9.49	0.47
Green (DA)	LX	0.154	0.468	15.51	9.49	0.46
	OK	0.155	0.467	15.51	9.49	0.47
Green (N)		0.152	0.479	15.51	9.49	0.42
Blue	All	0.206	0.446	16.47	8.53	0.55
Yellow	All	0.189	0.493	16.27	8.73	0.32

Illumination	Observer Type	L (or $L^{'}$ in PA)	M (or $M^{'}$ in DA)	Luminance
Red (N)	N	5.2%	−9.8%	0.0%
Red (P)	$⟨ N ⟩$	$⟨ 4.7 % ⟩$	$⟨ - 9.2 % ⟩$	$⟨ - 0.1 % ⟩$
	P	-	−9.2%	−9.2%
	$⟨ PA ⟩$	$⟨ - 4.9 % ⟩$	$⟨ - 9.2 % ⟩$	$⟨ - 6.5 % ⟩$
Red	$⟨ N ⟩$	$⟨ 5.0 % ⟩$	$⟨ - 9.7 % ⟩$	$⟨ - 0.1 % ⟩$
(D, DA)	D	5.0%	-	5.0%
	DA	5.0%	1.3%	3.6%
Green (N)	N	−5.2%	9.8%	0.0%
Green (P)	$⟨ N ⟩$	$⟨ - 4.8 % ⟩$	$⟨ 9.4 % ⟩$	$⟨ 0.1 % ⟩$
	P	-	9.4%	9.4%
	$⟨ PA ⟩$	$⟨ 5.0 % ⟩$	$⟨ 9.4 % ⟩$	$⟨ 6.6 % ⟩$
Green	$⟨ N ⟩$	$⟨ - 5.1 % ⟩$	$⟨ 9.8 % ⟩$	$⟨ 0.1 % ⟩$
(D, DA)	D	−5.1%	-	−5.1%
	DA	−5.1%	−1.3%	−3.7%
Illumination	Observer Type	S		Luminance
Blue	N	26.3%		0.1%
	P	26.3%		−1.0%
	$⟨ PA ⟩$	$⟨ 26.3 % ⟩$		$⟨ - 0.8 % ⟩$
	D	26.3%		0.8%
	DA	26.3%		0.5%
Yellow	N	−26.1%		−0.1%
	P	−26.1%		1.1%
	$⟨ PA ⟩$	$⟨ - 26.1 % ⟩$		$⟨ 0.8 % ⟩$
	D	−26.1%		−0.8%
	DA	−26.1%		−0.6%

Illumination	Maximum	Minimum	$Average (\pm SD)$
D65	23.0	20.6	$21.7 \pm 0.5$
Red (N)	26.4	19.4	$22.3 \pm 1.7$
Green (N)	24.4	18.4	$21.2 \pm 1.5$
Blue	23.7	20.4	$21.8 \pm 0.6$
Yellow	23.0	20.3	$21.7 \pm 0.7$

Illumination	$k_{L, trans}$	$k_{M, trans}$	$k_{S, trans}$
Red (N)	1.05	0.90	1.05
Red (P)	1.05	0.91	0.83
Red (D, DA)	1.05	0.90	0.93
Red ( $DA : M^{'}$ )	1.05	1.01	0.93
Green (N)	0.95	1.10	0.95
Green (P)	0.95	1.09	1.17
Green (D, DA)	0.95	1.10	1.07
Green ( $DA : M^{'}$ )	0.95	0.99	1.07
Blue	1.01	0.99	1.26
Yellow	0.99	1.01	0.74

Illumination	L		M
Illumination	$k$	$R^{2}$	$k$	$R^{2}$
Red (N)	1.09	0.87	1.17	0.91
Green (N)	1.04	0.77	0.98	0.87
Blue	1.04	0.95	1.05	0.97
Yellow	0.96	0.98	1.02	0.98

Color constancy of color-deficient observers under illuminations defined by individual color discrimination ellipsoids

Abstract

1. INTRODUCTION

2. METHODS

A. Observers

B. Apparatus and Calibration

C. Stimulus

D. Procedure

3. RESULTS

A. Color Constancy Performance

B. Von Kries Model Prediction at the Photoreceptoral Stage

C. Prediction at Postreceptoral Stages

4. DISCUSSION

A. Blue–Yellow Mechanism in the Color Constancy

B. Color Constancy of Anomalous Observers

C. Effect of the Adjustment Line in the Paper Match Task

D. Comparison of Results with Those of Previous Studies and Limitations of This Study

Funding

Acknowledgment

REFERENCES

Cited By

Figures (11)

Tables (7)

Equations (7)

Journal of the Optical Society of America A

Illumination	M (P)		L (D)		L (DA)		M (DA)		$M^{'} (DA)$
Illumination	$k$	$R^{2}$	$k$	$R^{2}$	$k$	$R^{2}$	$k$	$R^{2}$	$k$	$R^{2}$
Red (P, D or DA)	1.07	0.95	1.17	0.80	1.13	0.79	1.39	−1.95	1.19	0.44
Green (P, D or DA)	1.01	0.94	1.03	0.79	0.98	0.85	0.89	−1.90	0.95	0.52
Blue	1.02	0.97	1.08	0.80	1.06	0.82	1.15	−2.03	1.06	0.56
Yellow	1.03	0.95	1.03	0.89	1.03	0.81	1.07	−2.21	1.03	0.53

Illumination	S(N)		S(P)		S(D)		S(DA)
Illumination	$k$	$R^{2}$	$k$	$R^{2}$	$k$	$R^{2}$	$k$	$R^{2}$
Red (N, P, D or DA)	1.18	0.96	1.25	0.99	1.22	0.98	1.21	0.99
Green (N, P, D or DA)	1.23	0.94	1.10	0.98	0.94	0.99	0.91	0.99
Blue	1.03	0.95	0.93	0.98	0.88	0.99	0.91	0.99
Yellow	1.39	0.99	1.42	0.99	1.21	0.98	1.28	0.99