Possible influences on color constancy by motion of color targets and by attention-controlled gaze

Lifang Wan; Keizo Shinomori

doi:10.1364/JOSAA.35.00B309

1. INTRODUCTION

Color constancy is a common phenomenon in which the color of a surface does not change significantly as the illuminant changes. A substantial amount of research has investigated color constancy as described by Foster [1]. Adaptation of photoreceptors in the retina described by the von Kries model [2] has been supported by a series of experiments [3–11]. These experiments were mostly performed under daylight or under red, blue, and yellow illuminants, using color appearance matches or paper matches in asymmetric color matching [3–9] or in achromatic matching [10,11]. Color constancy is more complex than can be easily explained by the action of a simple visual mechanism; however, experiments have also been performed with a 3D setup of illuminated surfaces under colored lights in achromatic matching [12]. The fact that the von Kries model cannot completely explain color and lightness constancy suggests that cortical mechanisms must underlie color constancy; it was demonstrated by the asymmetric color-matching experiment in which subjects were asked to hold the color of stimuli under chromatic lights in memory and to match the color of the stimuli under daylight [13]. The visual identity of an object, which has a measurable effect on color perception, indicates an additional mechanism for color constancy [14].

The adaptation of the cone photoreceptors in the retina and cognitive mechanisms occurring at a higher level are considered as two main factors contributing to color constancy. Ma et al. [15] mentioned the higher-level mechanisms for color constancy underlying the statistical operation of the scene [16,17] and illuminant-by-illuminant estimation strategy [18–20], which depend on the appearance of the scene. The contribution of these complex mechanisms to color constancy should be compared to that of the von Kries adaptation mechanism using simple adaptation and/or gain control. In this study, we did not specify the certain mechanism as the higher-level process contributing to the color constancy. Instead, we summarized these mechanisms as the illumination estimation effect [15].

However, it was not clear whether the utilization of these two types of color constancy mechanisms depends on different observing modes: random viewing and fixation. The eye movement, which was mentioned in this study as the difference between free viewing of the stimulus (random state) and fixating on the stimulus with smooth pursuit eye movement (fixation state), would affect retinal adaptation differently. Werner [21] and Ebner [22] reported that color constancy improves by object motion. In these experiments, the color constancy was tested in the achromatic matching experiment in which observers fixated on stable objects or pursued moving objects, although attention was not considered. However, attention would also have different effects on retinal adaptation, and eye movements combined with the attentional state would subsequently influence color constancy. Suchow and George [23] reported that motion could cause failure to detect changes of color and size, except in the case when the observer attended to the objects and noticed the objects’ changes. Golz [24] found that viewing behavior also has influences on color constancy; the exploration of the visual field increases color constancy compared to fixating on just one spot of the test field.

The main purpose of this study was to investigate the possible influences of object motion and attention controlled by the gaze mode on retinal adaptation and the illumination estimation effect in color constancy. The contribution from retinal adaptation to color constancy was estimated with the amount of the overall color constancy through data and model analysis; the difference between them would be attributed to the contribution of the illumination estimation effect occurring at a higher-level mechanism.

2. METHOD

A. Apparatus and Calibration

All experiments were performed in a dark room. The stimuli were presented on a 19-inch (47.5 cm) cathode ray tube (CRT) color monitor (Trinitron G420, Sony). A visual stimulus generation (VSG) system (ViSaGe, Cambridge Research Systems [CRS], Inc.) provided 14-bit resolution for each RGB phosphor at $1280 \times 960$ resolution with a 60-Hz frame rate. A black paper board ( $90 cm \times 60 cm$ ) was vertically 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 distance between the observer’s eye and the monitor was 90 cm, and his/her head was supported by a chin rest. A handheld six-button box (CB-6, CRS) was used for observers to adjust the color. The gamma correction of the monitor was carried out by the calibration software of the VSG system (VSG-Desktop, CRS) with a calibrator (ColorCAL, CRS). The chromaticity coordinates and luminance of all colors in the stimuli were measured by a colorimeter (CS-200, Konica-Minolta, Inc.) and a spectral radiometer (CS-1000, Konica-Minolta, Inc.). These calibrations confirmed that the error of screen presentation was less than 3% in Commission Internationale de l’Éclairage (CIE) 1931 $x y$ chromaticity coordinates, and less than 5% in luminance.

B. Visual Stimulus

Figure 1 shows the visual stimulus consisting of a 5-deg square reference pattern virtually illuminated by standard daylight illumination, D65 (6505 K) and a 5-deg square test pattern virtually illuminated by red, green, blue, and yellow test illuminations; the standard and test patterns were presented side-by side, separated by a 1-deg black strip made with a black paper board frame covering the entire monitor screen except for the two square patterns. The paired patterns had an identical spatial arrangement, consisting of a 1.2-deg colored patch at the center surrounded by background ellipses. The background of both 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 in the shorter axis and between 1.6 and 2.0 deg in the longer axis. These two patterns were observed in haploscopic view; the left pattern and the right pattern were observed by the left and right eye, respectively; the black paper board was placed vertically from the screen to the subject separating the two patterns to prevent pattern viewing by the fellow eye.

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 [25,26] and the spectral radiance of the illumination. 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 a neutral gray of 20% flat-reflectance surface. Figure 2 shows chromaticity coordinates of these test colors under D65 illumination.

Fig. 2. CIE 1976 $u^{'} v^{'}$ chromaticity coordinates of the 12 central colored patches under 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 angular distance of approximately 45 deg in the hue circle of the Munsell Color System. 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 12 surfaces used in the central colored patch. The eight surfaces forming one background pattern were different for each of the six sessions, obtained by rotating the Munsell hue circle clockwise.

The standard illuminant was D65 and invariant in all experimental sessions. The test illuminant was constructed by a linear combination of the daylight spectral basis functions [27]. The color of illuminants was shifted from D65 by the specific color difference defined by CIE 1976 $L^{*} u^{*} v^{*}$ , $Δ E_{u v}^{*}$ on/in an equiluminant plane. The colors of red and green were shifted equally by the color difference of 53 $Δ E_{u v}^{*}$ along the L-M axis. The colors of blue and yellow illuminants were shifted by the color difference of 45 $Δ E_{u v}^{*}$ along the S − (L + M) axis. The intensity of each illumination was adjusted to achieve the same luminance on the 20% flat-reflectance surface illuminated to $25 cd / m^{2}$ . Table 1 summarizes the CIE 1931 $x y$ chromaticity coordinates, the color difference, $Δ E_{u v}^{*}$ , and the difference of L- or S-cone excitation for each color illuminant.

Table 1. Illumination Condition (CIE1931 $x y$ Chromaticity Coordinates, Color Difference, $Δ E_{u v}^{*}$ , and Difference of L- or S-Cone Excitation)

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The chromaticity coordinates of the Munsell surfaces under D65 and the test illuminations were calculated using CIE 1931 standard color-matching functions [28]. Spectra were sampled at 5-nm intervals and integrated from 380 to 780 nm. We used Smith–Pokorny’s cone fundamentals [29] and the CIE 1931 color-matching functions with the cone matrix by Kaiser and Boynton [30]. Figure 3 shows $u^{'} v^{'}$ chromaticity coordinates of 12 colors under red, blue, yellow, green, and D65 illuminants. Figure 4 shows the stimulus configuration in the haploscopic appearance under red, green, blue, and yellow using the 20% flat-reflectance surface as the central square target.

Fig. 3. $u^{'} v^{'}$ chromaticity coordinates of 12 test colors under the D65, red, green, blue, and yellow illuminants denoted by black, red, green, blue, and yellow circles, respectively. Slanted vertical and horizontal black lines denote S − (L + M) and L − M axes, respectively. The large triangle denotes the gamut of the monitor.

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Fig. 4. Stimulus configuration under red (top left), green (bottom left), blue (top right), and yellow (bottom right) test illuminant conditions in haploscopic view. The central target is a 20% flat-reflectance surface.

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C. Observers

Six observers (two male and four female) in the range from 21 to 28 years old participated in this study. We confirmed that all observers had normal or corrected-to-normal visual acuity better than 0.6 (1.67 min. of visual angle), and they had normal color vision assessed by Ishihara-test plates (International 38 plates edition) and the Farnsworth D-15 test. The procedures of the experiment 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 prior to testing.

D. Procedure

In order to control the state of the color constancy measurement, we set two parameters. The first parameter was about the motion of the color target: target-static, target-motion, and target-rotation conditions. In the case of the target-static condition, the test patch stayed at the center of the static background, as shown in Fig. 1. In the target-motion condition, the central test patch (in right or left) and standard patch (in left or right) in the paired patterns flipped simultaneously between the top and the bottom of the patterns at the rate of 3 deg/s. This vertical movement was continuously looped along the vertical lines at the horizontal center of each background pattern. In the target-rotation conditions, the test patch rotated clockwise or counterclockwise in a circular fashion, staying within 2 deg of the center of the background at the rate of 60 deg/s (10 rotations per min).

Then, to modulate attention, the second parameter was set for how the observer viewed the stimuli, either fixation of the eye or random viewing conditions. In the fixation condition, observers were asked to fixate their eyes on the test and standard patches. We would like to mention that in the target-motion condition, fixation means the pursuit eye movement to the moving target, and it does not mean that the eye is not moving. In the random viewing condition, the observer was asked to explore the entire stimulus. The observers were instructed before the experiment as follows: in the random condition, “make a paper match and explore the screen by looking around at the surrounding colors; move your gaze to different parts of the scene.” In the fixation condition, “Make a paper match and fixate your eyes on the test and standard patches during the matching task, even if the patches move vertically” [31,32].

At the beginning of the session, the observer adapted to the D65, $25 cd / m^{2}$ white screen for 5 min, binocularly. Before starting trials, the observer adapted to the backgrounds for 5 min with the colored illuminant for one eye and with the D65 illuminant for the other. After adaptation, the observer started to conduct the paper match for static targets, motion targets, and rotation targets. In the first part of the experiment, the observer was instructed to explore the surround in the random viewing condition. In the second part, the observer was instructed to fixate eyes on the target in the fixation condition. The one trial continued until the observer completed the matching and the averaged time for one trial was about 1 min. There were 12 trials performed in pseudo-random order for each condition, corresponding to the 12 different surfaces of the central test patches. Each matching point was averaged over six sessions for one observer. In three sessions, the left side was the standard pattern illuminated by D65, and the right side was the test pattern illuminated by the test illuminant; the reverse arrangement was used in the other three sessions. Each session took about 90 min.

3. RESULTS

A. Color Constancy Performance

1. Color Constancy Index

For color normal observers, the color constancy index proposed by Arend et al. [32] was used to quantitatively evaluate the degree of color constancy. The index $C I$ is defined as

C I = 1 - \frac{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 color constancy would be perfect in the sense of predicting the color appearance of the patch under the test illumination, and

a

denotes the Euclidean distance from the standard point (the chromaticity coordinates of the patch under the D65 illuminant) to the theoretical point. In this study, the distance was defined as the difference of

u^{'}

and

v^{'}

values (

\sqrt{Δ u^{' 2} + Δ v^{' 2}}

) in two-dimensional CIE 1976

L^{*} u^{*} v^{*}

color space, since we considered the chromatic difference and only the ratio of the distance was required in Eq. (1). 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.

Figures 5–8 show constancy indices of the color normal observers on 12 color patches under red, green, blue, and yellow illuminants, respectively. The results indicated that the target-motion did not improve color constancy, and it is different from the result of the achromatic matching experiment [21]. Each result could be influenced by different conditions of cone adaptation and illumination estimation effect (see Discussion section). Figure 9 shows mean constancy indices of six observers and 12 color surfaces with random viewing (denoted as “eye-free”) and fixation (denoted as “eye-fix”) conditions under red, green, blue, and yellow illuminants. The tendency of the data will be analyzed in the next subsection.

Fig. 5. Constancy indices of color normal observers on 12 color patches under the red illuminant with eye-free (random viewing) conditions and eye-fix (fixation) conditions. Light color, dark color, and patterned bars represent the data of target-static, target-motion, and target-rotation conditions, respectively. The value for each color patch was averaged over six observers. Error bars represent the $\pm 2 SEM$ .

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Fig. 6. Constancy indices under the green illumination. All other details are the same as in Fig. 5.

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Fig. 7. Constancy indices under the blue illumination. All other details are the same as in Fig. 5.

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Fig. 8. Constancy indices under the yellow illumination. All other details are the same as in Fig. 5.

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Fig. 9. Mean constancy indices of six observers and 12 color surfaces with eye-free (random viewing) and eye-fix (fixation) conditions under red, green, blue, and yellow illuminants. Light, dark, and patterned bars represent target-static, target-motion, and target-rotation conditions, respectively. Error bars denote $\pm 2 SEM$ .

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2. Three-Way Repeated Measures Analysis of Variance (ANOVA)

We conducted three-way repeated-measures analysis of variance (ANOVA) in parameters of the attention control (random viewing and fixation conditions) and target motion status (target-static, target-motion, and target-rotation conditions) with test illuminants (red, green, blue, and yellow) within subject factors. Foremost, we found significant main effects of target motion status ( $F = 11.808$ , $p = 0.021 < 0.05$ ). Attention control by gaze-state ( $F = 5.203$ , $p = 0.071 > 0.05$ ) and test illuminants ( $F = 6.043$ , $p = 0.087 > 0.05$ ) were not significant. Note that there was no significant interaction.

Multiple comparisons using Bonferroni’s correction (significance level: 0.05) revealed that color constancy indices, $C I$ s, were significantly higher in the static and rotation conditions than those in the motion condition. Color constancy of the four illuminants were in the order of green > red > yellow > blue, and there were significant differences between red and blue, green and blue, and green and yellow comparisons. Statistical analyses are summarized in Table 2.

Table 2. Multiple Comparisons Using Bonferroni’s Correction (Significance Level: 0.05)

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B. von Kries Model Prediction at the Receptoral Stage

We compared the performance of static, motion, and rotation targets with random viewing (eye-free) and fixation (eye-fix) conditions by using the von Kries model [2], and calculated the predicted L-, M-, and S-cone responses at the photoreceptor stage. The process and equations of the model analysis are described in Appendix A; further details for the analysis were explained in our previous study [15]. Figure 10 shows the comparison between L-cone responses to 12 color patches matched by the color-normal observers (ordinate) and those predicted by the von Kries model (abscissa) in red, green, blue, and yellow illuminants. Figures 11 and 12 show the comparison in M- and S-cone responses, respectively. In Figs. 10–12, the prediction under D65 (denoted by solid black lines) and that of the perfect adaptation line (denoted by diagonal dotted lines) were also presented. 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 the diagonal line; 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 (no constancy) were close to the diagonal line (perfect adaptation). In the case that the response predicted by the von Kries model and those under D65 illumination overlapped, the diagonal line indicates both perfect adaptation and no adaptation [15]. The fitted lines (denoted by the red, green, and blue lines for target-static, target-motion, and target-rotation conditions, respectively) to the data points were obtained by multiplying the von Kries model prediction by the constant coefficient to minimize the sum of the squared error between the prediction and the match. The slope coefficient $k$ and coefficient of determination $R^{2}$ for the fitted lines in each panel are shown in Table 3. It is well known that the von Kries-type adaptation is often modest under certain experimental conditions for a variety of reasons, meaning that the adaptation effect is not necessarily at maximum (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, the prediction under D65, and that of the perfect adaptation line.

Fig. 10. Comparison between L-cone matched results by observers (ordinate) and those predicted by the von Kries model (abscissa) of 12 color patches under red, green, blue, and yellow illuminants in eye-free (top four panels) and eye-fix (bottom four panels) conditions. Diagonal dotted black lines indicate perfect von Kries-type adaptation. The red square, green circles, and blue triangles are matched results of target-static, target-motion, and target-rotation conditions, respectively. The red, green, and blue lines denote the best fits of target-static, target-motion, and target-rotation conditions, respectively. Each data point was averaged over six observers and six sessions. Black lines denote the prediction under D65 (see text for details).

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Fig. 11. Comparison between the M-cone matched results (ordinate) and those predicted by the von Kries model (abscissa). All other details are the same as in Fig. 10.

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Fig. 12. Comparison between the S-cone matched results (ordinate) and those predicted by the von Kries model (abscissa). All other details are the same as in Fig. 10.

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Table 3. Slope Coefficient $k$ and Coefficient of Determination $R^{2}$ for Fitted Lines in Figs. 10, 11, and 12

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Figures 10–12 and Table 3 show that with the red illuminant, the L-cone response could not be explained even qualitatively by the simple application of the von Kries model alone; however, color constancy in the red illuminant was reasonably good, and the matched points were sufficiently close to the reflectance model data, meaning that the observers were good at illumination estimation. The comparison of the results between eye-fix and eye-free conditions, M- and S-cone responses were substantially higher in the eye-fix condition than those matched results in the eye-free condition. It indicates that fixation might cause the increase in M- and S-cone responses; one reason for this may be explained by adaptation. The adaptation response of L-cones was suppressed under the red illuminant, but M-cones could still be excited. Interestingly, the deviation of the S-cone matching was mainly caused by the strong S-cone stimulation in some of the color patches (5P5/6, 7.5PB5/6, and 10B5/6): this phenomenon agrees with those of previous studies [5,6,13,15], and this deviation increases especially in the target-motion condition with the eye-fix.

With the green illuminant, the L-cone response could not be explained even qualitatively by the simple application of the von Kries model alone, again. However, the color constancy performance calculated by the $C I$ as presented in Fig. 9 was even better than that with other illuminants; one possible explanation for this is that the observers could perform better illumination estimation under the green illuminant. In comparison between eye-fix and eye-free conditions, L- and S-cone responses were substantially higher in the eye-fix condition than those in the eye-free condition. It indicates that the gaze of eye-fix increased the L- and S-cone responses in the matched points. It also indicates a stronger adaptation effect; with the green illuminant, the adaptation response of M-cones was suppressed but L-cones were more excited.

With the blue illuminant, the matched L-, M-, and S-cone responses approached diagonal lines and M-cone response had a slightly upward deviation; this deviation was smaller in the eye-fix condition than that in the eye-free condition. With the yellow illuminant, the matched L-, M-, and S-cone responses approached diagonal lines, but S-cone responses had a slightly upward deviation, and this deviation was smaller in the eye-free condition than that in the eye-fix condition.

The analysis of cone responses indicated that the influence of motion on color constancy was relatively small, regardless of the cone-type, the color of illuminant, and gaze.

C. von Kries Model Prediction at the Postreceptoral Stage

In the next step, we applied the von Kries type of adaptation to the response of the postreceptoral stage. We first calculated the simple adaptation on each cone type separately by the von Kries model, and then calculated the red-green chromatically opponent channel response represented by ( $L - 2 M$ ) and the blue-yellow chromatically opponent channel response represented by [ $S - u_{n} * (L + M)$ ] using postadapted cone responses. In the blue-yellow response, we introduced a constant coefficient for neutralization to white, $u_{n}$ to make the balance of blue-yellow equivalence. In this study, $u_{n}$ was set to 0.0175 for all illumination conditions; more details were explained elsewhere [15]. We compared the color constancy performance in static, motion, and rotation conditions with eye-free and eye-fix conditions.

Figure 13 shows the comparison between the matched L-2M response by observers (ordinate) and the predicted L-2M response by the von Kries model (abscissa) in red, green, blue, and yellow illuminants with eye-free and eye-fix conditions. If the opponent color signals would be described in terms of simple adaptation on each cone type separately, the data points would fall on the diagonal lines, as explained in the previous subsection. In Fig. 13, we also calculated the prediction by the perfect illumination estimation effect as denoted by gray lines. Although the illumination estimation effect should be complex and has not been clear yet, as described in the Introduction, it can be simplified if only the “perfect” illumination estimation effect is considered, which should aim the “perfect” matching point calculated simply from the spectral radiance of the illumination and the reflectance of the surface. We defined the reflectance model to predict this perfect matching point as described in Appendix A. In red and green illuminants, the data points almost conform to the diagonal lines with just a small deviation from them. In red and green illuminants, the no-adaptation prediction (denoted by black lines), which is the cone response under D65 illuminant and indicates no color constancy, was different from the perfect-adaptation prediction (denoted by the diagonal lines); the data in red and green illuminants indicate that opponent channel responses reflected color constancy.

Fig. 13. Comparison between the matched L-2M response by observers (ordinate) and the predicted L-2M response by the von Kries model (abscissa) under red, green, blue, and yellow illuminants in eye-free (top four panels) and eye-fix (bottom four panels) conditions. Diagonal dotted black lines indicate perfect von Kries-type adaptation. The red squares, green circles, and blue triangles are the matched result of static, motion, and rotation conditions, respectively. The red, green, and blue lines denote the best fits of static, motion, and rotation conditions, respectively. Black lines denote the prediction under D65, and gray lines denote the prediction by perfect illumination estimation effect (see text for details). Each data point was averaged over six observers and six sessions.

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Although the difference was not sufficiently large, in the red illuminant, the matched points were on the gray line rather than on the diagonal line in both eye-free and eye-fix conditions. This indicates that the effect of illumination estimation must contribute to color constancy; it also means that the red-green opponent signal matched by the observers tended not to follow the von Kries model adaptation, but looked more like the reflectance model [15]. This result is consistent with results reported in the literature [6]. In blue and yellow illuminations, the reflectance data overlapped the diagonal line, meaning that it is impossible to separate the mechanism between the von Kries type of adaptation and the illumination estimation effect.

Figure 14 shows the matched blue-yellow chromatically opponent color responses for 12 color patches. The color patches of 5P5/6, 7.5PB5/6, and 10B5/6 had much larger responses with the red illuminant, which resulted in a relatively large deviation from the diagonal line, than other stimuli in the eye-fix and eye-free conditions under red, green, and yellow illuminations. Except for these three patches, the matched data for all observers under all illumination conditions could be well fitted by the von Kries model, meaning that the blue-yellow opponent responses can be predicted well by the von Kries model; this result is consistent with the previous study [15]. Table 4 shows the slope coefficient, $k$ , and coefficient of determination $R^{2}$ for each fitted line in Figs. 13 and 14.

Fig. 14. Comparison between the matched blue-yellow $S - u_{n} (L + M)$ response (ordinate) and the predicted blue-yellow by the von Kries model (abscissa). All other details are the same as in Fig. 13, except gray lines were not shown.

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Table 4. Slope Coefficient $k$ and Coefficient of Determination $R^{2}$ for Fitted Lines in Figs. 13 and 14

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In the analysis of the postreceptoral responses, the influence of motion on color constancy was still small, regardless of the cone types, the color of illuminants, and gaze. It may reflect that the effect of target motion is small compared to the von Kries type of adaptation. However, the possibility still remains that color constancy in this stimulus configuration is mainly determined by the illumination estimation effect like the reflectance model, and the effect of motion on adaptation and on illumination estimation tends to cancel each other. This possibility will be discussed in the next subsection.

D. Correlation between von Kries Model and Reflectance Model

In previous subsections, the von Kries model was used to explain the adaptation of the photoreceptors (cones) and/or postreceptoral process (red-green and blue-yellow opponent channels) at the retina and early stage color processes [i.e., lateral geniculate nucleus (LGN) and the primary visual cortex], and the reflectance model was used to explain cognitive and/or computational mechanisms occurring at a higher level; these two strategies are considered the two main factors contributing to color constancy [1,15] and could partly explain the matching data.

Our experiments were set up to investigate whether motion and attention would influence the contribution of retinal adaptation and high-level cognitive mechanisms (illumination estimation effect) to color constancy. Thus, we quantitatively compared the distance of the matched points of the results to the predicted points by the von Kries model to the distance of the matched points to the predicted points by the reflectance model. We calculated the distance, $D_{von Kries}$ , which is from matched points to the von Kries model prediction points and the distance, $D_{reflectance}$ , which is from the matched points to the reflectance model prediction points:

D_{von Kries} = \frac{1}{N} \sum_{i = 1}^{N} \sqrt{{(L_{i, Match} - L_{i, v K})}^{2} + {(M_{i, Match} - M_{i, v K})}^{2} + {(S_{i, Match} - S_{i, v K})}^{2}},

D_{reflectance} = \frac{1}{N} \sum_{i = 1}^{N} \sqrt{{(L_{i, Match} - L_{i, Ref.})}^{2} + {(M_{i, Match} - M_{i, Ref.})}^{2} + {(S_{i, Match} - S_{i, Ref.})}^{2}},

where

N = 12

;

L_{i, Match}

,

M_{i, Match}

, and

S_{i, Match}

are the matched results,

L_{i, v K}

,

M_{i, v K}

, and

S_{i, v K}

are the predicted points by the von Kries model,

L_{i, Ref.}

,

M_{i, Ref.}

, and

S_{i, Ref.}

are the predicted points by the reflectance model data;

i

represents the 12 test color patches. The results are shown in Fig. 15.

Fig. 15. Distance of matched points to the predicted points by von Kries model and reflectance model with four illuminants. Dark color and patterned bars denote the distance between matched points to the von Kries model and the reflectance model predictions, respectively. The red, green, blue, and yellow bars denote red, green, blue, and yellow illuminants, respectively. Error bars represent $\pm 1 SD$ .

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Figure 15 shows that with the red and green illuminants, the matched points were much closer to the reflectance model prediction points in the eye-free condition. Conversely, in the eye-fix condition, the matched points in the target-motion and target-rotation conditions were closer to the von Kries model prediction points. With the blue and yellow illuminants, the matched points were closer to the von Kries model prediction points in both eye-free and eye-fix conditions. It can be implied that since the blue and yellow illuminants along the daylight locus were familiar to the observers, color constancy by adaptation could be reasonably sufficient, and the observers felt the color appearance of the right and left sides were close to each other. Thus, the observers weakly conducted the illumination estimation, causing the matched points to be closer to the von Kries model prediction points. This result conforms to the previous study, which points out that the illumination changes along the daylight locus are primarily mediated by the S-cone responses; the human color system may have developed a higher sensitivity along the blue/yellow dimension [1].

In the eye-free condition, however, with the red and green illuminants that are not along the daylight locus, the observers felt that the color appearance of right and left sides was much different, and the observers needed stronger illumination estimation, which involves a higher-level mechanism. The observers had to make matches under the color illuminants that would mainly reflect possible mechanisms of the statistical operation of the scene [16,17] or the illuminant-by-illuminant estimation strategy [18–20]. This resulted in the matched points being closer to the reflectance model prediction points.

4. DISCUSSION

A. Mathematical Model Comparison

During the experiments, the observers reported that the color difference of the two patterns was not the same between the static and motion conditions in the eye-fix condition. Figure 16 shows the conceptual simulation of the perceived appearance in the haploscopic view. In the static and eye-fix conditions (left side), the observer felt the color of the two sides was more different, since the fixation on the static target and the background caused less modulation in the retinotopic (local) adaptation and rather caused more attention to the background colors; the adaptation effect became weaker, but the illumination estimation effect could be stronger and dominant, causing better color constancy overall. On the contrary, in the motion and eye-fixation condition (pursuit eye movement) (right side), the observers reported that the color of the two patterns was almost the same, since more modulation of the retinotopic adaptation by the spatio-temporal averaging to the background color and the background color was less attended; the adaptation effect could be stronger but the illumination estimation effect was weaker, causing overall poorer color constancy. According to the phenomenon that the observer perceived and the results of the color constancy shown clearly in Fig. 15 as the difference between eye-free and eye-fix conditions, we conducted a model analysis so as to explain the influence of the eye movements.

Fig. 16. Perceived appearance under haploscopic view in static and motion conditions.

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In this study, we compared performances of Ebner’s computational model and double opponency model. There are various color constancy models as “white patch,” “gray world assumption,” “retinex,” and so on [1]; however, these models are all based on the static-state condition of objects. Ebner’s model is aimed to calculate and compare the color constancy of motion and static objects; its basic theory is similar to the retinex model and the hypothesis of the gray world assumption. It was found that the result of the Ebner’s model calculation conformed to our experimental results. We expect that this is because the Ebner model has considered two factors, illumination estimation and adaptation. In addition, we applied the double-opponency model so as to find the difference of contribution effects of the illumination estimation and adaptation to color constancy, since the double-opponency model is only based on the chromatic adaptation theory to simulate the double opponent (DO) cells. We found that the double-opponency model does not exactly conform to the experiment’s results. It suggests that the adaptation effect cannot explain the entire results of our experiment, and the difference of the color constancy should be mainly ascribed to the illumination estimation.

1. Ebner’s Computational Model

Many potential algorithms have been proposed for color constancy. Land [33] developed the retinex theory of color constancy; then it was developed to numerous computational approaches, as the white-patch retinex algorithm [34] and the gray world assumption [35]; both algorithms assume that the illumination of the scene is uniform. Ebner [36] has shown the approaches that estimating the illuminant locally by the gray world assumption, and Ebner [22] also developed a computational model for the color constancy of motion object. In this sub-subsection, Ebner’s model [22] is used to simulate the color constancy of the static-state and motion conditions in four colored illuminants.

In the model, color stimuli were converted from LUV color space to RGB space. The three-channel RGB system corresponds to the retinal receptors, which absorb light in the long-, middle- and short-wave parts of the spectrum. The intensity, $O_{i, retina} (x, y)$ measured by the retinal receptors for three channels of red, green, and blue, $i \in {r, g, b}$ at position $(x, y)$ in the image, is given by

O_{i, retina} (x, y) = \log R_{i} (x, y) + \log L_{i} (x, y),

where

R_{i} (x, y)

is the reflectance, and

L_{i} (x, y)

is the intensity of the irradiance; here the assumption is that even though the illuminant is nonuniform, it varies smoothly over the image. The local space average color

a_{i} (x, y)

is computed by convolution with a kernel function [37]:

a_{i} (x, y) = k \iint O_{i, retina} (x, y) g (x - x^{'}, y - y^{'}) d x^{'} d y^{'} .

The constant, $k$ , is used to normalize the result. Here the kernel function is a normalized two-dimensional Gaussian:

g (x, y) = \frac{1}{2 π δ^{2}} e^{- \frac{x^{2} + y^{2}}{2 δ^{2}}} (δ = 30) .

In the motion condition with the pursuit eye moment, a temporal averaging of local space average color was calculated. Figure 17 (top row) shows the same stimuli observed in the aspect of the retinal receptors. The eyes essentially track the test patch, maintaining it exactly at the fovea, and seven frames were taken to make convolution with the Gaussian spatial filter, as shown at the middle row of Fig. 17. Local space averaged colors, $a_{i} (x, y)$ computed by Ebner’s mathematical model for static-state condition was the center image of the calculation result (Fig. 17, bottom row, left) and that for motion condition was obtained from the temporal average of the sequential seven frame results (bottom row right). The color bias of induction and adaptation may be explained by the local space-averaged color in Ebner’s model.

Fig. 17. Visual stimulus for experiment as measured at the aspect of retinal receptors under motion condition with pursuit eye movement (top row) and the result of convolution with Gaussian spatial filter (middle row). Local space averaged colors, $a_{i} (x, y)$ computed by Ebner’s mathematical model for static-state condition (bottom row, left) and that for motion condition (bottom row right) (see text for details).

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The color constancy descriptor, $O_{i, cc} (x, y)$ is computed by essentially subtracting local space-averaged color, $a_{i} (x, y)$ from the measured color, $O_{i, retina} (x, y)$ , as explained in Fig. 18. This process is like the process of illumination estimation, which needs to offset the bias color caused by the induction and adaptation:

O_{i, cc} (x, y) = O_{i, retina} (x, y) - a_{i} (x, y) = \log R_{i} (x, y) + 1 .

Fig. 18. Color constancy descriptor, $O_{i, cc} (x, y)$ , computed for motion and static-state conditions.

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The estimated reflectance at the position ( $x$ , $y$ ) in the image, $\vec{R} (x, y)$ is now obtained by applying the inverse of the cube root function:

\vec{R} (x, y) = {(\vec{O_{cc}} (x, y))}^{3} .

The color of the illuminant, $\vec{C}$ , can be estimated by taking the average of estimated reflectance over all pixel values of the test patch under the gray world assumption.

Thus, the angular error of the estimated reflectance of a surface was introduced by authors and is defined as

e = \cos^{- 1} \frac{\vec{R_{c}} \vec{R_{D}}}{| \vec{R_{c}} | | \vec{R_{D}} |},

where

\vec{R_{D}}

is the estimated reflectance of the surface under the D65 illuminant, and

\vec{R_{c}}

is the estimated reflectance of the surface under a nonstandard (colored) illuminant. Figure 19 shows the angular error,

e

between the estimated reflectances under the standard D65 illuminant and the nonstandard illuminant, obtained from the modified version of Ebner’s model. From this simulation result, the angular error of motion condition is slightly smaller than that of the static-state condition, and the order of the angular error is blue < yellow < red < green. This computational model result conforms to our experimental results, shown in Fig. 9. Ebner’s model predicts better color constancy in the blue and yellow illuminants compared to that in the red and green illuminants.

Fig. 19. Angular error calculated by Ebner’s model. Light gray and black bars denote static-state and motion conditions, respectively.

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2. Double-Opponency Model

The DO color sensitive cells in the primary visual cortex (V1) of the human visual system have been recognized as one of the important components of color constancy. The hypothesis is based on the responses of DO cells to the color-biased images providing clear information about the scene illuminant [38]. Thus, we use the double-opponency model by Gao [38] to simulate the results of this experiment.

The color information received by the retina is first transferred to L-, M-, and S-cones in a trichromatic way, and propagates to the retinal ganglion cell layer and LGN via single-opponent (SO) neurons. The DO neurons in V1 respond to the color information. We simulated this process by using double-opponency model, as shown in Fig. 20: the image is first transformed to cone LMS space. Then, the signal will be transformed from LMS space to the L − M (red-green), L + M − S (yellow-blue) and L + M (luminance) channels; we call it the SO space. We would like to mention that in the original description of the model [38], L + M was treated as “brightness (b+ and b−)” as “the black-white opponency of luminance;” however, we modified some part of the original description to much the current theory of color vision. The channel inputs from cones are

[\begin{matrix} o_{l m} \\ o_{y s} \\ o_{L +} \end{matrix}] = [\begin{matrix} 1 & - 1 & 0 \\ 1 & 1 & - 1 \\ 1 & 1 & 0 \end{matrix}] [\begin{matrix} l \\ m \\ s \end{matrix}], [\begin{matrix} o_{m l} \\ o_{s y} \\ o_{L -} \end{matrix}] = - [\begin{matrix} o_{l m} \\ o_{y s} \\ o_{L +} \end{matrix}] .

Fig. 20. Double-opponency model for color constancy. First, the stimulus information is converted from LMS-cone to ganglion layer and LGN as red-green (L − M), blue-yellow (L + M − S), and luminance (L + M) channels, and then propagated to DO cells in V1 with convolution by difference-of-Gaussian functions.

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The SO cell was found in ganglion and LGN cells and responds at the lowest spatial frequencies [39]. The SO codes the color information within their receptive fields (RFs) in the way of red-green, blue-yellow, and black-white opponency, although recently black and white have not been considered as color opponents [40,41]. Here we treated it as luminance plus-minus in terms of intensity. The RF spatial structure of each component of a type-II SO cell is described as a Gaussian function,

RF (x, y, δ) = \frac{1}{2 π δ^{2}} e^{\frac{x^{2} + y^{2}}{2 δ^{2}}},

where

δ

is the standard deviation of Gaussian function reflecting the size of RFs. The response of SO cell with red-on/green-off (L + M −) opponency is given by

{SO}_{l + m -} (x, y; δ)

:

{SO}_{l + m -} (x, y; δ) = O_{l m} (x, y) \otimes RF (x, y; δ),

where

\otimes

denotes the convolution.

{SO}_{m + l -} (x, y; γ δ)

is computed for green-on/red-off SO cells, and

{SO}_{s + y -} (x, y; δ)

and

{SO}_{y + s -} (x, y; γ δ)

are for the responses of blue-yellow SO cells,

{SO}_{L +} (x, y; δ)

and

{SO}_{L -} (x, y; γ δ)

are for the response of luminance (intensity)-sensitive cells,

γ

is the parameter to reflect the RF size difference between SO cells in the same channel (but in opposite response), and the signs “+” and “−” denote excitation and inhibition, respectively.

In area V1, most DO cells are orientation-selective for both achromatic and chromatic stimuli, and their RFs would be approximated by a difference-of-Gaussian function [42]. The responses of DO cells are computed as

{DO}_{l m} (x, y) = {SO}_{l + m -} (x, y; δ) + k_{1} \cdot {SO}_{m + l -} (x, y; γ δ),

{DO}_{s y} (x, y) = {SO}_{s + y -} (x, y; δ) + k_{2} \cdot {SO}_{y + s -} (x, y; γ δ),

{DO}_{L} (x, y) = {SO}_{L +} (x, y; δ) + k_{3} \cdot {SO}_{L -} (x, y; γ δ),

where

k_{1}

,

k_{2}

, and

k_{3}

are weights of excitatory and inhibitory responses that control the contribution of the RF surround in each DO cell response, although the weights were originally referred to as “relative cone weights” [38]. Here, we used

k_{1} = k_{3} = 0.3

and

k_{2} = 0.1

. Since the size of RF surround is roughly 3 times larger than the RF center, here we set

γ = 3

, and

δ = 10

. Then, the estimated reflectance,

\vec{R}

, is

\vec{R} = [\begin{matrix} {DO}_{l m} \\ {DO}_{s y} \\ {DO}_{L} \end{matrix}] .

The angular error, $e$ , is defined as Eq. (9), where $\vec{R_{D}}$ is the estimated reflectance under illuminant D65, and $\vec{R_{c}}$ is the estimated reflectance under a nonstandard illuminant. Figure 21 shows the angular error between the estimated reflectance under the standard D65 illuminant and the nonstandard illuminant obtained from the modified DO model. From this simulation result, the angular error of the motion is smaller than that in the static condition, and the order of the angular error is $blue ≒ yellow < red ≒ green$ . Although the color constancy in the motion is better than that in the static condition, the difference is still small. The adaptation effect to the color constancy itself cannot explain the entire results. The difference of the color constancy in this study should mainly be ascribed to the illumination estimation effect.

Fig. 21. Angular error for double-opponency model. Light gray and black bars represent static-state and motion conditions, respectively.

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B. Effect of Motion to Color Constancy

The results of this asymmetric color-matching experiment indicate that the motion of the target could not improve color constancy; rather, the motion generally decreased the color constancy. The random view (free eye movement) increased the color constancy through better attention, which conforms to previous studies [24], but the difference was not statistically significant. The comparison of the results between eye-fix and eye-free conditions indicates that in the random view (eye-free) condition, the observers could pay more attention to the colors of the surround, and color constancy was determined more dominantly by the illumination estimation. Thus, the color constancy in the random view (eye-free) condition was better than that in the fixation (eye-fix) condition. There are several studies showing that the color constancy in the paper match is better than that in the appearance match [6,31,32]; the possible reason is that the illumination estimation effect is stronger in the paper match, and the appearance match is more influenced by the induction and adaptation with the weaker illumination estimation effect. The amount of the color constancy in four illuminants was in the order of green > red > yellow > blue. The difference of the illuminants was significant between red and blue, green and blue, and green and yellow illuminants.

The results in this study have some potential association with the result of previous studies [21,43,44]. Werner [21] found that color constancy was improved when the object moved in an achromatic adjustment experiment. The result might be due to a lower memory contribution, and the illumination estimation effect was little required in the task; thus, the main effect contributing to the color constancy could be the induction and adaptation. Hurlbert [43] reported that the induction of the moving target is stronger than that in the stationary condition. However, in the simultaneous asymmetric matching experiment in this study, the observers could see the two illuminant backgrounds simultaneously; this means that the observer had a very accurate memory of the color in different illuminations, causing the observers to rely more on the illuminant estimation strategy in this situation. In addition, as described in the previous subsection, the target in the motion could have stronger induction and adaptation; it could increase the color bias of the test target. Hurlbert [43] also found that the strength of contrast induction was weakened in the order of blue > yellow > red > green substantially, and the induction of a blue-yellow texture is stronger than a red-green texture. Conversely, in this study, the observer needed to estimate illumination and had to offset the bias color caused by the induction and adaptation; this means that the illumination with high induction and adaptation may involve poorer color constancy. As reported by Pearce and Hurlbert [44], the ability to discriminate highly chromatic illuminations may involve strong estimation effect. The results in this study also correspond to the result in the previous study [44] showing that chromatic illumination discrimination ability of blue is poorest and green is best.

In our study, color constancy depends on both the illumination estimation effect and the von Kries-type adaptation effect; the difference of influences between conditions could be explained by stronger contribution of the illumination estimation effect to the color constancy. However, in most natural environments, the color constancy is more influenced by the adaptation and induction effect with less contribution of the illumination estimation effect. Thus, human color constancy is optimized for blue daylight illumination [44], except in the situation where the observers retain good memory and can do better illumination estimation.

C. Limitation of This Study

In this study, the experiments were conducted in a two-dimensional multielement background, and the results demonstrate some differences from the previous literature [18], which was performed in a three-dimensional background; the three-dimensional background may involve some more information to contribute to the color constancy. The second limitation of this study is the limited range of the color gamut of the CRT monitor. Because the color difference of the four illuminants to D65 was limited, the difference of the matched points predicted by the reflectance model and obtained just by the D65 background (no color constancy) were not obviously different, especially in the blue and yellow illuminants, which may cause the matched points conforming to the von Kries model or to no color constancy; if we will increase the color difference of the illuminants in the simulation, the observers will have to conduct the illumination estimation for the matching. These limitations have to be investigated in the future.

APPENDIX A

This appendix summarizes the two core models of color constancy used in this study: the von Kris model and the reflectance model.

A. von Kries Model

The von Kries model is a chromatic adaptation model that 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; this means that the influence of each object’s surface is not included. For one patch, the postadapted 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 & k_{M, T} & 0 \\ 0 & 0 & k \end{matrix}}_{S, T}) (\begin{matrix} L_{T} \\ M_{T} \\ S_{T} \end{matrix}) = ({\begin{matrix} k_{L, D 65} & 0 & 0 \\ 0 & k_{M, D 65} & 0 \\ 0 & 0 & k \end{matrix}}_{S, D 65}) (\begin{matrix} L_{D 65} \\ M_{D 65} \\ S_{D 65} \end{matrix}),

where

L_{post-adapted}

,

M_{post-adapted}

and

S_{post-adapted}

are the postadapted cone signals obtained independently by the coefficients

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 of the 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; this means that the illumination determines the constants multiplied in each of the three kinds of cones. Thus,

{\begin{matrix} k_{L, T} = \frac{1}{L_{W, T}} \\ k_{M, T} = \frac{1}{M_{W, T}} \\ k_{S, T} = \frac{1}{S_{W, T}} \end{matrix}, {\begin{matrix} k_{L, D 65} = \frac{1}{L_{W, D 65}} \\ k_{M, D 65} = \frac{1}{M_{W, D 65}} \\ k_{S, D 65} = \frac{1}{S_{W, D 65}} \end{matrix},

where

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. (A1) and (A2), 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} \frac{k_{L, D 65}}{k_{L, T}} & 0 & 0 \\ 0 & \frac{k_{M, D 65}}{k_{M, T}} & 0 \\ 0 & 0 & \frac{k_{S, D 65}}{k_{S, T}} \end{matrix}) (\begin{matrix} L_{D 65} \\ M_{D 65} \\ S_{D 65} \end{matrix}),

{\begin{matrix} \frac{k_{L, D 65}}{k_{L, T}} = \frac{L_{W, T}}{L_{W, D 65}} \\ \frac{k_{M, D 65}}{k_{M, T}} = \frac{M_{W, T}}{M_{W, D 65}} \\ \frac{k_{S, D 65}}{k_{S, T}} = \frac{S_{W, T}}{S_{W, D 65}} . \end{matrix}

B. Reflectance Model

The reflectance theoretical model depends on the surface properties of the object and on the light that illuminates the object. The color constancy of the theoretical point would be perfect in the sense of predicting the color appearance of the patch under the test illumination. The $X$ , $Y$ , $Z$ tristimulus values of theoretical color under the test illumination could be calculated by

{\begin{matrix} X = k \int_{λ} R (λ) S R (λ) \bar{x} (λ) d λ \\ Y = k \int_{λ} R (λ) S R (λ) \bar{y} (λ) d λ \\ Z = k \int_{λ} R (λ) S R (λ) \bar{z} (λ) d λ \end{matrix},

k = \frac{100}{\int_{λ} S R (λ) \bar{y} (λ) d λ},

where

S R (λ)

is the spectral radiance of the light source,

R (λ)

is the surface reflectance,

k

is the normalization constant, and

λ

is the wavelength of the light.

\bar{x} (λ)

,

\bar{y} (λ)

,

\bar{z} (λ)

are the color-matching functions of a CIE 1931 standard colorimetric observer. We used Smith–Pokorny’s cone fundamentals as described in the Method section.

Funding

Kochi University of Technology (KUT) (Focused Research Laboratory Support Grant).

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	$x$	$Y$	$Δ E$	$Δ L$ or $Δ S$
R	0.3562	0.3074	53	$+ 4.9 % Δ L$
G	0.2635	0.3516	53	$- 4.8 % Δ L$
B	0.2907	0.2736	45	$- 43.0 % Δ S$
Y	0.3393	0.3923	45	$- 37.5 % Δ S$

Compare Conditions	Mean Difference $Δ C I$	Sig
Static-motion	0.039	0.009**
Rotation-motion	0.031	0.037*
Static-rotation	0.008	0.97
Red-blue	0.169	0.031*
Green-blue	0.195	0.035*
Yellow-blue	0.108	0.171
Red-yellow	0.061	0.073
Green-yellow	0.087	0.033*
Green-red	0.026	0.178
Eye free-eye fix	0.053	0.071

		Red Illuminant
		Eye-Free		Eye-Fix
Status		$K$	$R^{2}$	$k$	$R^{2}$
Static	L	1.657	0.916	1.507	0.932
	M	1.013	0.914	1.206	0.969
	S	1.115	0.978	1.124	0.987
Motion	L	1.492	0.894	1.407	0.945
	M	0.959	0.971	1.323	0.956
	S	1.147	0.982	1.400	0.969
Rotation	L	1.480	0.883	1.370	0.939
	M	1.044	0.933	1.054	0.948
	S	1.171	0.986	1.187	0.984
		Green Illuminant
		Eye-Free		Eye-Fix
Status		$K$	$R^{2}$	$k$	$R^{2}$
Static	L	0.540	0.489	0.595	0.513
	M	0.970	0.992	1.031	0.979
	S	0.981	0.972	1.057	0.960
Motion	L	0.405	0.325	0.689	0.506
	M	0.937	0.976	1.074	0.988
	S	0.917	0.964	0.997	0.977
Rotation	L	0.497	0.381	0.546	0.610
	M	1.004	0.979	0.998	0.957
	S	0.935	0.958	0.996	0.984
		Blue Illuminant
		Eye-Free		Eye-Fix
Status		$K$	$R^{2}$	$k$	$R^{2}$
Static	L	1.005	0.922	0.926	0.891
	M	1.287	0.984	1.163	0.978
	S	1.015	0.981	0.964	0.994
Motion	L	0.998	0.928	0.765	0.899
	M	1.251	0.986	1.220	0.983
	S	0.984	0.986	0.975	0.991
Rotation	L	0.848	0.975	0.856	0.924
	M	1.180	0.980	1.134	0.976
	S	0.987	0.997	0.976	0.993
		Yellow Illuminant
		Eye-Free		Eye-Fix
Status		$K$	$R^{2}$	$k$	$R^{2}$
Static	L	0.871	0.905	0.907	0.938
	M	1.036	0.992	1.084	0.983
	S	1.160	0.983	1.274	0.979
Motion	L	0.761	0.928	0.868	0.851
	M	1.009	0.981	1.138	0.989
	S	1.129	0.981	1.321	0.984
Rotation	L	0.810	0.954	0.925	0.947
	M	0.917	0.972	1.014	0.989
	S	1.208	0.983	1.338	0.987

		Red Illuminant
		Eye-Free		Eye-Fix
Status		$K$	$R^{2}$	$k$	$R^{2}$
Static	L−2M	1.2703	0.9911	1.2640	0.9802
	$S - u_{n} (L + M)$	1.1202	0.9607	1.1233	0.9794
Motion	L−2M	1.2455	0.9787	1.2720	0.9932
	$S - u_{n} (L + M)$	1.1431	0.9778	1.3915	0.9692
Rotation	L−2M	1.2091	0.9843	1.2098	0.9822
	$S - u_{n} (L + M)$	1.1517	0.9816	1.1878	0.9835
		Green Illuminant
		Eye-Free		Eye-Fix
Status		$K$	$R^{2}$	$k$	$R^{2}$
Static	L−2M	0.7184	0.9786	0.7537	0.9813
	$S - u_{n} (L + M)$	0.9807	0.9639	1.0449	0.9500
Motion	L−2M	0.6766	0.9807	0.7522	0.9892
	$S - u_{n} (L + M)$	0.9199	0.9532	0.9831	0.9642
Rotation	L−2M	0.7130	0.9803	0.7933	0.9747
	$S - u_{n} (L + M)$	0.9286	0.9447	0.9838	0.9766
		Blue Illuminant
		Eye-Free		Eye-Fix
Status		$K$	$R^{2}$	$k$	$R^{2}$
Static	L−2M	1.1182	0.9962	1.0187	0.9942
	$S - u_{n} (L + M)$	0.9915	0.9866	0.9435	0.9939
Motion	L−2M	1.1081	0.9968	1.0160	0.9886
	$S - u_{n} (L + M)$	0.9608	0.9910	0.9552	0.9893
Rotation	L−2M	1.1333	0.9935	1.0567	0.9969
	$S - u_{n} (L + M)$	0.9732	0.9966	0.9571	0.9943
		Yellow Illuminant
		Eye-Free		Eye-Fix
Status		$K$	$R^{2}$	$k$	$R^{2}$
Static	L−2M	0.8961	0.9958	0.9306	0.9913
	$S - u_{n} (L + M)$	1.1517	0.9886	1.2551	0.9823
Motion	L−2M	0.8923	0.9944	0.9516	0.9961
	$S - u_{n} (L + M)$	1.1173	0.9909	1.2767	0.9833
Rotation	L−2M	0.8860	0.9889	0.9387	0.9927
	$S - u_{n} (L + M)$	1.2063	0.9926	1.3290	0.9919

	$x$	$Y$	$Δ E$	$Δ L$ or $Δ S$
R	0.3562	0.3074	53	$+ 4.9 % Δ L$
G	0.2635	0.3516	53	$- 4.8 % Δ L$
B	0.2907	0.2736	45	$- 43.0 % Δ S$
Y	0.3393	0.3923	45	$- 37.5 % Δ S$

Possible influences on color constancy by motion of color targets and by attention-controlled gaze

Abstract

1. INTRODUCTION

2. METHOD

A. Apparatus and Calibration

B. Visual Stimulus

C. Observers

D. Procedure

3. RESULTS

A. Color Constancy Performance

1. Color Constancy Index

2. Three-Way Repeated Measures Analysis of Variance (ANOVA)

B. von Kries Model Prediction at the Receptoral Stage

C. von Kries Model Prediction at the Postreceptoral Stage

D. Correlation between von Kries Model and Reflectance Model

4. DISCUSSION

A. Mathematical Model Comparison

1. Ebner’s Computational Model

2. Double-Opponency Model

B. Effect of Motion to Color Constancy

C. Limitation of This Study

APPENDIX A

A. von Kries Model

B. Reflectance Model

Funding

REFERENCES

Cited By

Figures (21)

Tables (4)

Equations (22)

Journal of the Optical Society of America A