Transition type influences the adaptation state under dichromatic illuminations

Shining Ma; Siqi Li; Ruoduan Sun; Yue Liu; Yongtian Wang; Weitao Song

doi:10.1364/OE.478648

1. Introduction

Chromatic adaptation is a perceptual phenomenon that the color appearance of objects can keep approximately constant across different illumination chromaticities. The human visual system can compensate for the color change of objects from different illuminations by rescaling the spectral sensitivity of three cones [1]. Besides adaptation to the mean retinal response of the visual field [2], the adaptation state of the visual system also depends on the spatial or temporal distribution of the adapting field [3,4], leading to other phenomena such as contrast adaptation [5,6] and simultaneous contrast [7,8] which mainly happen at post-receptoral stages. While chromatic adaptation mainly refers to the multiplicative gain control of cone signals operating at the retinal level. As other post-receptoral phenomena are beyond the scope of this study, the experiment design tries to exclude their interferences on visual data.

Chromatic adaptation is quantified by corresponding colors the colors with the same color appearance under different adapting conditions [1]. Over the years, many Chromatic Adaptation Transforms (CAT), such as CMCCAT97 [9], CAT02 [10], CAT16 [11], etc., were developed for spatially uniform illumination. Most CAT models follow the von Kries coefficient law, whereby each of the cone sensitivities (signals) is independently scaled [12]. The typical von Kries CAT has a common structure as shown below:

(1)$$\left( \begin{array}{l} {L_\textrm{c}}\\ {M_\textrm{c}}\\ {S_\textrm{c}} \end{array} \right) = \left[ {D\left( {\begin{array}{{ccc}} {{L_{rw}}/{L_w}}&{}&{}\\ {}&{{M_{rw}}/{M_w}}&{}\\ {}&{}&{{S_{rw}}/{S_w}} \end{array}} \right) + 1 - D} \right]\left( \begin{array}{l} L\\ M\\ S \end{array} \right)$$

L, M, S represent the cone signal of the L, M, S cone, respectively; the subscripts 0 and c refer to the corresponding color under the test and reference illumination, respectively; the subscripts w and rw refer to the color signals of the reference white under the test and reference illumination, respectively; D represents the degree of adaptation. Most CAT models take the adapting luminance as the only parameter in the degree of adaptation formula [1]. Ma et al. proposed a more comprehensive D model integrated with the von Kries CAT (see Eq. (1)), by taking the luminance, chromaticity, and size of the adapting field into account, as shown below [13]:

(2a)$${D_{Ma}}({u^{\prime},v^{\prime},{L_a},fo{v_H},fo{v_V}} )= {f_F}(fo{v_H},fo{v_V}) \times {f_L}({L_a}) \times {f_{Chroma}}(u^{\prime},v^{\prime})$$

(2b)$${f_F}(fo{v_H},fo{v_V}) = 0.867 \cdot [1 - \exp ( - \textrm{ }0.060\, \cdot \sqrt {fo{v_H} \times fo{v_V}} )]$$

(2c)$${f_L}({L_a}) = 1 - \left( {\frac{1}{{3.6}}} \right)\exp (\frac{{ - {L_a} - 42}}{{92}})$$

(2d)$$\begin{array}{l} {f_{Chroma}}(u^{\prime},v^{\prime}) = \exp ( - \frac{{{{[r(u^{\prime},v^{\prime}) + 0.73]}^2}}}{{2{{(0.11)}^2}}} - \frac{{{{[b(u^{\prime},v^{\prime}) + 0.10]}^2}}}{{2{{(1.33)}^2}}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \frac{{0.58[r(u^{\prime},v^{\prime}) + 0.73][b(u^{\prime},v^{\prime}) + 0.10]}}{{0.11 \times 1.33}}) \end{array}$$

With u’, v’ the adapting chromaticity, r(u’, v’) and b(u’, v’) the log-compressed MacLeod Boynton coordinates as a function of the u’, v’ adapting chromaticity, L_a the adapting luminance, (fov_H, fov_V) horizontal and vertical field of view of the adapting field. In Ma’s model, the degree of adaptation is calculated as the multiplication of the size factor (f_F), luminance factor (f_L), and chromaticity factor (f_Chroma). The chromaticity factor was estimated by a bivariate Gaussian function in the log-compressed MacLeod Boynton space with optimized parameters, as proposed by Smet et al. Note that the background in the study of Ma et al. was uniformly illuminated by a single light source with a constant chromaticity.

However, in realistic scenes, it is common to have multiple illuminations varying in luminance or chromaticity. Think for example of daylight mixing with artificial light in an office setting, or interior lighting mixing with the local lighting provided by the table lamp. As these models were derived from several published corresponding color datasets collected under single illumination, they fail to account for chromatic adaptation under multi-illumination conditions. To extend the applicability of the CAT model, we need to figure out how to estimate the adapted white point and the degree of adaptation according to the spatial distribution of illuminations.

Several algorithms have been proposed to estimate the adaptation state in a complex field. One method, called ‘grey world assumption’, is to take the average cone signals of the whole scene as the adapted white point [14]. A similar principle has been adopted by the von Kries - Helson model [15,16], where the rescaling factor of each cone is inversely proportional to the average cone signals of the visual field. However, the study conducted by Ma et al. indicates its failure for predicting the adapted white point of sharp-transitioned dichromatic illuminations [17]. Retinex theory, introduced by Edwin Land, was used to describe three independent spatial processing mechanisms (corresponding to three cones) that explain color constancy and color contrast [14,18]. Later, various Retinex models varying in the number of path directions were formulated from the original theory, which can guide better image reproduction [19–21]. But the Retinex model has some flaws in the physiological implementation and it is never designed as a complete model of image appearance and quality [22]. In addition, it is still unknown if the Retinex model can accurately estimate the color appearance under spatially complex illuminations.

Only a few studies were conducted for investigating chromatic adaptation or color constancy under multi-illumination conditions. Yang et al. conducted an asymmetric matching experiment in the rendered virtual scene with two different illuminants. The visual data were collected at different illuminant integration levels, from complete separation to partial mix [23]. The results show that the cues from two illuminants simultaneously appearing on one side can reduce the degree of color constancy. Ma et al. investigated the effect of the spatial distribution of dichromatic illuminations on the adaptation state using the achromatic matching method [17]. Three color pairs (Red-B2K, Binf-B2K, Red-Green) and two transition types (sharp, gradient) have been selected to compose six different dichromatic illuminations varying in the spatial distribution of chromaticity. Note that the B2K and Binf refer to the Planckian illuminations with CCT at 2000K and infinite K, respectively. The results indicated that the equivalent illumination chromaticity of the gradient-transitioned dichromatic illumination could be estimated by the average chromaticity of the whole adapting field. While for sharp transition, the ‘grey world assumption’ fails to predict the equivalent illumination chromaticity as the adaptation state is always less influenced by the B2K than other illuminations (such as Binf, and Red). While for the Red-Green pair, the equivalent illumination chromaticity doesn’t change with the transition type. The biased contributions of the wo illuminations could be due to their discrepancies in the degree of adaptation, but this assumption has not been experimentally demonstrated yet. In addition, an illumination pair with a sharp transition shows a less complete adaptation compared to those with a gradient transition, except for the B2K-Red pair.

The study of Ma et al. preliminary shows the impact of the transition type on the adaptation state, but due to insufficient color pairs, there are still many questions to be answered. For example, the interaction between the illumination color pair and transition type is still unclear, and we still don’t know how to estimate the equivalent illumination chromaticity of the dichromatic illumination if the spatial distribution is already known. Furthermore, Ma’s work doesn’t provide a model or solution to predict the degree of adaptation which can be embedded in the von Kries CAT. As an extension of previous Ma’s work, this study further investigated how the spatial distribution of illumination influences the adaptation state. The corresponding color data were collected for 30 dichromatic illuminations (15 color pairs × 2 transition types). Then the adapted white point model integrated with a von Kries CAT was proposed by considering the unequal contribution of two sharp-transitioned illuminations as a weighting factor.

2. Experiment design

2.1 Apparatus

Similar to the setup described in Smet et al. [24], this experiment used a 3D stage as the adapting background whose depth cues can enhance the scene realism. The stage was covered by the non-fluorescent white paper, with a field of view (FOV) of 80° (horizontal) × 72° (vertical) from the observer’s position. The stimulus was a grey cube (FOV = 6°) located at the center of the background. The reflectance spectra of the grey cube and white paper were measured by an X-rite 7000A spectrophotometer, as shown in Fig. 1(a). Within the visible wavelength range, their reflectance curves are approximately spectrally flat. The illuminations of the background and the grey cube were independently provided by an Epson-CB-2265U data projector, mounted above the position of the observer. Figure 1(b) shows the spectra of the R, G, B channels of the projector with the peak normalized to 1. The color of the grey cube could be adjusted by changing the RGB drive values of the corresponding pixels. As the FOV of the grey cube is larger than 4°, the CIE 1964 10° color matching functions were adopted for further colorimetric calculations. The projector was calibrated by the Look-Up-Table (LUT) which builds the bidirectional mapping between the RGB drive values and output luminance of each channel. To further improve the calibration accuracy of the background, the RGB drive values of the projector illuminating the background were optimized by minimizing the chromaticity difference in terms of u’₁₀v’₁₀ between the measured chromaticity and the target chromaticity. The reflectance spectra of the stimulus and background were measured by a Konica Minolta CS-2000 spectrometer.

Fig. 1. (a). Reflectance spectra of the stimulus (grey cube) and the white background represented by the dashed line and solid line, respectively. (b). The normalized spectra of the three channels (R, G, B) of the data projector, where the maximum radiance of each channel was scaled to 1.

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2.2 Experiment conditions

The background was illuminated by two horizontally-aligned illuminations with different chromaticities, with each one occupying half of the background. Ten different illuminations were selected including a neutral one – equal-energy-white (EEW), four low chromatic ones – Planckian radiators of 3000 K (B3K) and 10000 K (B10K), Green (G_LC), Red (R_LC), and five high chromatic ones – Planckian radiators of 2000K (B2K) and infinite CCT (Binf), Green (G_HC), Red (R_HC), Purple (P_HC). Note that the selected greenish (G_LC, G_HC) and reddish illuminants (R_LC, R_HC) have the hue angle at 20.14° and 164° in the CAM02-UCS space [25], corresponding to the hue angle of the unique green and unique red, respectively. Table 1 summarizes the u’₁₀ v’₁₀ chromaticities of the ten selected illuminations. In total, 15 illumination color pairs were derived from the ten illuminants in this experiment, as divided into four groups: (1). Two illuminations with the same or similar hue, but different chroma, including a high chromatic one and a low chromatic one (B3K-B2K, B10K-Binf, G_LC-G_HC, R_LC-R_HC); (2). Two illuminations including a neutral one and a high chromatic one (EEW-B2K, EEW-Binf, EEW-G_HC, EEW-R_HC); (3). Two high chromatic illuminations with different hues (G_HC-Binf, G_HC-B2K, R_HC-Binf, P_HC-B2K); (4). Two low chromatic illuminations with different hues (B3K-R_LC, B3K-G_LC, B3K-B10K). Note that the four color pairs in the first and second groups correspond to four different hues (yellow, green, red, and blue). The chromaticity distribution of each illumination color pair in the u’₁₀ v’₁₀ chromaticity diagram was plotted in Figs. 2(a), 2(b), 2(c), 2(d), corresponding to Group 1, Group 2, Group 3, Group 4, respectively.

Fig. 2. The measured u’₁₀ v’₁₀ chromaticities of two illuminants in each illumination color pair. The color of each circle is approximately consistent with the color of the corresponding illuminant. The two circles located on the same black dashed line compose of an illumination color pair. (a). Group 1 including B3K-B2K, B10K-Binf, G_LC-G_HC, R_LC-R_HC color pairs. (b). Group 2 including EEW-B2K, EEW-Binf, EEW-G_HC, EEW-R_HC color pairs. (c). Group 3 including G_HC-Binf, G_HC-B2K, R_HC-Binf, P_HC-B2K color pairs. (d). Group 4 including B3K-R_LC, B3K-G_LC, B3K-B10K color pairs.

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Table 1. The u’₁₀ v’₁₀ chromaticities of the ten illuminations selected for dichromatic illumination experiments

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For each color pair, to further quantify the adaptation state of the dichromatic illumination, the achromatic matching experiments were conducted under three uniform illuminations whose chromaticities in the u’₁₀v’₁₀ chromaticity diagram were equally distributed along the line connecting the u’₁₀v’₁₀ coordinates of the illuminant pairs, as shown in Fig. 2. The average measured luminance of the background (using the CIE 1964 10° color matching functions) is 245 cd/m² with a standard deviation of 8 cd/m². The illuminance of the stimulus is the same as that of the background. The average luminance of the grey cube is 91 cd/m² with a standard deviation of 2 cd/m².

For each illumination pair, there were two non-uniform spatial configurations varying in transition type, including sharp transition and gradient transition. By taking the R_HC-Binf color pair as an example, Fig. 3 presents the two types of dichromatic illuminations. The relative background areas of the two horizontally aligned illuminations are the same (50/50) with the transition region located at the center of the field of view. The gradient transition region occupies one-third of the illuminated adapting field where the illumination chromaticity linearly changes from one illuminant (I1) to the other (I2) in the u’₁₀ v’₁₀ chromaticity diagram. Note that the relative position of the grey cube (stimulus) is coincident with the transition area. In total, there are thirty non-uniform dichromatic illuminations.

Fig. 3. Non-uniform background scenes (adapting fields) under the dichromatic illuminations (R_HC-Binf), with a centrally located stimulus. Each illuminant in one pair illuminates 50% background areas. (a) Two horizontally aligned illuminants with a gradient transition which occupies one-third of the illuminated field. (b) Two horizontally aligned illuminants with a sharp transition in the center.

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2.3 Experiment procedures

Before the experiment, the experimenter gave a brief instruction on the experiment task to the observer. Then the observer sat in front of the adapting scene, at a horizontal distance of approximately 1 m from the grey cube. After adapting to the dark environment for 5 min, the formal experiment started and a combination of the background illumination and the starting point was presented. Before performing the matching task, the observer was required to adapt to the illumination condition for 45 s [26,27]. To ensure an unbiased adaptation state, the observer was asked to stare at a black moving spot whose track was symmetrically distributed across the scene. Therefore, during the adaptation period, the observer adapted to each illuminant chromaticity for the same amount of time. When the moving black spot disappeared, the observer can start adjusting the color of the stimulus until it appears neutral grey by using the keyboard. The four arrow keys represent the four directions along the coordinate axis of the u’₁₀ v’₁₀ color diagram. Note that the type of match that observers were asked to make is color appearance match, instead of a surface match. When the observer made a satisfactory match, the illumination changed to another combination of the background illumination and starting point. The visual experiments have been divided into four sessions and each session includes both dichromatic and uniform illuminations. After the observer finished all the matches in one session, the radiance spectra of visual matches were measured by a Konica Minolta CS-2000 spectrometer. The order of four sessions was randomized for each observer. To avoid the matching bias from the initial color [28], the achromatic matching task was repeated from three highly chromatic starting points (red, green, blue) with a symmetrical distribution centered at the EEW chromaticity.

2.4 Observers

Ten observers (5 males and 5 females) with the age of either twenty-two or twenty-three years participated in the experiment. Participants were graduate or undergraduate students recruited from Beijing Institute of Technology. They all have normal color vision, as tested by the Ishihara 38-plate test.

3. Analysis

3.1 Observer variability

The mean color difference from the mean (MCDM) in u’₁₀v’₁₀ space was used to evaluate the inter-observer and intra-observer variability for all the illumination conditions [29]. For each observer, the chromaticity of the visual match under each illumination was calculated as the average chromaticity of the matches made from three starting points. The inter-observer variability for each illumination was calculated as the mean color difference between the visual match of individual observer and the mean visual match averaged over ten observers. For each illumination and each observer, the intra-observer variability was estimated as the mean color difference between the visual match from each starting point and the mean visual match over the three starting points. Note that the intra-observer variability for one illumination was obtained as the mean MCDM values of all ten observers. A larger MCDM value signifies a larger inter-observer or intra-observer variability.

Table 2 summarized the mean intra- and inter-observer MCDM values under 30 dichromatic illuminations and the corresponding uniform backgrounds. For each illumination pair, there are three uniform backgrounds, denoted as U1, U2, U3, whose chromaticities linearly change from that of I1 to that of I2. The mean inter-observer MCDM values of the gradient-transitioned and sharp-transitioned dichromatic illuminations in four groups are (0.010, 0.008, 0.009, 0.010) and (0.010, 0.009, 0.009, 0.009), respectively; the intra-observer MCDM values are (0.012, 0.013, 0.011, 0.011) and (0.014, 0.011, 0.010, 0.013), respectively. There is no substantial difference in inter- and intra-observer variability between the two transition types.

Table 2. The inter- and intra-observer variability in terms of MCDM in the u’₁₀v’₁₀ chromaticity diagram under each illumination condition. For each illumination pair, the two dichromatic illuminations ‘S’ and ‘G’ refer to that with a sharp transition and gradient transition, respectively. The three uniform backgrounds denoted as U1, U2, U3, have chromaticities linearly changing from that of I1 to that of I2. The abbreviation of each illumination pair is formed as ‘I1-I2’. For example, the illumination pair ‘B3K-B2K’ has B3K as I1, B2K as I2.

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3.2 Impact of the transition type on the equivalent illumination chromaticity

As defined in Ma’s previous paper [17], equivalent illumination refers to a single illumination that can achieve the same chromatic adaptation state as the dichromatic illumination. As the additive mixture of any two colors can be represented by a certain point on the straight line connecting the chromaticities of two colors in the u’₁₀ v’₁₀ chromaticity space, the equivalent illuminant chromaticity was assumed to be located on the straight line joining the two points representing the two illuminants. Some studies have suggested that this law is not always valid as the adaptation state could be biased by the preceding adapting illumination [30,31]. Therefore, the presentation order of the illumination was randomized for each observer to counterbalance the bias from the previously adapted illumination. We still keep using the assumption to estimate the equivalent illuminant chromaticity. For each illuminant pair, the equivalent illumination chromaticity was estimated by taking the three visual matches made under three uniform backgrounds as the reference. Note that u’₁₀v’₁₀ chromaticities of the three uniform backgrounds were linearly spaced between those of the two illuminants. As explained in Ma’s paper, the equivalent illumination chromaticity can be determined by a linear interpolation using a ratio derived from the location of its visual match relative to that under two neighboring uniform backgrounds [17]. As the calculation procedure has been thoroughly described in Ma’s paper, it will not be repeated in this paper. The only difference is that the number of uniform backgrounds selected by this study (= 3) is less than that in Ma’s experiment (= 7). The following analysis focuses on how the equivalent illumination chromaticity is interactively influenced by the illumination color pair and the transition type, in preparation for the development of a more comprehensive equivalent illumination chromaticity model.

For each color pair, the mean visual matches and the 95% confidence ellipses in the u’₁₀ v’₁₀ chromaticity diagram under three illuminations with the same chromaticity averaged over the adapting scene but different spatial distributions (U2, S, G) have been plotted in Fig. 4. Fifteen subfigures in Fig. 4 correspond to fifteen color pairs and the four rows correspond to four groups. It can be found that for each color pair in Group 1 and Group 4, the visual match ellipses of two dichromatic illuminations are almost overlapped with that of U2. But for each color pair in Group 2, compared to the confidence ellipse of U2, the confidence ellipse of the dichromatic illumination with sharp transition (S) is closer to the EEW illumination than the other high-chromatic one, especially for the EEW-B2K pair. For the G_HC-Binf pair in Group 3, the three confidence ellipses representing three spatial distributions largely overlapped with each other, while for the other three color pairs in this group, a substantial difference in the visual match between the two transition types can be observed.

Fig. 4. 95% confidence ellipses for three illumination conditions with the same average chromaticity over the scene including two dichromatic illuminations (sharp, gradient) and one uniform illumination (U2). Fifteen subfigures correspond to fifteen color pairs and the four rows correspond to four groups. The gradient color filling the background of each subfigure is roughly consistent with the color represented by its chromaticity coordinate.

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In a color pair, the contribution of I1 to the adaptation state could be quantified in terms of the relative color difference (Δ_meq,r1), defined as the ratio of the chromaticity difference between the equivalent illumination and I1, and that between I2 and I1. If Δ_meq,r1 is equal to 0.50, two illuminants in one color pair have equal contributions to the adaptation state, which follows the grey world assumption. If Δ_meq,r1 is smaller than 0.5, I1 contributes more to the adaptation state than I2; and vice versa. Theoretically, the Δ_meq,r1 value ranges from 0 to 1, but a small fluctuation in visual match induced by inter-observer variability can probably result in a Δ_meq,r1 value beyond the theoretical range. Table 3 summarized the Δ_meq,r1 values of the dichromatic illuminations with two transition types calculated from the mean visual match over 10 observers and the corresponding inter-observer standard error for the fifteen illumination color pairs. It can be observed that the B3K-B2K color pair has a Δ_meq,r1 value smaller than 0 (out of the theoretical range) for both two transition types. This could be due to the small chromaticity difference in the visual match between B2K and B3K (DEu’₁₀ v’₁₀ = 0.0067) which can make the Δ_meq,r1 value sensitive to the inter-observer variability of the achromatic match. It also explains the large standard error of Δ_meq,r1 value for the B3K-B2K color pair.

Table 3. The Δ_meq,r1 values of the dichromatic illuminations with two different transition types for 15 illumination color pairs. The data are presented as a ± b, where a is the Δ_meq,r1 value based on the average visual match over 10 observers and b is the corresponding inter-observer standard error.

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As shown in Table 3, the Δ_meq,r1 values for the 8 dichromatic illuminations in Group 1 always deviate from 0.50 – the equivalent illumination chromaticity equals to the average chromaticity of the whole scene, and they vary with color pair and transition type from -0.53 to 0.89. Note that the inter-observer standard errors of Δ_meq,r1 value for illuminations in this group are substantially larger than that in other groups, especially for the B3K-B2K color pair. In the R programming language version 3.6.1, a generalized linear mixed model (GLMM) from the ‘glmer’ package was adopted to test the differences in Δ_meq,r1 between the four color pairs in Group 1 and two transition types at the 95% confidence level. There are two fixed factors including illumination color pair and transition type. The random factor is the observer (intercept-only). The significant effect of illumination color pair (p < 0.05) and the insignificant effect of transition type (p > 0.05) were confirmed by a GLMM test. There is no significant interaction between the two fixed factors (p > 0.05). For each dichromatic illumination in Group 1, Δ_meq,r1 is insignificantly different from 0.50 (p > 0.05) regardless of the transition type, as indicated by a one-sample Wilcoxon signed rank test. The results suggest that the two illuminants with the same or similar hue angle but different chroma make approximately equal contributions to the adaptation state under a centrally transitioned dichromatic illumination, irrelevant to the transition type.

For all the color pairs in Group 2, the equivalent illuminant chromaticity of the sharp-transitioned dichromatic illumination is always closer to illumination EEW than the other highly chromatic one (Δ_meq,r1 < 0.50). While for the gradient transition, the Δ_meq,r1 values of most color pairs in Group 2 slightly deviate from 0.50 except for the EEW-R_HC pair (Δ_meq,r1 = 0.26) with a large inter-observer standard error. Both the transition type and color pair have a significant effect on Δ_meq,r1 (p < 0.05), as statistically confirmed by a GLMM analysis. No significant interaction between the two fixed factors has been found (p > 0.05). To further test the difference in Δ_meq,r1 between dichromatic illuminations and the ‘U2’ uniform background (Δ_meq,r1 = 0.50), a one-sample Wilcoxon signed rank test was performed. For each gradient-transitioned dichromatic illumination in Group 2, Δ_meq,r1 is insignificantly different from 0.50 (p > 0.05). While for sharp transition, Δ_meq,r1 is significantly lower than 0.50 (p < 0.05) for the EEW-B2K, EEW-G_HC, EEW-R_HC color pairs, but insignificantly different from 0.50 for the EEW-Binf color pair (p > 0.05).

For the G_HC-Binf color pair in Group 3, the Δ_meq,r1 values for two transition types are very close to 0.50, suggesting that the illumination G_HC and Binf have almost equal contributions to the adaptation state. For the G_HC-B2K color pair, the equivalent illuminant chromaticity is always closer to illumination G_HC than B2K, and the illumination bias is more pronounced for the sharp transition than for the gradient transition. For the R_HC-Binf color pair, the large Δ_meq,r1 value indicates that the equivalent illuminant chromaticity is much closer to illumination Binf than R_HC, especially for the dichromatic illumination with a sharp transition. For the P_HC-B2K color pair with a sharp transition, the illumination P_HC has a larger contribution to the adaptation state than the illumination B2K, but it is not the case for the gradient transition. For the visual data in Group 3, a significant impact of the transition type and color pair on Δ_meq,r1 value has been confirmed by a GLMM analysis (p < 0.05). As there is a significant interaction between the transition type and color pair (p < 0.05), a GLMM test was performed for each color pair in Group 3 to test the difference in Δ_meq,r1 between the two transition types. It has been found that the impact of transition type on Δ_meq,r1 is significant for the G_HC-B2K, R_HC-Binf, and P_HC-B2K color pair, but not for the G_HC-Binf color pair. Consistent with the finding in Ma’s study, the B2K illumination tends to have less influence on the equivalent illuminant chromaticity compared to the others. For sharp-transitioned dichromatic illuminations in Group 1, Δ_meq,r1 is significantly different from 0.50 (p < 0.05) for the G_HC-B2K, R_HC-Binf, and P_HC-B2K color pair, but not for the G_HC-Binf color pair (p > 0.05), as suggested by a one-sample Wilcoxon signed rank test. For the gradient transition, Δ_meq,r1 values of the G_HC-Binf and P_HC-B2K color pair are insignificantly different from 0.50 (p > 0.05), while those of G_HC-B2K, R_HC-Binf are significantly different from 0.50 (p < 0.05).

As shown in Table 3, for the dichromatic illuminations in Group 4, it could be found that the Δ_meq,r1 values are close to or slightly smaller than 0.50, indicating a similar or less contribution of the yellowish illumination to the adaptation state than the other one. A GLMM analysis confirms the insignificant impact of transition type on Δ_meq,r1 value (p > 0.05). No significant interaction between the color pair and transition type has been found (p > 0.05). In addition, the Δ_meq,r1 values of dichromatic illuminations in Group 4 with either gradient or sharp transition have no significant difference from 0.5 (p > 0.05), as indicated by a one-sample Wilcoxon signed rank test. The results suggest that the two low-chromatic illuminations with different hues almost equally contribute to the adaptation state under the dichromatic illumination with a horizontal transition in the center, regardless of the transition type.

3.3 Performance of the von Kries CAT

By taking the EEW as the reference illumination in the one-step von Kries CAT, for each color pair, 5 corresponding color pairs can be derived from 2 dichromatic illuminations varying in transition type, and three uniform backgrounds (U1, U2, U3). Note that the equivalent illuminant chromaticity derived from the three uniform backgrounds was adopted as the white point of the dichromatic illumination in a regular von Kries CAT. The X₁₀Y₁₀Z₁₀ tristimulus was converted to cone signals (L₁₀M₁₀S₁₀) using the Hunt-Pointer-Estevez (HPE) matrix [32,33]. The degree of adaptation for each illumination was optimized by minimizing the prediction error in terms of DEu’₁₀v’₁₀ between the visual match and the predicted corresponding color under the reference illumination, and it is denoted as D_optim. Table 4 summarized the D_optim values and the minimized prediction error DEu’₁₀v’₁₀ calculated from the average match of 10 observers and their inter-observer standard errors, under five test illumination conditions for the fifteen illuminant pairs.

Table 4. D_optim and the minimized prediction errors DEu’₁₀v’₁₀ under 5 illuminations (2 dichromatic ones and 3 uniform ones) for the 15 illuminant pairs which were divided into four groups. The reference illumination is the EEW illumination, whose D_optim value is fixed at one. The data are presented as a ± b, where a is the D_optim or DEu’₁₀v’₁₀ calculated by the average visual match of 10 observers, and b is the corresponding inter-observer standard error.

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As shown in Table 4, the D_optim values of dichromatic illuminations do not systematically change with transition type. To further test the impact of illumination spatial distribution on the D_optim value, a GLMM analysis was performed for three illuminations with the same average chromaticity but different spatial distributions (G, S, U2) in each color pair group. The fixed factors include the illumination color pair and spatial distribution. The observer (intercept only) is taken as the random factor. The GLMM test confirms that there is no significant difference (p > 0.05) in D_optim between the three illumination conditions with different spatial distributions (G, S, U2), regardless of the color pair group. And the interaction between illumination spatial distribution and transition type is insignificant (p > 0.05) for each group, as indicated by a GLMM test. The results imply that for dichromatic illuminations, its adaptation state can be well represented by a uniform background with a specific chromaticity (equivalent illumination chromaticity).

For the uniform and dichromatic illuminations, the degree of adaptation can be estimated by the D model proposed by Ma (denoted as D_Ma) which considers the effect of the background chromaticity, luminance, and FOV, as shown in Eq. (2). Note that for the dichromatic illumination, the chromaticity of adapting field is calculated as the equivalent illuminant chromaticity derived from the U1-U3 uniform background experimental results. The relationship between D_Ma and D_optim was presented in Figs. 5(a) and 5(b), corresponding to dichromatic and uniform illuminations, respectively. In each subfigure, the black dashed line represents a linear function y = x. It could be observed that most of the data points are always located above or very close to the black dashed line, indicating that D_Ma always overestimates the degree of adaptation compared to D_optim, no matter for uniform or dichromatic illuminations. Even though several studies have demonstrated that D_Ma can predict the degree of adaptation under uniform illuminations quite well, D_Ma values have a large deviation from D_optim values in the present study, with the root-mean-square-error (rmse) around 0.23, corresponding to a prediction error approximately at 23% of the total scale (D ranges from 0 to 1).

Fig. 5. The relationship between D_optim and D_Ma for all the uniform and dichromatic illuminations in this experiment. Each red filled spot represent one illumination condition. The black dash line represents a linear function y = x. (a). Dichromatic illuminations. (b). Uniform illuminations.

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The experiment method and apparatus used in this study are almost identical to that in the study conducted by Ma et al. where the D_Ma model was derived. The only difference in experiment apparatus between the two experiments is the reflectance of the stimulus surface – the reflectance of the grey cube (R ≈ 20%) used in this experiment is substantially smaller than that in Ma’s study (R ≈ 35%), leading to a lower lightness level of the grey cube. With the reduced lightness level, a given difference in chromaticity corresponds to a smaller perceived color difference [34]. In other words, in the present study, the observer is less sensitive to the chromaticity change of stimulus during the matching task as the lightness of the stimulus is lower than that in Ma’s study. Therefore, the low lightness of the grey cube can lead to a larger inter-observer or intra-observer variability in terms of DEu’₁₀v’₁₀, but it cannot explain the systematic shift in the degree of adaptation for various illuminations. The other possible explanation could be the systematic difference in achromatic settings between two groups of observers recruited from two countries (China and Belgium), which deserves further investigation.

To compensate for the overestimation of D_Ma and better predict the degree of adaptation of the illuminations in the present study, the D_Ma model was modified by adding a multiplicative rescaling factor (c), as given below:

(3)$${D_{Ma - M1}} = c \cdot {D_{Ma}}$$

Where D_Ma-M1 represents the modified D_Ma model, c is the rescaling constant. By taking the D_optim as the benchmark, the rescaling factor c was optimized by minimizing the root-mean-square (rms) of the D prediction error for all illuminations. The optimized c value in Eq. (3) is equal to 0.76, leading to the fitting error of D_Ma-M1 (rmse = 0.113) approximately at 11% of the total scale. But this modification method cannot improve the correlation between the predicted D value and D_optim (the benchmark), and the large dispersion of data points still exists.

4. Model of equivalent illumination chromaticity

From the above analysis, it could be found that for most color pairs in Group 2 and Group 3 with a sharp transition, the Δ_meq,r1 values significantly deviate from 0.5, suggesting unequal contributions of two illuminants to the adaptation state. For sharp-transitioned dichromatic illuminations in Group 2, the adaptation state is always biased by EEW which has a higher degree of adaptation than the other chromatic one. Similarly, for the R_HC-Binf color pair with a sharp transition, its adaptation state is more influenced by Binf whose degree of adaptation is higher than that of R_HC (see Table 4). A similar trend can also be observed for other color pairs except that in Group 1 where the Δ_meq,r1 value is very sensitive to the variation of achromatic matches. It could be preliminarily deducted that the equivalent illumination chromaticity of a sharp-transitioned dichromatic illumination is related to the degree of adaptation ratio of I1 to I2, and the illuminant with a higher degree of adaptation always contributes more to the adaptation state. To further verify this assumption, Fig. 6 shows the relationship between the Δ_meq,r1 value derived from the visual match and the Δ_meq,r1-D value estimated by the ratio of the degree of adaptation between the two illuminants:

(4)$${\Delta _{meq,r1\_D}} = \frac{{{D_{I2}}/{D_{I1}}}}{{1 + {D_{I2}}/{D_{I1}}}}$$

With Δ_meq,r1-D the predicted Δ_meq,r1 value, D_I2 and D_I1 the degree of adaptation for the uniform illuminant I1 and I2, respectively. For illuminant I1 and I2, both D_optim and D_Ma-M1 can be used to estimate their degrees of adaptation. The Δ_meq,r1-D values predicted by D_optim and D_Ma-M1 are denoted as Δ_{meq,r1-Doptim} and Δ_meq,r1-DMa1, respectively. The relationship between the Δ_meq,r1 value derived from the U1-U3 uniform backgrounds and the Δ_meq,r1-D (Δ_{meq,r1-Doptim} or Δ_meq,r1-DMa1) as calculated in Eq. (4) was presented in Figs. 6(a) and 6(b) for the sharp-transitioned and gradient-transitioned dichromatic illuminations, respectively. The error bar of each data point represents the corresponding inter-observer standard error of the Δ_meq,r1 value. Note that the dichromatic illuminations in Group 1 were excluded from Fig. 6 due to their large standard errors in Δ_meq,r1 value. It could be found that for the dichromatic illuminations with a sharp transition, the Δ_meq,r1 value is linearly correlated with Δ_{meq,r1-Doptim} and Δ_meq,r1-DMa1, and their relationship could be fitted by a function given below:

(5)$${\Delta _{meq,r1}} = \exp ( - a \times {({\Delta _{meq,r1 - D}})^b} + a)$$

Where the variable Δ_meq,r1-D could be Δ_{meq,r1-Doptim} or Δ_meq,r1-DMa1 in Fig. 6. Both Δ_meq,r1-D estimated by Eq. (4) and Δ_meq,r1 predicted by Eq. (5) theoretically range from 0 to 1. Note that the linear function was not selected to fit the data as the predicted value of Δ_meq,r1 could be beyond its theoretical boundary (0 ∼ 1). The exponential fitting function is very close to a linear function in the practical working range, as shown in Fig. 6. For sharp transition, the coefficient of determination (r²) values of Δ_{meq,r1-Doptim} and Δ_meq,r1-DMa1 are 0.58 and 0.65, respectively, indicating that the Δ_meq,r1 value can be well estimated by Eq. (5). While for the gradient transition, as shown in Fig. 6, the Δ_meq,r1 value is irrelevant to the Δ_meq,r1-D value determined by the D ratio with a low r² between the Δ_meq,r1 and Δ_meq,r1-D (Δ_{meq,r1-Doptim} or Δ_meq,r1-DMa1) value. According to the analysis in the previous section, for most of the dichromatic illuminations with a gradient transition, the Δ_meq,r1 value is not significantly different from 0.5, consistent with the grey world assumption. Therefore, for the dichromatic illumination with a gradient transition, the Δ_meq,r1 value could be fixed at 0.5, regardless of the color pair.

Fig. 6. The relationship between the Δ_meq,r1 value derived from three uniform backgrounds and the Δ_meq,r1-D value as calculated in Eq. (4). There are 11 red blank-filled circles in each subfigure, as a representation of 11 color pairs in Group 2∼4. The black dashed line represents the exponential function (see Eq. (5)) fitted for the data points. In each subfigure, the coefficient of determination (r²) between the Δ_meq,r1 value and Δ_meq,r1-D (Δ_{meq,r1-Doptim} or Δ_meq,r1-DMa1) is also presented on the top left side. (a). For the dichromatic illuminations with a sharp transition, the subfigures in the first row and the second row have the Δ_{meq,r1-Doptim}, and Δ_meq,r1-DMa1 as the x-axis, respectively. (b). For the dichromatic illuminations with a gradient transition, the Δ_meq,r1 value was plotted against the Δ_{meq,r1-Doptim} and Δ_meq,r1-DMa1 in the subfigure of the first and the second row, respectively.

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The estimation formula of Δ_meq,r1 and equivalent illumination chromaticity (XYZ_Equi) for a dichromatic illumination with the transition area in the center (see Fig. 3) are given below:

(6)$$\left\{ \begin{array}{l} {\Delta _{meq,r1}} = \exp ( - a \times {(\frac{{{D_{I2}}/{D_{I1}}}}{{1 + {D_{I2}}/{D_{I1}}}})^b} + a)\qquad\textrm{ For sharp transition}\\ {\Delta _{meq,r1}} = 0.5\qquad\qquad\qquad\qquad\qquad\qquad\textrm{ For gradient transition} \end{array} \right.$$

(7)$$\left( \begin{array}{l} {X_{Equi}}\\ {Y_{Equi}}\\ {Z_{Equi}} \end{array} \right) = (1 - {\Delta _{meq,r1}})\left( \begin{array}{l} {X_{I1}}\\ {Y_{I1}}\\ {Z_{I1}} \end{array} \right) + {\Delta _{meq,r1}}\left( \begin{array}{l} {X_{I2}}\\ {Y_{I2}}\\ {Z_{I2}} \end{array} \right)$$

With a and b the constants in the fitting function, XYZ_I1 and XYZ_I2 the CIE 1964 XYZ tristimulus of I1 and I2 in an illumination color pair. As D_Ma-M1 has a similar performance in predicting Δ_meq,r1 to D_optim, it can be adopted to estimate the D_I2 and D_I1 value as a function of illumination chromaticity, which determines the Δ_meq,r1 of a dichromatic illumination (see Eq. (6)). Therefore, for any illumination color pair, the proposed model can be used to predict Δ_meq,r1 and XYZ_Equi, instead of collecting data from a psychophysical experiment. Using the degree of adaptation model D_Ma-M1, the constant a and b in Eq. (6) were optimized, to minimize the rms of the prediction error DEu’₁₀v’₁₀ of all the 30 (15 × 2) dichromatic illuminations. The optimized a and b values are 3.11 and 0.37, respectively. The corresponding color under the reference illumination could be predicted by the von Kries CAT (see Eq. (1)) integrated with the degree of adaptation model D_Ma-M1 and the equivalent illumination chromaticity model.

(8)$$\left( \begin{array}{l} {L_\textrm{c}}\\ {M_\textrm{c}}\\ {S_\textrm{c}} \end{array} \right) = \left[ {D\left( {\begin{array}{{ccc}} {{L_{rw}}/{L_{Equi}}}&{}&{}\\ {}&{{M_{rw}}/{M_{Equi}}}&{}\\ {}&{}&{{S_{rw}}/{S_{Equi}}} \end{array}} \right) + 1 - D} \right]\left( \begin{array}{l} L\\ M\\ S \end{array} \right)$$

With L_Equi, M_Equi, S_Equi the cone signals converted from XYZ_Equi. Using the equivalent illumination chromaticity model as proposed in Eq. (6) and Eq. (7), XYZ_Equi could be estimated with the chromaticities of two illuminants (XYZ_I1, XYZ_I2) and the transition type. Then the D value in Eq. (8) was calculated using the D_Ma-M1 model by taking XYZ_Equi as one of the input variables.

Table 5 summarized the mean prediction error over the three or four dichromatic illuminations (the same transition type) in each color pair group for one-step von Kries CAT adopting different equivalent illumination chromaticity (EIC) and D models: (1) a benchmark composed of the optimized degree of adaptation D_optim and the EIC calculated from the visual matches of the U1-U3 uniform backgrounds (henceforth refer to as the D_optim + EIC_U1-U3 model), (2) the degree of adaptation D_Ma-M1 and the EIC derived from the U1-U3 uniform backgrounds (D_Ma-M1+ EIC_U1-U3 model), (3) the degree of adaptation D_Ma-M1 and the EIC model as defined in Eqs. (6)–(7) (D_Ma-M1+ EIC_M model), (4) the degree of adaptation D_Ma-M1 and the EIC model of the grey world assumption where the EIC is estimated as the average chromaticity of the whole background (D_Ma-M1+ EIC_GW model).

Table 5. A summary of the performance (mean DEu’₁₀v’₁₀) of four one-step von Kries CATs varying in EIC and D models for four color pair groups. The four models include D_optim + EIC_U1-U3, D_Ma-M1 + EIC_U1-U3, D_Ma-M1 + EIC_M and D_Ma-M1 + EIC_GW which has been explicitly explained in the text

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As shown in Table 5, it can be found that the von Kries CAT adopting D_Ma-M1 and EIC_U1-U3 model has a higher prediction error compared to the benchmark one (D_optim + EIC_U1-U3). As for the EIC model, when adopting D_Ma-M1 to calculate the degree of adaptation, the von Kries CAT with the applicable EIC_M model has a slightly higher prediction error than the EIC_U1-U3 model for the four groups, indicating the prediction accuracy of the EIC_M model. Compared to the D_Ma-M1 + EIC_GW model, the D_Ma-M1 + EIC_M model performs substantially better for the dichromatic illuminations with a sharp transition, especially in Group 2 and Group 3. While for the dichromatic illuminations with a gradient transition, the D_Ma-M1 + EIC_GW model has a similar performance to the D_Ma-M1 + EIC_M model. The results suggest a failure of the grey world assumption when estimating the equivalent illuminant chromaticity of the sharp-transitioned dichromatic illumination for this dataset. Given its good performance, the EIC_M model provides a preliminary model for estimating the white point of the dichromatic illumination in the von Kries CAT.

Note that the EIC_M proposed in this paper can only be applied to the dichromatic illuminations where I1 and I2 equally occupy the adapting field (50/50) with the transition in the center (as shown in Fig. 3). For a more complex spatial distribution of the illumination chromaticity, the Δ_meq,r1 model as shown in Eq. (5) can be integrated with the spatial weighting factor modeled by a Gaussian model as proposed in [35] when estimating the equivalent illumination chromaticity of a dichromatic illumination. In addition, the impact of the spatial luminance distribution on chromatic adaptation was not investigated in this study, and the validity of this model for adapting fields with a non-uniform luminance distribution needs further examination.

5. Conclusions

To investigate the joint effect of transition type and illumination color pair on the adaptation state, achromatic matching experiments were conducted under various dichromatic illuminations, involving fifteen color pairs. A sharp and gradual horizontal transition between the two illuminations is located right behind the grey cube (stimulus). To further identify the adaptation state under a dichromatic illumination, achromatic matches were also collected under three uniform illumination whose chromaticities are equally distributed between that of the two illuminants in one color pair.

For the illumination color pair with a small chromaticity difference (in Group 1 and Group 4), the equivalent illuminant chromaticity is not significantly different from 0.5, regardless of the transition type. While for the dichromatic illuminations with a sharp transition in Group 2 and Group 3, the equivalent illuminant chromaticity tends to be unequally influenced by two illuminants with different hues, with the contribution ratio strongly depending on the color pair. As for the degree of adaptation of the dichromatic illumination, it can be well represented by that of illumination U2 which has the same average chromaticity of the adapting field as the dichromatic illuminations in the present study. Therefore, it can be concluded that the degree of adaptation of the dichromatic illuminations (except the sharp transition) can be well represented by an equivalent uniform background.

Further analysis indicates that for the sharp transition, the contribution of one illuminant to the adaptation state is positively related to the degree of adaptation ratio of the two illuminants. While for the gradient transition, the equivalent illuminant chromaticity can be estimated as the average of the adapting field. Then an equivalent illuminant chromaticity model (EIC_M model) was proposed using the estimated contribution ratio of two sharp-transitioned illuminants which was modeled by the degree of adaptation ratio. The proposed von Kries CAT model integrating the EIC_M and D_Ma-M1 has a good performance (DEu’₁₀v’₁₀ = 0.0077) for the corresponding color data in this experiment, and it substantially outperforms the von Kries CAT using the average chromaticity of the whole field as the white point of the dichromatic illuminations, especially for those with a sharp transition. To further verify if the EIC_M model can be extended to a non-uniform luminance distribution, more visual experiments need to be investigated under the dichromatic illuminations varying in color pair and luminance distribution.

Funding

National Natural Science Foundation of China (62205018, 62002018, 61960206007).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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	Neutral	Low chroma				High chroma
	EEW	R_LC	G_LC	B10K	B3K	R_HC	G_HC	Binf	B2K	P_HC
u’₁₀	0.2095	0.2307	0.1716	0.1863	0.2534	0.2701	0.1542	0.1771	0.2838	0.2318
v’₁₀	0.4714	0.4779	0.4929	0.4333	0.5213	0.4850	0.5011	0.3988	0.5337	0.3925

		Δ_meq,r1
	Color pair	Sharp transition	Gradient transition
Group 1	B3K-B2K	-0.53 ± 0.75	-0.03 ± 0.59
	B10K-Binf	0.89 ± 0.18	0.70 ± 0.20
	R_LC-R_HC	0.34 ± 0.24	0.22 ± 0.26
	G_LC-G_HC	0.61 ± 0.20	0.72 ± 0.21
Group 2	EEW-B2K	0.18 ± 0.07	0.43 ± 0.08
	EEW-Binf	0.46 ± 0.10	0.61 ± 0.04
	EEW-R_HC	0.18 ± 0.12	0.26 ± 0.14
	EEW-G_HC	0.38 ± 0.05	0.55 ± 0.05
Group 3	G_HC-Binf	0.51 ± 0.04	0.47 ± 0.03
	G_HC-B2K	0.24 ± 0.03	0.34 ± 0.04
	R_HC-Binf	0.74 ± 0.04	0.62 ± 0.04
	P_HC-B2K	0.27 ± 0.04	0.43 ± 0.03
Group 4	B3K-R_LC	0.45 ± 0.15	0.25 ± 0.15
	B3K-G_LC	0.43 ± 0.07	0.43 ± 0.07
	B3K-B10K	0.42 ± 0.10	0.51 ± 0.06

		D_optim					DEu’₁₀v’₁₀ (×10⁻³)
		S	G	U1	U2	U3	S	G	U1	U2	U3
Group1	B3K-B2K	0.42 ± 0.09	0.34 ± 0.06	0.34 ± 0.05	0.26 ± 0.03	0.20 ± 0.02	8.3 ± 2.3	8.2 ± 2.0	8.3 ± 1.4	12.7 ± 1.2	15.6 ± 2.0
	B10K-Binf	0.55 ± 0.03	0.55 ± 0.05	0.57 ± 0.06	0.56 ± 0.04	0.56 ± 0.05	0.7 ± 1.3	1.3 ± 1.7	0.3 ± 1.1	1.6 ± 1.3	0.2 ± 1.6
	R_LC-R_HC	0.42 ± 0.11	0.38 ± 0.13	0.27 ± 0.10	0.45 ± 0.07	0.37 ± 0.06	1.3 ± 2.4	4.7 ± 3.6	1.6 ± 2.4	1.1 ± 2.0	3.5 ± 2.2
	G_LC-G_HC	0.50 ± 0.04	0.51 ± 0.04	0.55 ± 0.06	0.50 ± 0.05	0.50 ± 0.05	2.7 ± 3.5	2.7 ± 2.8	1.1 ± 2.2	1.9 ± 2.7	6.3 ± 2.4
Group2	EEW-B2K	0.43 ± 0.14	0.40 ± 0.10	1.00 ± 0.00	0.37 ± 0.04	0.20 ± 0.02	2.8 ± 2.5	0.5 ± 1.9	0.0 ± 0.0	5.1 ± 1.3	15.6 ± 2.0
	EEW-Binf	0.43 ± 0.09	0.47 ± 0.06	1.00 ± 0.00	0.45 ± 0.06	0.56 ± 0.05	2.1 ± 2.2	1.5 ± 1.7	0.0 ± 0.0	1.7 ± 0.0	0.2 ± 1.6
	EEW-R_HC	0.50 ± 0.13	0.49 ± 0.13	1.00 ± 0.00	0.48 ± 0.05	0.37 ± 0.06	1.0 ± 3.2	0.1 ± 2.9	0.0 ± 0.0	1.7 ± 1.9	3.5 ± 2.2
	EEW-G_HC	0.52 ± 0.08	0.49 ± 0.05	1.00 ± 0.00	0.55 ± 0.04	0.50 ± 0.05	0.9 ± 3.3	4.2 ± 3.2	0.0 ± 0.0	1.6 ± 2.2	6.3 ± 2.4
Group3	G_HC-Binf	0.55 ± 0.04	0.60 ± 0.04	0.50 ± 0.05	0.53 ± 0.05	0.56 ± 0.05	3.4 ± 2.2	3.8 ± 1.8	6.3 ± 2.4	3.0 ± 1.3	0.2 ± 1.6
	G_HC-B2K	0.43 ± 0.08	0.39 ± 0.06	0.50 ± 0.05	0.37 ± 0.05	0.20 ± 0.02	3.5 ± 3.0	1.9 ± 2.4	6.3 ± 2.4	1.9 ± 0.0	15.6 ± 2.0
	R_HC-Binf	0.51 ± 0.06	0.50 ± 0.07	0.37 ± 0.06	0.51 ± 0.06	0.56 ± 0.05	4.8 ± 2.2	3.6 ± 2.7	3.5 ± 2.2	2.4 ± 1.9	0.2 ± 1.6
	P_HC-B2K	0.36 ± 0.05	0.41 ± 0.04	0.54 ± 0.04	0.48 ± 0.05	0.20 ± 0.02	4.7 ± 2.8	0.0 ± 2.6	4.3 ± 1.5	0.2 ± 2.2	15.6 ± 2.0
Group4	B3K-R_LC	0.65 ± 0.08	0.49 ± 0.07	0.34 ± 0.05	0.68 ± 0.09	0.27 ± 0.10	1.2 ± 2.2	5.0 ± 2.3	8.3 ± 1.4	3.4 ± 1.8	1.6 ± 2.4
	B3K-G_LC	0.61 ± 0.06	0.70 ± 0.09	0.34 ± 0.05	0.68 ± 0.08	0.55 ± 0.06	2.7 ± 1.7	0.4 ± 2.4	8.3 ± 1.4	3.1 ± 1.5	1.1 ± 2.2
	B3K-B10K	1.00 ± 0.10	1.00 ± 0.08	0.34 ± 0.05	1.00 ± 0.10	0.57 ± 0.06	9.7 ± 5.3	12.9 ± 5.9	8.3 ± 1.4	12.5 ± 6.2	0.3 ± 1.1

		D_optim + EIC_U1-U3	D_Ma-M1 + EIC_U1-U3	D_Ma-M1 + EIC_M	D_Ma-M1 + EIC_GW
Group 1	Sharp	0.0033	0.0053	0.0065	0.0078
Group 1	Gradient	0.0042	0.0066	0.0084	0.0084
Group 2	Sharp	0.0017	0.0037	0.0034	0.0114
Group 2	Gradient	0.0016	0.0047	0.0058	0.0060
Group 3	Sharp	0.0041	0.0062	0.0085	0.0150
Group 3	Gradient	0.0023	0.0049	0.0081	0.0092
Group 4	Sharp	0.0045	0.0086	0.0070	0.0119
Group 4	Gradient	0.0061	0.0096	0.0135	0.0119

	Neutral	Low chroma				High chroma
	EEW	R_LC	G_LC	B10K	B3K	R_HC	G_HC	Binf	B2K	P_HC
u’₁₀	0.2095	0.2307	0.1716	0.1863	0.2534	0.2701	0.1542	0.1771	0.2838	0.2318
v’₁₀	0.4714	0.4779	0.4929	0.4333	0.5213	0.4850	0.5011	0.3988	0.5337	0.3925

Transition type influences the adaptation state under dichromatic illuminations

Abstract

1. Introduction

2. Experiment design

2.1 Apparatus

2.2 Experiment conditions

2.3 Experiment procedures

2.4 Observers

3. Analysis

3.1 Observer variability

3.2 Impact of the transition type on the equivalent illumination chromaticity

3.3 Performance of the von Kries CAT

4. Model of equivalent illumination chromaticity

5. Conclusions

Funding

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (6)

Tables (5)

Equations (11)

Optics Express