1. Introduction

Chromatic adaptation refers to the ability of the human visual system to (partially) adapt to the retinal stimulus of the adapting field. It is achieved by a heterogeneous group of mechanisms occurring at various visual processing stages [1]. At the retinal level, the related processes comprise multiplicative gain control [2] and subtractive adjustments due to the antagonistic center-surround receptive-fields [3–5]. It is often described by linear mathematical models involving multiplicative gain control, such as the von Kries model [6], and involving subtractive shifts that operate at the receptoral and post-receptoral stages. In addition to adaptation to the mean chromaticity of the visual field, the visual system also adapts to the spatial or temporal contrast properties of the visual field. Simultaneous contrast [7,8] and contrast adaptation [9,10] happen at several stages of the visual pathway. For example, reported sites for simultaneous contrast are the bipolar and ganglion cells [11], as well as the striate cortex [12].

1.1 von Kries chromatic adaptation transform

To quantify the impact of the chromatic adaptation on the perceived color appearance, over the years, many Chromatic Adaptation Transforms (CAT) have been developed to predict corresponding colors (CC) under different illumination conditions. Corresponding colors are colors that look the same under different viewing (adaptation) conditions. Most CATs adopt the von Kries coefficient law [13–16], where the sensitivity (signal) of each cone type is independently scaled relative to the cone signal generated by the time-averaged incident light:

The von Kries – Ives model [17] assumes that adaptation is a function of the cone signals to the least chromatic area in the center of the visual field (which typically corresponds to the white point of the illumination), with the rescaling coefficients given below:

In the von Kries - Helson model [18,19], the renormalization coefficient is taken to be inversely proportional to the average cone excitation generated by the total visual field, as given below:

As a next step, a parameter controlling for the degree of adaptation (D) was introduced into CAT models, such as CMCCAT2000 [14], CAT02 [15], CAT16 [16,20], as follows:

To estimate the chromaticity of the illuminant illuminating the scene, many theories have been proposed. One simple illuminant estimation method is based on the average cone excitation of the entire scene, referred to as the ‘grey world assumption’ [19,22], which is also the intrinsic principle of the von Kries - Helson model. It states that, in general, the average spectral reflectance of the whole visual field is neutral. However, this estimation method sometimes fails because the ‘grey world assumption’ cannot hold under all circumstances, especially when the average reflectance is indeed far from neutral. Another method is based on the ‘bright-is-white’ assumption [21,22], where the highest luminance in a scene, such as from a specular highlight, may serve as the reference. The critical assumption is that the brightest element is indeed dominated by the spectrum of the illuminant, but this also is not always the case.

1.2 Mixed adaptation

If a complex scene is illuminated by a single uniform illuminant, chromatic adaptation is typically modeled with a CAT using the illuminant chromaticity estimated by one of the approaches mentioned earlier (e.g. assuming near neutral average object reflectance in the scene). However, if a second, differently colored, illuminant is added to the scene, which is a common occurrence in many artificially lit environments (i.e. daylight mixing with artificial light in an office setting), there are new questions to be answered, such as: can the observer distinguish the two illuminations? How to evaluate the impact of the two illuminations on the adaptation state? Can the adaptation under such dual illuminant conditions be approximated by that of a single, spectrally and spatially averaged, illumination condition? If so, could the white point of this single illumination provide an adequate illuminant estimator in a von Kries CAT? Finally, how is the degree of adaptation defined and affected under multi-illumination conditions?

The impact of dichromatic illuminations has been investigated in several color constancy studies. Khang and Zaidi [23] conducted a color estimation and an asymmetric matching experiment with a spotlight illuminating chromatic materials presented on a dark, dim and bright background. The results show that when only one illuminant is in the field of view, the color estimation of the spotlight is obviously biased by the chromaticity of the illuminated surface. While if the surround is illuminated by a second light source, the spotlight matches are less influenced by the illuminated surface, especially for the bright surrounding. In an asymmetric matching experiment, Yang et al. [24] investigated color perception in rendered 3-dimensional scenes split in half by an opaque dividing wall, and with the two scene sections illuminated by two different illuminants. The mixing level of the two illuminations ranged from completely separate (no mixing) to partially mixed by varying the height of the dividing wall (isolated wall, high wall, low wall, and no wall). They found that the degree of color constancy decreased when a region on one side of the wall in the rendered scene had cues to both illuminants. Color constancy could be improved when specular cues consistent with the two illuminations were provided. However, none of these studies proposed CAT models to predict corresponding colors under these mixed adaptation conditions.

CAT models for mixed adaptation conditions were, however, developed in several studies focusing on cross-media reproduction, where a mixed adaptation to two illuminants occurs when a self-luminous display is viewed in a lit environment. Hunt [25] conducted an experiment to investigate mixed adaptation when viewing softcopy images on a monitor with a correlated color temperature (CCT) of 6500 K under ambient illumination provided by an incandescent lamp with a CCT of 2900 K. It was found that the adaptation white point moves to the display’s white point as the luminance level of the ambient lighting decreases. In the S-LMS model proposed by Katoh [26–28], the white point of the display under such mixed adaptation conditions was estimated by linearly weighing the cone signals of the reference white of the print and monitor with a complicated weighting function. Sueeprasan and Luo [29] simplified the complicated weighting factor function in the S-LMS model, by adopting the degree of adaptation formula used in the CMCCAT97, CMCCAT2000, CIECAT94 models, and found that the simplified models could perform equally, or even slightly better than the original S-LMS model. Similarly, Seok [30] also proposed a linear model with yet another weighting factor function to predict the white point of the display under the ambient illumination varying with luminance level and chromaticity. Kwak [31] investigated the impact of the CCT of the ambient lighting on the preferred display white for two types of display (emissive transparent and opaque) and proposed a preferred white model which integrates the degree of adaptation based on CIECAT02 [32] and a formula for the adapting white point.

However, all the models mentioned above only consider the impact of the luminance level of the ambient illumination on the adapted white point, or on the degree of adaptation of the mixed adaptation condition, ignoring the chromaticity of the illuminations whose impact on the degree of adaptation has already been indicated by many studies [33–36]. In addition, the models were derived from cross-media matching experiments between soft copy and hard copy images, which does not guarantee usefulness of the models for other applications, such as lighting. Finally, these models were derived based on results from experiments where only one of the two adaptation conditions was mixed (display + ambient). Therefore, these adaptation models cannot be generalized to scenes with two or more illuminations and several illuminated objects under the test adapting conditions (i.e., conditions with the same spatial configuration, but with changed chromaticity of the adapting field). Moreover, further improvements are needed to account for additional factors affecting the degree of adaptation, such as the chromaticity of the illumination conditions and the (relative) sizes of the adapting fields. In this study, we systematically investigated how the spatial distribution of the adapting fields generated by several dichromatic illuminants influenced the adaptation state. We developed a model to determine the mixed adaptation white point and the degree of adaptation and integrated this into a von Kries CAT to predict the corresponding color in dual illumination conditions.

2. Experiment design

2.1 Apparatus

In the experiment, the background scene was a 3D stage covered with non-fluorescent white paper, with a field of view (FOV) of 80° (horizontal) × 66° (vertical) from the observer’s position. The stimulus was a grey cube with a 6° FOV, from the observer’s position, and centrally positioned in the background scene. The setup was similar to the one described in Smet et al. [33,34]. The reflectance factors of the non-fluorescent white background and grey cube were approximately constant over the entire visible wavelength range. A data projector was used as a spatial- and color-tunable light source to provide the background illumination (surrounding the cube) and to simultaneously control the illumination of the stimulus (the grey cube) by adjusting the RGB drive values of the pixels of the calibrated projector. As the FOV of the stimulus is larger than 4°, the CIE 1964 10° cone-fundamental-based color matching functions [37], available in the range between 360 nm and 830 nm, were used for all colorimetric calculations in the analysis. The reflected spectral radiance of the stimulus was measured by a calibrated OceanOptics QE65Pro tele-spectroradiometer at a 2 m measuring distance. At the start of each experiment day, the RGB drive values of the projector to illuminate the background were optimized, such that the measured u’₁₀v’₁₀ chromaticity difference with the target value was less than 0.001. In the experiment design and subsequent analysis of this study, the CIE 1976 u’₁₀v’₁₀ uniform color space was adopted, as it is a uniform color space without involving any intrinsic chromatic adaptation transforms. However, it is important to note that while this color space is considered uniform, it does exhibit a noticeable compression in the yellow region. In other words, equal distances in the u’₁₀v’₁₀ space do not correspond to equal perceived color differences, particularly in the yellow region where changes in chromaticity have a greater impact on perceived color.

A series of achromatic matching experiments were conducted with the background adaptation field either illuminated by two illuminations with different chromaticities and different non-uniform spatial configurations or illuminated uniformly by a single illuminant. In the former condition, three color pairs were selected: Yellow-Red (YR), Yellow-Blue (YB), and Green-Red (GR). Each color pair consists of two illuminations – I1 (Illumination 1) and I2 (Illumination 2), whose u’₁₀v’₁₀ chromaticities are summarized in Table 1. In the latter condition, for each color pair, five u’₁₀v’₁₀ chromaticities were equally distributed along straight lines in the u’₁₀v’₁₀ chromaticity diagram, connecting the chromaticity coordinates of the two illuminants in the spatially dichromatic illumination. As shown in Fig. 1, the Y(ellow) and B(lue) illumination chromaticities are located on the Planckian locus, while the line connecting the R(ed) and G(reen) illumination chromaticities is approximately perpendicular – in the u’₁₀v’₁₀ chromaticity diagram – to the Planckian locus. Matching data for three illuminants pairs were determined, each organized in a separate session. The luminance of the background (calculated from the CIE 1964 10° color matching functions) has a mean value of 167 cd/m² and a standard deviation of 10 cd/m². The reflectance of the grey cube and the background (white paper) corresponds to approximately 35% and 85%, respectively, as measured by a Hunterlab UltraScan Pro colorimeter. The illuminance on the stimulus is the same as that on the background. The average measured luminance of the grey cube is 117 cd/m² and the standard deviation is 7 cd/m². Note that the measured luminance of the grey cube is higher than the luminance calculated from the reflectance ratio between background and grey cube (69 cd/m²), due to the different measurement geometries of cube and background. In the experiment setup, the surface of the background behind the cube was tilted, while the front surface of the cube was almost perpendicular to the measured direction.

 

Fig. 1. The measured chromaticities of all the uniform backgrounds in u’₁₀v’₁₀ diagram. The four bigger filled circles represent the chromaticities of the four illuminants adopted in the three illumination pairs: Y and R, Y and B, G and R. For each illumination pair, the five smaller filled circles represent the uniform backgrounds with chromaticities that are linearly distributed between those of the pairs. The color of each circle is approximately consistent with its chromaticity.

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Table 1. The chromaticities (u’₁₀v’₁₀) of two highly chromatic illuminations (I1, I2) for each color pair selected for the dichromatic illumination experiments.

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For each illuminant pair, there were 12 non-uniform background configurations varying in transition type (sharp/gradient), perceptual relative background area (25/75, 50/50, 75/25), and relative position of the matching stimulus to a transition (coincident, non-coincident). The various backgrounds for the YB pair, grouped per relative background area, are illustrated in Fig. 2. In total, there are 36 non-uniform illumination conditions. The impact of transition type was only tested for the equal proportion of two illuminations (50/50) and with the transition at the center of the stimulus (coincident). In each color pair, there are three levels of illumination proportion (25/75, 50/50, 75/25), corresponding to three different average chromaticities of the adapting field. For instance, when referring to a background ratio of 25/75, it represents a dichromatic illumination with I1 illuminating 25% of the background area and I2 illuminating 75% of the background area, respectively. To further investigate how the spatial distribution of the background illumination affects the adaptation state, two configurations were selected with the transition between the illuminants located at: (1) the center of the adapting field, which is identical to the position of the stimulus (coincident), (2) the peripheral region of the adapting field, which is distant from the stimulus (non-coincident). Note that for the dichromatic illumination with a horizontal gradient transition, the gradient region is always located at the center of the field of view, which occupies one-third of the illuminated area. In this central gradient region, the chromaticity changes linearly from the left illuminant to the right illuminant.

 

Fig. 2. Non-uniform background scenes (adapting fields) under the dual illuminant conditions (Y, B), with a centrally located stimulus. The text at the top of each picture describes the local adaptation field surrounding the grey cube. For example, ‘G12’ and ‘S12’ represent gradient transition and sharp transition from I1 (left) to I2 (right), respectively; ‘I1’ and ‘I2’ represent non-coincident conditions with a uniform illuminant I1 and I2 in the background center (surrounding the grey cube), respectively. Note that ‘G12’ and ‘G21’ are mirror-symmetrical in each subfigure, as are ‘S12’ and ‘S21’. (a). Y and B illuminate 25% and 75% of the scene area, respectively (b). Y and B illuminate 50% and 50% of the scene area, respectively (c). Y and B illuminate 75% and 25% of the scene area, respectively

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If the ‘grey world assumption’ holds, dichromatic illuminations with the same spatial illumination proportions, and hence the same average chromaticity, should have an equivalent effect on the chromatic adaptation state, even when their specific spatial arrangements differ. To further investigate the impact of the local adaptation field surrounding the stimulus on the chromatic adaptation, for each session, the achromatic matching experiment was conducted under several dichromatic illuminations with the same average chromaticity, but with different local adaptation chromaticities. As shown in Fig. 2(a), for the dichromatic illuminations with 25% I1 and 75% I2 (25/75), the local field is either the uniform field illuminated by I2 or the gradient transition from I1 to I2. For the dichromatic illuminations with 50% I1 and 50% I2 (50/50), there are four types of local fields as shown in Fig. 2(b): a uniform field illuminated by I1, a uniform field illuminated by I2, the gradient transition from I1 to I2, and the sharp transition from I1 to I2. For the dichromatic illuminations with 75% I1 and 25% I2 (75/25), as shown in Fig. 2(c), there are two types of the local field surrounding the stimulus: a uniform field illuminated by I1 or the gradient transition from I1 to I2. Note that for each background with an asymmetrical chromaticity distribution, the achromatic matching experiment was repeated twice under the two mirror-symmetrical dichromatic illuminations, such as ‘G12’ and ‘G21’ in Fig. 2(a), to counterbalance the left-right bias.

To determine the equivalent illuminant chromaticity of a uniform background condition that provides the same adaptation state as the dichromatic illuminations, for each illumination pair, a matching experiment was also conducted under seven uniform backgrounds with their u’₁₀v’₁₀ chromaticities evenly distributed along a line connecting the two illumination chromaticities, as shown in Fig. 1. Additionally, an achromatic match under a uniform white background, with the chromaticity of equal-energy-white (EEW), and the same luminance level as other conditions, was collected for each observer to obtain a baseline for the CAT.

2.2 Experiment procedure

Observers were instructed to sit in front of the 3D scene at a horizontal distance of about 1 m from the stimulus. The data projector illuminated the 3D stage and the stimulus. Before the experiment, observers were given a brief instruction of the experimental task and adapted to the EEW illumination of the projector in the otherwise dark room for 2 min [38,39]. They were instructed to make color appearance matches by taking the neutral grey as the internal reference, rather than surface matches. In the former task, observers need to match the color appearance of the stimuli, regardless of other areas in the adapting field. While in a surface match, the observers need to adjust the stimulus until the surface appears to be cut from the same piece of material as that of the Ref. [40]. Then each combination of the starting point and background was presented (illuminated by the projector). Before making each achromatic match, observers adapted to the lighting condition for 45 s to ensure a steady adaptation state [38,39]. To ensure all the observers were exposed to approximately identical adaptation conditions, during the adaptation period, they were required to follow a black dot that moved across the scene in such a way that observers had their gaze and focuses towards the left and right halves of the scene for the same amount of time. When the black dot disappeared from the scene, observers could start to adjust the color of the stimulus (grey cube) until it appeared neutral grey by using the arrow keys of a keyboard to navigate in the u’₁₀v’₁₀ space. Note that the adjustment only changed the chromaticity of the stimulus but kept its luminance (Y₁₀) relatively constant. After a satisfactory match (as determined by the observer), the illumination switched to the next condition, and a new round started. On average, the observers took approximately 1 minute to complete each achromatic matching task. Together with the additional 45-second pre-adaptation period, the total adaptation time for each condition was considered adequate to attain a complete or near-complete state of adaptation [38,39]. To avoid visual fatigue, one experiment session was split into two parts, each part taking approximately one and half hours. During each part of the experiment session, the observer was allowed to take a break whenever they felt tired. After all the matching tasks in one part had finished, the radiance spectrum of each match was measured using a calibrated tele-spectroradiometer. In each part, to avoid order bias, the presentation order of the starting point and background was randomized. Similarly, the order of the six sub-sessions (3 sessions × 2 parts) was randomized for each observer. Note that the illuminants G12 (50/50), G21 (50/50) for three color pairs, EEW, Red, Blue, and Yellow appear in two sessions to test the observer’s repeatability.

For each illumination condition, the matching task was repeated four times, each time starting from a different, highly saturated stimulus chromaticity to avoid a matching bias from the initial color of the stimulus [41]. A red, yellow, green, and blue chromaticity, symmetrically distributed along the hue circle, centered at the chromaticity of the EEW in u’₁₀v’₁₀ diagram, were chosen.

2.3 Observers

Ten observers (4 males and 6 females) with normal vision, as tested by the Ishihara 24-plate test, participated in the experiment. Their average age was 27.5 years with a standard deviation of 3.3 years. Participants were graduate students with a variety of nationalities and cultural backgrounds, recruited from Flemish universities.

3. Analysis

3.1 Variability

Under each dichromatic illumination, the achromatic matches of ten observers follow a multivariate normal distribution in u’₁₀v’₁₀ space, as confirmed by the Mardia’s test [42], with the Skewness and Kurtosis p-value larger than 0.05. The normality of achromatic matches justifies the use of the average value to represent the results. The inter- and intra-observer variability were evaluated for all the illumination conditions in terms of the mean color difference from the mean (MCDM) [43] in u’₁₀v’₁₀ space. A larger MCDM value corresponds to a larger variation, either within (intra) or between (inter) observers. For each illumination, inter-observer variability was estimated by calculating the mean of the u’₁₀v’₁₀ color difference between the matching result of individual observer and the mean matching result over all ten observers. Note that the chromaticity averaged over the match results for the four starting points was taken as the matching result for each observer under a certain illumination condition when calculating the inter-observer variability. For each observer, intra-observer variability in terms of MCDM refers to the mean color difference between the matching results starting from one of the four initial chromaticities and the mean chromaticity over those matching results. Then, the mean was taken of all ten observer intra-MCDM values to obtain an estimate of the average intra-observer variability for one illumination condition.

Table 2 summarizes the mean intra- and inter-observer MCDM values and their standard deviations under 12 dichromatic illuminations and 8 uniform backgrounds for the three illuminant pairs. For the conditions with 50% I1 and 50% I2, the mean inter- and intra-observer MCDM values of the two mirror-symmetrical illuminations with a gradient transition for the three illuminant pairs are (0.010, 0.007, 0.007) and (0.009, 0.010, 0.007), respectively; the values for a sharp transition are (0.011, 0.012, 0.009) and (0.011, 0.013, 0.009), respectively. The significant impact of the transition type on the inter- and intra-variability has been tested with the generalized linear mixed model (GLMM) from the ‘glmer’ package in the R programming language version 3.6.1. For each illuminant pair, the larger inter- and intra-observer variability under the dichromatic illumination (50% I1, 50% I2) with the sharp transition indicates that it is more difficult to get stable and consistent matches compared to the gradual transitions. That may be from the differences in eye movements across the scene during achromatic matching or from short-term after-images appearing during the eye movements. The adaptation state under the sharp transitioned illumination is susceptible to the gaze point as the stimulus is located right on the borderline of the two illuminants. The mean inter- and intra-observer MCDM values of all the 20 illumination conditions for the three illuminant pairs are similar, with values of (0.009, 0.008, 0.008) and (0.009, 0.010, 0.008), respectively. Both the inter- and intra-observer MCDM values are not significantly different between the three color pairs, as tested by a GLMM test.

Table 2. Inter- and intra-observer variability, in terms of the mean color difference from the mean (MCDM) in the u’₁₀v’₁₀ chromaticity diagram, under each adapting condition. The seven uniform backgrounds denoted as U1, U2, …, U7, have chromaticities that linearly change from that of I1 to that of I2. The YR and YB pair have the same illumination ‘U1’, and the YR and GR pair have the same illumination ‘U2’. The EEW illuminations for three color pairs are identical. In the table, the data is presented as a ± b; for inter-observer variability, a is the mean and b is the standard deviation of the color differences from each observer to the mean; for intra-observer variability, a is the mean and b is the standard deviation of MCDM values over all observers.

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The repeatability of the achromatic matches under dichromatic illumination was also checked by the two identical repetitions for G12 (50/50) and G21 (50/50) for three color pairs. The achromatic match for each repetition was calculated as the average of four starting points. The MCDM value of repeatability was calculated in u’₁₀v’₁₀ space and then averaged over ten observers. The mean MCDM value ranges from 0.0015 to 0.0030, which is much smaller than the inter-observer variation, indicating an overall high repeatability of achromatic matches under dichromatic illuminations. But there is still a slight variation in the two matches, and the maximum individual MCDM of repeatability reaches 0.0066, which could be caused by the inconsistent adaptation state for two repeated tasks. To approximate the natural viewing condition, the observer was also allowed to look around the adapting scene when they performed the matching, which could lead to a slight change in the adaptation state of each task round.

3.2 Equivalent illumination chromaticity (EIC)

An analysis was performed to investigate the possibility of representing adaptation to the non-uniform dichromatic illumination scenes by adaptation to a uniform, single illumination scene of a specific chromaticity. This equivalent illuminant chromaticity was assumed to be located on the line connecting the chromaticity of the two illuminants (cfr. additive mixture). If the ‘grey world assumption’ holds, the equivalent illuminant chromaticity should be equal to the average chromaticity of the visual field in the dichromatic illumination. If the von Kries coefficient law holds, the equivalent illuminant chromaticity for the dichromatic illumination and its degree of adaptation can be estimated by minimizing the prediction error between the visual match and the predicted chromaticity to zero. However, past research has shown that the chromatic adaptations of some illuminants with small S-cone excitation (such as yellowish illuminants) have a fairly large deviation from the von Kries law [44], indicating that the von Kries CAT cannot always well predict the visual data. Therefore, minimization of the prediction error using a von Kries based CAT may result in an inaccurate estimation of the equivalent illuminant chromaticity.

For each illuminant pair, the matching results under seven uniform backgrounds with u’₁₀v’₁₀ chromaticities linearly spaced between those of the two illuminants were used as references to estimate and verify the equivalent illuminant chromaticity of the dichromatic illuminations. For example, if the matching chromaticity for a dichromatic illumination is the ‘same’ as that for a uniform background, then its equivalent illuminant chromaticity equals the chromaticity of that uniformly illuminated background. Similarly, if the matching result is located between that of two neighboring uniform backgrounds, its equivalent illuminant chromaticity is somewhere on the line connecting that of the two uniform backgrounds. For the analysis, the equivalent illuminant chromaticity of a dichromatic illumination is determined in four steps, as illustrated in Fig. 3. Firstly, for each illuminant pair, a line was fitted (see the red dashed line in Fig. 3), using a least-squares method, to the u’₁₀v’₁₀ matching chromaticities (averaged over all observers) for the seven uniform backgrounds. Secondly, for each dichromatic illumination, the matching chromaticity (averaged over all observers) was perpendicularly projected onto the fitted line of step 1, with the projected point denoted as u’v’_dichromatic (in the u’₁₀v’₁₀ chromaticity diagram). The seven uniform background matching chromaticities were also projected onto the line and denoted as u’v’_j (in the u’₁₀v’₁₀ chromaticity diagram) with j equal to 1, 2, …, 7 corresponding to ‘U1’, ‘U2’, …, U7’ (as shown in Table 2), respectively. Thirdly, the position of u’v’_dichromatic was specified in terms of the two neighboring projected matches for the uniform background (i.e. u’v’_j and u’v’_j + 1) as the ratio of the chromaticity differences Δ between (u’v’_dichromatic, u’v’_j) to that between (u’v’_j, u’v’_j + 1), i.e. Δ(u’v’_dichromatic, u’v’_j) / Δ(u’v’_j, u’v’_j + 1), correspond to the ratio of d2 to d1 (d2/d1) as depicted in Fig. 3. Fourthly, the equivalent illuminant chromaticity of the dichromatic illumination was derived by a linear interpolation between the u’₁₀v’₁₀ chromaticities of ‘Uj’ and ‘Uj + 1’ using the ratio derived in step three. The achromatic matching results under seven uniform illuminations and the corresponding illumination chromaticities were plotted in Fig. 4 where three subfigures correspond to three illumination pairs.

 

Fig. 3. Illustrates the calculation method for determining the chromaticity of the equivalent illuminant based on the ratio of chromaticity differences by taking the ‘S21’ in YB color pair as an example. The solid red circle and blank-filled red circle represent the achromatic match under uniform and dichromatic illumination, respectively.

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Fig. 4. The u’₁₀v’₁₀ chromaticity distribution of seven achromatic matches and the corresponding uniform backgrounds. Three subfigures represent three illumination pairs. The achromatic match and background were marked in circles and squares, respectively. In each subfigure, the color of each symbol is consistent with that in Fig. 1. Note that the black asterisk in each subfigure represents the EEW illumination.

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The white point of the equivalent illuminant is regarded as the white point of the dichromatic illuminations in a von Kries CAT (cfr. L_w, M_w, S_w in Eq. (4)). For accuracy and verification purposes, the determination of the equivalent illuminant chromaticity values (EIC_U1-U7) in this section depends on the visual matches under a number of uniformly illuminated backgrounds with chromaticities located between those of the two illuminants. However, in real applications, it is necessary to have a model to predict the equivalent illuminant chromaticity, instead of collecting matching data from a psychophysical experiment. In the following sections, the possible factors that can influence the equivalent illuminant chromaticity will be investigated, including left-right bias, transition type, and spatial distribution of the chromaticity of the adapting field. The results of this analysis are then used to guide the development of an equivalent illuminant chromaticity model that can be embedded in a regular von Kries CAT.

3.3 Left-right bias

As mentioned in Section 2.1, for the asymmetrical scenes with a gradient or a sharp transition, the achromatic matching experiments were repeated twice in mirror-symmetrical configurations to counterbalance any potential left-right bias. If this bias does not exist, the matching results under the two mirror-symmetrical scenes should be the same (within matching accuracy). If this bias does exist and the observer pays more attention to the left side, the two scenes will generate different matching results, which are biased towards the left side. Left side visual attention bias [45] has been consistently demonstrated in both perceptual judgment and visuospatial attention tasks, such as the chimeric face task [46], and line bisection task [47]. One widely accepted explanation is that the right hemisphere (controls the left visual field) is more activated than the left hemisphere (controls the right visual field) [48]. And the right hemisphere is dominant for visuospatial processing, which has been indicated by many neuroimaging studies [49–51]. In addition, some studies have shown that reading habits can influence the visuospatial asymmetry - for example, the left-side bias of the right-to-left readers can be reduced compared with the left-to-right readers [52]. Therefore, as the sharp dichromatic illuminations in our study have larger left-to-right chromaticity differences in the narrow field surrounding the grey cube than the gradient conditions, this discrepancy can generate quite different matching results under the two mirror-symmetrical scenes. To determine if such left-right bias exists in our results, the mean (across all 10 observers) of the average (across the 4 starting points) matching points under these asymmetrical scenes have been plotted in the u’₁₀v’₁₀ chromaticity diagram in Figs. 4(a), and 4(b). The three graphs in each subfigure correspond to pairs YR, YB, GR, respectively. The 95% confidence ellipses on the mean matches have plotted as well. Figure 5(a) plots the matching results under the illuminations with different transition types but the same illumination proportion (i.e., 50/50), and Fig. 5(b) plots the matching results under the illuminations with the same transition type (gradient) but different illumination proportions. From Fig. 5(a), it can be observed that for the YR pair, the mean match for the 50/50 sharp transition with Y on the left is indeed closer to the Y chromaticity than that with R on the left, resulting in a left-right chromaticity bias of approximately 0.0062 u’₁₀v’₁₀ units. On the other hand, as shown in Fig. 5(b), for nearly all gradient transitions, the mean matches under the two mirror-symmetrical scenes tend to be closer together (DEu’₁₀v’₁₀ = 0.0011). Similar trends, although less pronounced, can be found for the other two illuminant pairs.

 

Fig. 5. Mean matching results and 95% confidence ellipses for the gradient and the sharp illumination-transition under the three illumination pairs (columns). The results have been split in a graph (a). for illuminations with different transition types but the same illumination proportion (i.e., 50/50), and a graph (b). for the illuminations with the same transition type (gradient) but different illumination proportions. In all graphs, the ellipses are determined based on the average (over all 4 starting points) matches of all 10 observers. The proportions and distributions of the illuminations in the adapting field are as indicated in the legend, where, for example, ‘G12 (25/75)’ refers to the gradient transition that changes from a 25% I1 to a 75% I2 (left to right). In each graph, the seven colorful points represent the matching results, averaged over all observers, of the seven uniform backgrounds. The color of each point is approximately consistent with the chromaticity of the corresponding uniform background.

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Indications for the presence of a potential left-right bias for the sharp transition can also be found when analyzing the matching data for all three illuminant pairs in terms of the absolute u’₁₀v’₁₀ chromaticity differences (Δ_meq,a) and the differences in relative chromaticity difference (Δ_meq,r1) between the equivalent illuminant chromaticities of the two mirror-symmetrical conditions (see Table 3). Note that the relative chromaticity difference Δ_meq,r1 is calculated as the ratio of the chromaticity difference between the equivalent illuminant chromaticity and I1, and that between I2 and I1, and it ranges from 0 to 1. From Table 3, it can be observed that for the gradient dichromatic illuminations, the Δ_meq,a values between two mirror-symmetrical conditions are small or even negligible, i.e. less than or very close to 0.0033 (about the radius of a 3-step MacAdam ellipse close to the Planckian locus which was suggested as one just-noticeable-difference by MacAdam [53]), while they are much larger for the sharply transitioned backgrounds, especially for the pairs containing Y. Similar trends can be observed in terms of the difference in Δ_meq,r1.

Table 3. The summary of the absolute u’₁₀v’₁₀ chromaticity difference Δ_meq,a and the left-right difference in the relative chromaticity difference Δ_meq,r1 (in %) between the equivalent illuminant chromaticities obtained in the gradient and sharp mirror-symmetrical conditions for the three illuminant pairs.

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In the R programming language version 3.6.1, a GLMM from the ‘glmer’ package was used to test the impact of several factors on the equivalent illuminant chromaticity in terms of Δ_meq,r1, such as left-right bias, transition type and illumination pair, at the 95% confidence level. Only the data for the 50/50 conditions with a gradient or sharp transition in the center was included. The fixed factors are left-right bias, transition type, illumination pair, and the random factor is observer (intercept only). To be valid, the GLMM requires the normality of the random effect which was confirmed by a Shapiro-Wilk test [54]. No significant interactions (p > 0.05) with the left-right bias factor were found. Furthermore, consistent with overlapping confidence ellipses shown in Fig. 5, no significant left-right bias (p > 0.05) could be statistically confirmed, despite the quite large indications mentioned earlier for the presence of a left-right bias for the sharp transition. For the remainder of the analysis, as a simplification, the matching data of the gradient and sharp mirror-symmetrical conditions were averaged (i.e. minimize/ignore any left-right impact), leaving eight dichromatic illuminations for each illuminant pair.

3.4 Impact of the transition type

In this section, the impact of the transition type from one illuminant to the other on chromatic adaptation has been investigated using asymmetrical scenes with a 50/50 spatial proportion. The transition occurs in the center of the visual field and consists of two types: a sharp transition and a gradient transition (as shown in Fig. 2(b)). As the background (adaptation) field is equally illuminated by both illuminants, the average chromaticity of the whole scene is equal to the average chromaticity of the two illuminants.

The contributions from the two illuminants to the adaptation state were analyzed in terms of the Δ_meq,r1 value, defined earlier. If Δ_meq,r1 is equal to 0.5, I1 and I2 have equivalent contributions to chromatic adaptation. If Δ_meq,r1 is smaller than 0.5, I1 has a larger impact than I2 on the adaptation state; and vice versa. Table 4 summarizes the Δ_meq,r1 values, calculated from the matching chromaticity averaged over observers, of the dichromatic illuminations with different spatial distributions for the three illuminant pairs. The inter-observer standard error, obtained from the Δ_meq,r1 values of each individual observer are also reported. For all the three illuminant pairs, the equivalent illuminant chromaticities of the dichromatic illuminations with a gradient transition are almost equal to the average chromaticity of the whole scene, and Δ_meq,r1 values are not substantially different from 0.5 (Table 4), which is consistent with the grey world assumption. However, for the dichromatic illuminations with a sharp transition in the center, the results do not always follow the grey world assumption. For example, for the YR pair, the equivalent illuminant chromaticity is much closer to R (Δ_meq,r1 = 0.741), suggesting that R has a contribution almost twice as large as Y. For the YB pair, the equivalent illuminant chromaticity is slightly closer to B (Δ_meq,r1= 0.544), indicating that B contributes a little more than Y. For the GR pair, the equivalent illuminant chromaticity is very close to the average chromaticity of the two illuminations, but slightly biased by R (Δ_meq,r1 = 0.527). Moreover, for the YR pair, Δ_meq,r1 has a larger standard error than for the other two pairs as shown in Table 4, which may result from the smaller distance in u’₁₀v’₁₀ space between the two illuminants.

Table 4. The Δ_meq,r1 values of the eight dichromatic illuminations for the three illuminant pairs: YR, YB, GR. The data are presented as a ± b, where a is the Δ_meq,r value based on the average matching chromaticity of 10 observers and b is the corresponding inter-observer standard error.

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When evaluating the individual Δ_meq,r1 data under each illumination condition, it was found that the Δ_meq,r1 values of ten observers always follows a normal or skewed normal probability distribution with a single peak, instead of a bimodal distribution. As tested by the Shapiro-Wilk test, the distribution of the Δ_meq,r1 data is not significantly (p > 0.05) different from a normal distribution for most dichromatic illuminations, but there are a few exceptions whose distributions are skewed. A GLMM was used to test the difference in Δ_meq,r1 between the two transition types at the 95% confidence level. The fixed factors are illumination pair and transition type, and the random factor is observer (intercept only). The normality of the random effect was confirmed by a Shapiro-Wilk test [54]. Because the interaction between illumination pair and transition type is significant (p < 0.05), the impact of transition type on Δ_meq,r1 was tested for each color pair by a further GLMM analysis. The results show that the difference of Δ_meq,r1 between two transition types is significant (p < 0.05) for the YR and YB pairs, but not for the GR pair.

As mentioned earlier, the equivalent illuminant chromaticity of the dichromatic illumination can be taken as the white point of the (mixed) test illuminant in a regular von Kries CAT model. For each illuminant pair, 15 corresponding color pairs can be derived from the 8 dichromatic illuminations, and 7 uniform illuminations, with the EEW as the reference illumination. The long (L), medium (M) and short (S) wavelength cone signals of the visual match, test illumination, and the reference illumination were converted from the X₁₀Y₁₀Z₁₀ tristimulus values by applying the Hunt-Pointer-Estevez (HPE) matrix [55,56]. The L₁₀M₁₀S₁₀ values of the corresponding color predicted by the von Kries model (Eq. (4)) were converted to the X₁₀Y₁₀Z₁₀ tristimulus values by the inverse of the HPE matrix.

For each illuminant, the D parameter was optimized by minimizing the DEu’₁₀v’₁₀ between the corresponding colors predicted by the one-step von Kries CAT model and the visual match made under the baseline (EEW) and is referred to as D_optim. The D_optim values and the minimized prediction errors DEu’₁₀v’₁₀, obtained using the matching chromaticity averaged over all observers, under the 15 illumination conditions for the three illuminant pairs are summarized in Table 5. The corresponding inter-observer standard errors, obtained from the optimized D values and DEu’₁₀v’₁₀ of each observer, are also given.

Table 5. D_optim and the minimized prediction errors DEu’₁₀v’₁₀ under the 8 dichromatic illuminations, and 7 uniform illuminations for the three illuminant pairs: YR, YB, GR. The baseline is the EEW illumination, whose D value is fixed to one. The data are presented as a ± b, where a is the D_optim or DEu’₁₀v’₁₀ obtained using the average matching chromaticity of all 10 observers, and b is the corresponding inter-observer standard error.

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The impact of the transition type on the D_optim value can be observed from Table 5. Gradient transitions can achieve substantially higher D_optim values than the sharp transitions for the YB and RG pairs, while for the YR pair, similar adaptation states were achieved for the 50/50 background condition. The significance (p < 0.05) of these results was tested using a GLMM, after verifying the assumptions using the tests mentioned earlier. The fixed factors are illumination pair and transition type, and the random factor is observer (intercept only). Due to the significant interaction between illumination pair and transition type (p < 0.05), the impact of transition type on D_optim was tested for each color pair separately. The results show that for the YB and GR pairs, the difference of D_optim between two transition types is significant (p < 0.05), but not for the YR pair.

3.5 Impact of spatial distribution

Under the grey world assumption, the equivalent illuminant chromaticity should only depend on the average chromaticity of the visual field, which is determined by the specific spatial configuration of the illumination. Table 4 summarizes the Δ_meq,r1 values of the dichromatic illuminations with different local adaptation fields and three levels of spatial illumination proportions for the three illuminant pairs. Firstly, it can be observed that the adaptation state is mainly influenced by the local field surrounding the stimulus, rather than the average chromaticity of the whole visual field, as for many conditions with I1 or I2 in the center, the Δ_meq,r1 values deviate substantially from 0.5 as shown in Table 4. If the center field is the gradient transition from I1 to I2, for all the possible spatial proportions and illuminant pairs, the Δ_meq,r1 values are close to 0.5 (0.397 to 0.588), which is consistent with the chromaticity of the local gradient field being equal to or close to the average chromaticity of I1 and I2.

Secondly, the adaptation state is also influenced by the peripheral regions in the adapting field. For the adapting fields with I1 in the center, the Δ_meq,r1 values decrease with increasing proportions of I1 in the central field (from 50% to 75%), due to a decreasing impact (smaller relative area) of I2 in the peripheral region on the adaptation state. Because of the definition of Δ_meq,r1, the opposite can be observed when I2 is in the center. The GLMM test confirms a significant impact of the size of the peripheral region on the Δ_meq,r1 values when the central illumination is I1 or I2. For the spatial configurations with a gradient in the center, the Δ_meq,r1 values increase slightly when the proportion of I2 increases from 25% to 75%, but no significant change in Δ_meq,r1 values could be confirmed by the GLMM test. Therefore, the peripheral region does have an impact on the adaptation state, just not as much as the central region of the visual field.

3.6 Predictive performance of von Kries CAT for dichromatic illuminations

To determine whether chromatic adaptation and its degree under dichromatic illuminations can actually be well-represented in a regular von Kries CAT by an equivalent illuminant chromaticity, the D_optim under the dichromatic illuminations were analyzed with respect to those of the uniform illuminations. For each illuminant pair, it is clear from Tables 5 and 4, respectively, that the D_optim and Δ_meq,r1 values change systematically going from uniform background ‘U1’ to ‘U7’. Plots of the D_optim values of the uniform backgrounds as a function of the corresponding Δ_meq,r1 have been made in the three subfigures of Fig. 6 for the YR, YB and GR illumination pairs, respectively. The data points for the dichromatic illuminations and a quadratic curve (and its 95% confidence band) fitted to the data points of the uniform backgrounds have been plotted as well. The fitting error of the quadratic curve, in terms of root-mean-square-error (rmse), is less than 0.05, which is approximately 5% of the total scale (D_optim ranges from 0 to 1). From Fig. 6, it is clear that the (degree of chromatic adaptation of the) dichromatic illuminations can indeed be well represented by a uniform background with a specific chromaticity characterized by Δ_meq,r1: all dichromatic illumination data points are within the 95% confidence band of the curve representing the uniform backgrounds, with the exception of the 50/50 sharp transition for the YB and GR conditions.

 

Fig. 6. D_optim versus Δ_meq,r1. The three subfigures correspond to the three illuminant pairs. The small blue points represent the data points for the seven uniform backgrounds. The blue squares represent the dichromatic illuminations with a sharp transition and the blue circles represent those with a uniform background in the center and a sharp transition on both sides. The red asterisks and the red circles correspond to the dichromatic illuminations with a gradient transition in the center, with 50/50 and 25/75 illuminant proportions, respectively. The blue band in each subfigure corresponds to the 95% confidence band of the quadratic curve representing the uniform backgrounds.

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In addition, like the degree of chromatic adaptation, the CAT prediction error for dichromatic illuminations also depends on its equivalent illuminant chromaticity, as shown in Table 5. For example, for the YR pair, a spatial configuration with Y in the center – resulting in a small Δ_meq,r1 value (high Y contribution to the adaptation state) – has a larger DEu’₁₀v’₁₀ prediction error than other configurations and illuminant pairs. This is consistent with the fact that the von Kries CAT performs poorly for illuminants with a small S cone excitation [44], such as illumination Y.

4. Discussion

4.1 New CAT for dichromatic illuminations

Given the above analysis, to apply the von Kries CAT in a dichromatic illumination, Eq. (4) can be written as:

The proposed von Kries based CAT for dichromatic illumination is composed of five steps. Steps 1-3 comprise a model for the determination of the equivalent illuminant chromaticity (EIC model), in step 4 the degree of adaptation is calculated (D-model), while the final step integrates the two models in a regular von Kries CAT.

In the first step of the EIC model, the spatial weighting factor at different viewing angles is calculated by a Gaussian distribution, as given below:

In the second step, the cone excitation of the equivalent illuminant chromaticity is calculated as the integration of the product of the weighting factor and the cone excitation of the background over the whole illuminated adapting field, as given below:

In the third step, the cone excitation of the equivalent illuminant from the second step is transformed to X₁₀Y₁₀Z₁₀ tristimulus values using the inverse of the HPE matrix. The result is denoted as XYZ_Equi, which is then further transformed to u’₁₀, v’₁₀ chromaticity coordinates.

The degree of adaptation of the dichromatic illumination is calculated using the D model proposed in 2020 by Ma [35] (denoted as D_Ma2020) – which considers the effect of the chromaticity, luminance, and FOV of the (adapting) background. Note that for dichromatic illuminations, the variable – chromaticity of the adapting field in D_Ma2020 is taken to be that of the equivalent illuminant. Using the degree of adaptation model D_Ma2020, the standard deviation of the Gaussian distribution σ (Eq. (6)) for each illuminant pair, was optimized to minimize the RMS (Root-mean-square) of the prediction error DEu’₁₀v’₁₀ of all the 24 (8 × 3) dichromatic illuminations. The optimized σ value is 0.33, corresponding to a viewing angle at 18°, and it can be taken as a constant in Eq. (6). Note that the optimized σ value (= 0.33) probably varies with viewing conditions, which deserves further investigations in the future.

By taking the EEW as the reference white, 24 (8 × 3) corresponding color pairs can be derived from the 24 dichromatic illuminations. The corresponding colors predicted by various von Kries CAT models were plotted in the u’₁₀v’₁₀ color space, as shown in Fig. 7. Table 6 lists the mean prediction errors over the 8 corresponding colors of each illuminant pair for a von Kries CAT adopting different EIC and D models:

 

Fig. 7. The predicted chromaticities of corresponding colors (under EEW) estimated by five von Kries models as listed in Table 6. The circles with different colors represent different prediction models. The black star represents the mean achromatic match collected under EEW. The larger distance between the black star and the colorful circles corresponds to a higher prediction error of the modeled von Kries CAT. The three subfigures correspond to the three illuminant pairs.

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Table 6. A comparison of the performance (mean DEu’₁₀v’₁₀) of the standard von Kries CAT with different EIC and D models including D_optim + EIC_U1-U7, D_Ma2020 + EIC_U1-U7, D_Ma2020 + EIC_M and D_Ma2020 + EIC_GW, and the modified von Kries CAT with D_optim + EIC_U1-U7 for the three illumination pairs and on average (across the three pairs).

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For the standard von Kries CAT with EIC_U1-U7, as expected, the prediction error with D_Ma2020 (mean DEu’₁₀v’₁₀ is 0.0088) is larger than the benchmark model with D_optim, (mean DEu’₁₀v’₁₀ is 0.0048) especially for the GR pair. However, it’s performance is quite comparable to that of the D_Ma2020 for the datasets D_Ma2020 was derived from (the mean DEu’₁₀v’₁₀, is 0.0074) [35], indicating the applicability of the D_Ma2020 for the dichromatic illumination data. The practically useful von Kries CAT with the EIC_M and D_Ma2020 model has similar but slightly worse performance than the EIC_U1-U7 + D_Ma2020 model for the three illuminant pairs. Compared to the D_Ma2020 + EIC_GW model, the D_Ma2020 + EIC_M model performs substantially better for all three color pairs. In Fig. 7, it can be observed that the magenta points (the D_Ma2020 + EIC_GW model) deviate more from the achromatic match under EEW than points in other colors, indicating its worse performance. The poor performance of the D_Ma2020 + EIC_GW model indicates a failure of the grey world assumption as an estimation of the equivalent illuminant chromaticity of the dichromatic illumination for our datasets. Given its quite good performance, the standard von Kries CAT (shown in Eq. (5)) integrating D_Ma2020 and EIC_M therefore provides a preliminary model to predict the corresponding colors under dichromatic illuminations by considering the spatial distribution of the chromaticity, size, and luminance of the adapting field.

The D_Ma2020 + EIC_M CAT model has a similar structure to the model developed by Kwak et al., but they adopted different methods to calculate the equivalent illuminant chromaticity and the D factor. For example, Kwak’s model adopted the D formula from CIECAT02 which only includes the background luminance as a variable, while the D formula proposed in this study considers the impact of the equivalent illuminant chromaticity, field of view, and luminance level on the D factor. In addition, Kwak’s model adopted a linear function with fixed weighting factors (optimized from the visual data) to estimate the equivalent illuminant chromaticity, while the equivalent illuminant chromaticity model in this study integrates the chromaticity over the spatial distribution of the dual illuminations. Therefore, compared with Kwak’s model, the proposed CAT model is more general and more comprehensive with a wider application range. When implementing the D_Ma2020 + EIC_M CAT model in the determination of the corresponding color under a dichromatic illumination, the equivalent illuminant chromaticity was calculated by integrating the product of the spatially varying background cone excitation and a spatial weighting factor over the whole illuminated scene, as shown in Eq. (6)–(7). Then the degree of adaptation D_Ma2020 can be estimated by using the equivalent illuminant chromaticity as the adapting chromaticity (white point) in the von Kries model. The proposed von Kries CAT which integrates the D_Ma2020 and EIC_M model can predict the corresponding color under dichromatic illumination, and it could be embedded into the color appearance model. Note that this study proposed a psychophysical model derived from a series of visual data, instead of a physiological model which explains the physiological mechanism under dichromatic illuminations.

In most conventional corresponding color datasets published in previous literature, the stimulus is a two-dimensional (2D) uniform color patch presented on a 2D uniform background. However, in the present study and in Smet’s and Ma’s previous work [34,35], the stimulus was a three-dimensional (3D) cube presented in a 3D scene. Yang and Shevell found that the binocular disparity in stereo vision can improve color constancy [58]. Hedrich et al. reported a higher color constancy index for a cue-rich 3-D setup than for a 2-D setup [59]. Similar conclusions have been drawn in the studies of Morimoto et al. and Mizokami et al. [60,61]. Therefore, compared to a 2D scene, a 3D scene is expected to lead to a more complete chromatic adaptation state. If applying the proposed model to a flat 2D scene without any depth cue, the degree of adaptation is expected to be attenuated by approximately 20% as summarized from previous results. But in a real 3D scene, the exact relationship in the degree of adaptation between the 2D and 3D stimulus is still not experimentally determined, which deserves more experiments with stimuli varying in dimensionality in the future.

To correct the unequal D values between the L/M and S cones for the yellowish and greenish illuminations, Ma et al. proposed a modified von Kries CAT by including a compression of the rescaling factor for the S cone [57]. For each color pair, the mean prediction errors DEu’₁₀v’₁₀ of the modified von Kries CAT which adopts the optimized D values and the equivalent illuminant chromaticity derived from the U1-U7 uniform backgrounds (D_optim + EIC_U1-U7) were summarized in Table 6. Only the optimized D values were adopted here as there is no available degree of adaptation model for the modified von Kries CAT. Comparing the performance of the two CAT models adopting D_optim + EIC_U1-U7, it can be observed that the modified von Kries CAT substantially outperforms the standard von Kries CAT for the YR pair, but for the YB and GR pairs, the two models have similar performance.

4.2 Limitations

However, there are still some limitations of the current model. Firstly, the model was derived for dichromatic illuminations with equal luminance, so the impact of non-uniform luminance distribution on chromatic adaptation was not investigated. Secondly, the dichromatic illuminations only include a linear chromaticity transition in the horizontal direction. Other orientations or more complicated spatial configurations (e.g. spotlights embedded in an ambient illumination and Mondrian-like backgrounds) deserve further investigation. Thirdly, in the proposed EIC_M model, the unequal contributions of the two sharp-transitioned illuminations were not considered due to the small number of illumination pairs (n = 3) for this condition. Therefore, more experiments with various illumination pairs are required to improve and extend the EIC model. Fourthly, this study has demonstrated the importance of the local surrounding field for chromatic adaptation, but doesn’t investigate how the local contrast influences the adaptation state. Further research on the local contrast of the adapting field or the non-uniform luminance distribution should be conducted in the future.

5. Summary

Achromatic matching experiments were conducted to investigate the chromatic adaptation under spatially complex dichromatic illuminations, involving a yellow-blue (YB), a yellow-red (YR) and a red-green (RG) illumination pair. Sharp and gradual horizontal transitions between the two illuminations were used at different locations relative to the stimulus, i.e., a neutral grey cube on an otherwise white background scene. The illumination pairs were swapped in position to check for a left-right bias. Furthermore, achromatic matches were also made under a uniform illumination, during which seven combinations for each of the three illumination pairs were used.

Firstly, the average matching results indicate a left bias for the sharp transition, but not for the gradient transition. This left bias, however, was not statistically significant.

Secondly, for illumination pairs with a sharp transition, the equivalent illuminant chromaticity tends to be less influenced by the yellowish illumination (low CCT) compared to the other illuminants. Furthermore, an illumination pair with a sharp transition at the center of the neutral grey cube shows a less complete adaptation to the equivalent illumination compared to those with a gradient transition.

Thirdly, the adaptation state is mainly influenced by the local field surrounding the stimulus, rather than the average chromaticity of the whole visual field. But the peripheral region also has some contributions which increase with its size.

Fourthly, the degree of chromatic adaptation under the dichromatic illumination can be well represented by an equivalent uniform background for most dichromatic illuminations, except the 50/50 sharp transition for the YB and GR illumination pairs.

Fifthly, a model for estimating the equivalent illuminant chromaticity (EIC_M model) was proposed which integrates the product of a horizontally oriented spatial weighting factor and the cone stimulation of the illumination over the adapting field. The spatial weighting factor was modeled by a Gaussian function (along horizontal direction) with the peak at the center of the visual field, i.e., the center of the stimulus.

Finally, a von Kries CAT for dichromatic illumination conditions was developed which considers the spatial distribution of chromaticity using the EIC_M model, and the size, luminance and chromaticity of the adapting field using the D_Ma2020 model. The proposed CAT model has good performance with a mean DEu’₁₀v’₁₀ prediction error of 0.0098 for the visual data collected in this experiment, and substantially outperforms the von Kries CAT adopting the average chromaticity of the whole illuminated scenes as the white point of the dichromatic illumination (mean DEu’₁₀v’₁₀ 0.0183).