1. Introduction

1.1 Development of chromatic adaptation transforms

Chromatic adaptation refers to the ability of the human vision system to adapt to the light stimulus, induced by the adapting field, and keep the color appearance of the illuminated object relatively constant. It is modeled by chromatic adaptation transforms (CATs), which are an important part of color appearance models (CAM) [1], and which can predict the adaptive shift due to a change in the chromaticity of the adapting illumination. CATs are developed based on corresponding color (CC) data (CCs are stimuli that appear equal in color under different illumination conditions [2]).

Over the years, many CAT models have been derived from corresponding color datasets published in 20^th century literature. In 1902, von Kries proposed the von Kries law [3] which defines that chromatic adaptation can be regarded as an independent rescaling of the cone signals. Based on the von Kries law, the von Kries – Ives model [4] and von Kries - Helson model [5] were proposed by providing the formula to calculate the rescaling factor for each individual cone type while assuming complete adaptation. Nayatani et al. proposed a non-linear CAT model with two-step transformations [6,7], accepted by CIE (Commission Internationale de l'Eclairage) as CIECAT94 [2], but which does not account for incomplete adaptation. Later on, based on the Lam & Rigg CC data [8], the Bradford (BFD) model [8], with a similar structure to Bartleson's model [9], was proposed in 1985, and further enhanced by Luo and Hunt [10]. The latter became the CMCCAT97 model, where the degree of adaptation increases with the background luminance. By considering the incomplete chromatic adaptation under colorful illuminations indicated by Hunt [11] and Breneman [12], Fairchild proposed a von Kries-type CAT model for imaging applications, called RLAB [13]. To correct the invertibility problem of CMCCAT97, caused by the inclusion of an exponential factor in the S cone signal, a new von Kries CAT, i.e. CMCCAT2000, was proposed by setting the exponential factor to one [14]. Further improvements of the performance of the CMCCAT2000 model– while excluding the McCann corresponding color data during model development – resulted in CAT02 [15,16]. More recently, Li et al. proposed CAT16 to correct some mathematical problems plaguing the CIECAM02 model due to the incompatibility of the CAT02 sensor space and the Hunt-Pointer-Estevez (HPE) cone space used later in the model [17,18], while at the same time simplifying the CAM’s structure [18]. CMCCAT2000, CAT02, CAT16 are all von Kries-type CATs, but with different degree of adaptation formulas and sensor spaces. CMCCAT97 has a structure very similar to the von Kries CAT, except for the exponential factor in the S cone response calculation. Smet et al. [19,20] has investigated the performance of several CAT models, including, among others, the Nayatani model, the RLAB model, the RLAB model with the optimized partial chromatic adaptation factor and the standard von Kries model with several ways of determining the degree of adaptation (D), such as using the D formula in CMCCAT97, CMCCAT2000, CAT02, and minimizing the prediction error for 26 datasets from eight corresponding color data sources. The results indicated a better performance of the von Kries model with the optimized D than when obtained from the CMCCAT97 and CAT02 models. The latter two models performed better than the RLAB model, while the Nayatani model had the worst performance. Note that all CAT models mentioned in this paper were developed for photopic viewing conditions.

Eight datasets are typically used for CAT development and benchmarking: the CSAJ-C (Colour Science Association of Japan) data [21], Kuo & Luo data [22], Lam & Rigg data [8], Helson data [5], LUTCHI data [23], Breneman-C data [12], Braun & Fairchild data [24], and McCann data [25]. For example, CMCCAT97 [8,10] was developed based on the Lam & Rigg data by minimizing the CIELAB color difference between the predicted and experimental CC. CMCCAT2000 [14] was developed with eight CC datasets, also based on CIELAB differences. CAT02 [15,16] and CAT16 [18], were both developed by minimizing the CMC(l:c) color difference (defined by the Colour Measurement Committee of the Society of Dyers and Colourists) [26] for the same data sources, but with the McCann data excluded. In the McCann study, highly chromatic colors and low illuminance levels were used, which are less relevant for typical lighting applications.

The CAT models developed from the eight data sources mainly aim for typical applications with neutral to near-neutral illuminations, such as daylight, tungsten and fluorescent tubes. To extend the application range of the current models, besides these eight data sources published in the 20th century, additional corresponding colors data sets were generated by Smet et al. [19,20], Wei et al. [27], and Ma et al. [28] for a variety of adaptation conditions, such as changes in chromaticity, field of view, and luminance of the adapting field. An overview of these datasets is provided later.

1.2 Aim

In the following sections, ten data sources were further analyzed and compared: seven data sources (21 datasets) from the 20^th century literature (the CSAJ-C, Kuo & Luo, Lam & Rigg, Helson, LUTCHI, Breneman-C, and Braun & Fairchild data), denoted as CC7; and the Smet data, Wei data, and Ma data. The McCann data was excluded from the following analysis because it was collected under colorful test illuminations at a low luminance level which is not common in typical lighting applications. The corresponding colors from all ten datasets mentioned above were collected under various illumination conditions using different experimental methods. As these CC sets provide the foundation for the development of CATs, it is important to know if similar illumination conditions from different data sources result in a consistent predictive CAT performance and similar D values. In addition, as there are still quite large prediction errors of the current CATs for the collected visual CC data, there is still room to improve their performance. Furthermore, a systematic investigation of the sources of the prediction error for several existing CAT models can provide insights and guidance to develop a new model with better performance. As the von Kries CAT has been shown to have a higher predictive performance than other CATs for the Smet data and CC7 [19,20], we used this one and compared the predictive performance of the regular von Kries model (independent scaling of sensor responses) and the D factors for the earlier mentioned CC sets. Furthermore, we have analyzed the origin of the prediction errors of the regular von Kries CAT for all datasets and proposed a modified von Kries model with better performance.

1.3 Overview of corresponding colors datasets

1.3.1 20^th century datasets

The details of the CC7 datasets are summarized in the Table 1, including the experiment conditions (e.g., sample, illumination, method, number of observers) and the type of available data (tristimulus values or spectral power distribution). The experimental conditions are slightly different from those summarized by Luo et al. [29], because of inconsistencies with the original publications. For each source, there are one or more datasets corresponding to different test-reference illumination pairs, and each dataset includes a number of CC pairs. One exception is the Braun & Fairchild data where the four datasets correspond to two test-reference illumination pairs and two test images. The CSAJ-C data [21], and Lam & Rigg data [8] include CCs between illumination A (test illumination) and D65 (reference illumination). The Helson data [5] includes illumination A as the test and illumination C as the reference. The LUTCHI data [23] is composed of three datasets with A, D50, and WF (white fluorescent) as the test illuminations and with D65 as the reference illumination. For the Kuo & Luo data [22], there are two datasets with CCs between two test illuminations, CIE illumination A and TL84, and a D65 reference illumination. The Breneman-C data includes nine datasets. There are five sets with five luminance levels from 15 cd/m² to 11100 cd/m² and with illumination A as the test illumination and illuminations D65 or D55 as the reference illumination, and four datasets with highly chromatic test illuminations. The Braun & Fairchild data [24], collected in a cross-media matching experiment with two test images, has test illuminations D93 and D30, and a D65 reference illumination. Each image contains 16 or 17 stimuli. Note that the CCs are XYZ tristimulus values specified using the CIE 1931 2° color matching functions (CMFs) in most datasets mentioned above except for the Lam & Rigg data which uses the CIE 1964 10° CMFs. For these seven data sources, there are 21 corresponding color datasets in total.

Table 1. A summary of the experimental conditions and available data for ten corresponding color data sources, including seven published in the 20^th century and three published in the recent ten years.

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The seven experiments adopted different psychophysical methods to collect corresponding colors. The CSAJ-C [21] and Breneman-C [12] data were collected in haploscopic matching experiments. The LUTCHI data [23], and Kuo & Luo data [22] were collected in magnitude estimation experiments. The Lam & Rigg data [8] and Helson data [5] were collected in memory matching experiments. The Braun & Fairchild data [24] was obtained in a cross-media matching experiment.

1.3.2 Smet et al. data (2017)

To investigate the impact of background chromaticity on the D factor, Smet et al. [19,20] conducted a memory color matching (MCM) experiment for five familiar objects under thirteen illuminations with chromaticities covering a large gamut in the CIE 1976 u₁₀’v₁₀’ chromaticity space. The internal reference of the matching is the familiar object’s memory color (the prototypical color associated with the object in long-term memory [30,31]). There are several advantages of the MCM method over other experimental methodologies: it has a natural viewing condition that avoids the binocular rivalry and the eye movement restriction in the haploscopic matching; it is less time-consuming than the haploscopic matching; it allows for more accurate matches than short-term ‘learned’ memory matching. As D65 is the reference illumination in most of 20^th century datasets, for comparison reasons, only the CC sets from the Smet data with D65 as the reference were retained in the following analyses. For each illumination, (spectral) matching data for the five familiar objects were collected, resulting in five corresponding color pairs per illumination pair. . As the viewing angle (at observer’s position) of the familiar objects were larger than 4°, the CIE 1964 10° CMFs were used for the colorimetric calculations.

1.3.3 Ma et al. data (2020)

Using a similar set-up as Smet et al. [20], Ma et al. did a series of achromatic matching experiments to investigate the influence of adapting field size on the D factor [28]. The same 13 background illuminations were used as for Smet et al. [19,20]. For each adapting field size, 156 CC pairs could be derived (same as for Smet et al. [19,20]). There are two groups of adapting field sizes. In the first group, the field sizes (horizontal × vertical) were 20° × 20°, 40° × 40°, 80° × 66°, with the adapting luminance at 180 cd/m². In the second group, the field sizes were 20° × 20°, 40° × 40°, 60° × 60°, while the vertical illuminance level at eye level (corneal illuminance) was kept constant at 7 lx. Similar with the Smet et al. data, for each adapting field size level, only CC sets with D65 as the reference illumination were kept for the further analyses; and the CIE 1964 10° CMFs were used for the colorimetric calculations.

1.3.4 Wei et al. data (2019)

Wei et al. investigated the joint impact of luminance and correlated color temperature (CCT) on the degree of adaptation in an achromatic matching experiment [27]. The illumination was provided by a lighting booth with a tunable light source and the stimulus was a square area displayed on an iPad Air 2. Note that the adapting luminance reported in the paper was not directly measured on the background configuration, but on a white tile. Consequently, it was impossible to determine the exact adapting luminance. As an approximation, the 40% reflection of the matt grey Munsell N7 sheet was used to estimate the adapting luminance. CCs were collected for 16 illumination conditions, composed of four luminance levels (115, 300, 600, 900 cd/m² measured on the white tile) and four CCTs (2700, 3500, 5000, 6500 K). In addition, for the white tile luminance of 900 cd/m², one illumination with a CCT of 9000 K was also included. For each background condition, there were six luminance levels of the iPad stimuli (100 ∼ 350 cd/m²). The CCs are the X₁₀Y₁₀Z₁₀ tristimulus values determined using the CIE 1964 10° CMFs. As before, for comparison reasons with other datasets, CC sets were composed for each luminance level with D65 as the reference illumination.

2. Analysis

2.1 Derivation of LMS responses from spectral or XYZ data

Since almost all the available XYZ data in the 20^th century data sets was calculated using the CIE 1931 2° CMFs, the Hunt-Pointer-Estevez (HPE) matrix [32,33] can be used to derive estimates of the associated long (L), medium (M) and short (S) wavelength cone signals. For the Lam & Rigg and Wei data, however, only the 10° tristimulus (X₁₀Y₁₀Z₁₀) values were available, whereas for the Smet and Ma data the spectral data is available, allowing to directly calculate the cone signals. To check whether the 2° and 10° data can be pooled in the analysis without incurring substantial errors, firstly a small subset of the Ma spectral data was analyzed. From the spectral data, the 2° and 10° cone signals can be calculated directly (with the CIE 2006 cone fundamentals [34]) or indirectly via the 2° and 10° tristimulus values followed by a conversion matrix to derive the cone signals. The optimized D values for the relevant combinations are summarized in Supplement 1. Given the relatively small differences in obtained D values, we decided to combine the 2° XYZ data with the 10° X₁₀Y₁₀Z₁₀ data, and converted the tristimulus values to cone signals using the HPE matrix for all datasets. Note that the cone fundamentals derived from the tristimulus values with the HPE matrix differ from the actual cone fundamentals, which might introduce substantial errors for light sources with narrow emission bands.

2.2 Analysis of cone signal ratios under the test and reference illuminations

Following the von Kries CAT model [3], the cone signals of the corresponding colors from the test illumination to the reference illumination are predicted using a one-step CAT, as shown below:

For a certain test-reference illumination pair, the multiplicative diagonal matrix in Eq. (2) will remain the same for different samples. If the von Kries model can perfectly explain the experimental data – ignoring color rendition shifts –, the ratios of the experimental stimulus cone signals L_r,c/L, M_r,c/M, S_r,c/S, henceforth referred to as eCSRs which are equivalent to the experimental gain control factors for each of the cone types, should all be the same and depend only on the chromaticity of the test and reference illumination. The cone signals of the CC collected from the visual experiment L_r,c, M_r,c, S_r.c are not directly compared with the predicted cone signals L’_r,c, M’_r,c, S’_r.c, because the L’_r,c, M’_r,c, S’_r.c depend on the CAT model chosen to estimate the CC. To test the consistency of the eCSRs values among samples, for each dataset, the experimentally determined CC cone signals L_r,c, M_r,c, S_r.c of all the samples under the reference condition were plotted against the L, M, S values under the test condition, respectively. Figure 1 shows two examples from all the datasets, where the three columns represent the L, M, S cones respectively and the two rows represent the CSAJ-C (Illumination A) [21], and LUTCHI data (white fluorescent) [23], respectively. These two datasets were selected because illumination A and white fluorescent (WF) were commonly used in commercial light sources and also used in CC7. The black line in each subfigure indicates the linear function with zero intercept (y = kx) fitted to these points. The points in each subfigure correspond to the samples in that dataset. The CSAJ-C (Illumination A) [21] and LUTCHI (white fluorescent test illumination) data [23] include 104 and 41 samples, respectively. As shown in Fig. 1, both two datasets show a good overall agreement (r² > 0.9) between the cone signals of these samples under the test illumination and the reference illumination, indicating the high consistency of each cone’s multiplicative rescaling coefficient for the different samples. For the M-cone of the CSAJ-C data, the data points were aggregated in three clusters which could have led to an inflated r² value. Note that the high consistency does not mean a constant rescaling coefficient for different samples, as can be observed from the spread of the data points around the overall trend in Fig. 1. Possible sources of the spread include, but may not be limited to, observer matching error, color discrimination threshold and color rendition differences between the test and reference illuminations.

 

Fig. 1. The CSAJ-C data (illumination A) and LUTCHI data (White fluorescent illuminant), where the three columns represent the signals from three sensors L_r,c, M_r,c, S_r.c plotted against L, M and S, respectively. The reference illuminations for the CSAJ-C and LUTCHI data are D65. In each subfigure, the data points are fitted by a linear model (black line) with zero intercept

(y = kx), and its r² value (coefficient of determination) and k value (slope) are shown in the top of the figure.

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Because Wei et al. [27] and Ma et al. [28] adopted an achromatic matching technique –which results in only a single CC pair per illumination pair–, the number of data points (= 1) is not sufficient to determine their internal sample consistency. For the Smet et al. data [19,20], there are five stimulus objects, corresponding to five data points under each illumination. Similarly as before, a linear fit (y = kx) to the five points shows that the L, M and S, are highly correlated (r² > 0.97) with the L_r,c, M_r,c, S_r.c,, again indicating high internal consistency of the multiplicative coefficients.

2.3 Relationship between the illumination cone signal ratio (CSRi) and experimental stimulus cone signal ratio (eCSR)

Equation (2) can be rewritten as below:

For each cone, the theoretical stimulus-induced cone signal ratio (L’_r,c/L, M’_r,c/M, S’_r,c/S), denoted as CSRs, can be regarded as a linear function of the ratio of the illumination-induced cone signals CSRi (L_rw/L_w, M_rw/M_w, S_rw/S_w). From Eq. (3), it can be observed that both the intercept and slope of this linear function depend on the D value and the functions with different D values intersect at point (1, 1). For a certain dataset, the CSRi values are constant for different samples, as obviously, they only depend on the white point of the illuminants. However, as discussed in section 2.2, the eCSRs (L_r,c/L, M_r,c/M, S_r,c/S) values vary with sample, even for the datasets with high r² value between the theoretical CSRi and eCSRs values, such as the CSAJ-C and LUTCHI data shown in Fig. 1. In what follows, for each cone, the eCSRs of a dataset is therefore represented by the median of those of all the samples. It is the ‘median’ instead of the ‘mean’ was chosen to represent a dataset because the ‘median’ is not skewed by a small proportion of extremely large or small values in the data group (especially for the S cone of the Breneman data with low luminance level).

The relationship between the CSRi and the eCSRs are compared in Fig. 2 for four groups of illuminations including yellowish and bluish illuminations on or near to the Planckian locus, and highly chromatic greenish and reddish illuminations, marked in yellow, blue, green, and red, respectively. In Fig. 2, the three graphs represent the three cone types (L, M, S). The ten types of marker symbol correspond to the ten data sources and the four colors of the marker correspond to the four groups of illuminations. The data for yellowish illuminations, with a CCT from 2000K to 3500 K, are obtained from 10 data sources: the CSAJ-C, Kuo & Luo, Lam & Rigg, Helson, LUTCHI, Breneman-C, Braun & Fairchild, Smet, Wei and Ma data. The data for bluish illuminations, with a CCT larger than 8000 K, are from four datasets: the Braun & Fairchild, Smet, Wei and Ma data. The data for greenish illuminations are from three data sources: the Breneman-C, Smet and Ma data. The data for reddish illuminations are from two data sources: the Smet and Ma data.

 

Fig. 2. The eCSRs values plotted against the CSRi for four groups of illuminations including yellowish illuminations near or close to the Planckian locus (2500 K < CCT < 3500 K), bluish illuminations near or close to the Planckian locus (CCT > 8000 K), greenish illuminations, and reddish illuminations, marked in yellow, blue, green, and red symbols, respectively. In each subfigure, the black dashed line represents a linear function y = x, corresponding to D = 1. The three subfigures correspond to the three cone types. (a). L cone (b). M cone (c) S cone

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Firstly, it can be observed that for L and M cones, most of the symbols are crowded in the same small region and have eCSRs and CSRi values around 1.0, except for the Helson and Lam & Rigg data. And the overall ranges of the eCSRs and CSRi values for the L and M cones are substantially smaller than those for the S cone.

Secondly, the eCSRs and CSRi are linearly correlated for the L (r² = 0.80) and M cones (r² = 0.71). The S cones, however, show a considerably larger spread in eCSRs and CSRi values (between 1.5 and 5) for yellowish illuminations, which can be partially attributed to the multiple viewing conditions that were investigated per illumination chromaticity in the Breneman-C, Wei and Ma data sets, such as different luminance levels (the Breneman-C and Wei data) and different field of views of the adapting fields (the Ma data). In addition, a discrepancy between the different datasets can be observed in Fig. 2(c). The symbols representing the CSAJ-C, Kuo & Luo, Lam & Rigg, Helson, LUTCHI, Braun & Fairchild data are located very close to the black dashed line (y = x). However, for the Breneman-C, Smet, Wei and Ma data, the corresponding symbols are well below the black dashed line (y = x), indicating that the S-cones eCSRs are smaller than the S-cone CSRi. As for the other three groups of illuminations (blue, red, and green), there is no obvious difference between these data sources.

Thirdly, for yellowish illuminations from the Breneman-C, Smet, Wei and Ma data and the greenish illuminations in general, the corresponding symbols in the subfigures for the L and M cones are located on or very close to the y = x line, while these symbols in the S-cone subfigure are well below the y = x line. While for the yellowish illuminations in the CSAJ-C, Kuo & Luo, Lam & Rigg, Helson, LUTCHI, Braun & Fairchild data, the corresponding symbols are always very close to the y = x line in the subfigures for the L, M, and S cones.

It has been mentioned earlier that the CSRi values for the L and M cones have a much smaller range than for the S cone. Possible causes could be a bias in the illumination selection, whereby those with larger CSRi for the L and M cones were omitted. To test the origin of the different CSRi ranges of the three cones, the 1/CSRi values for the three cone types (L_w/L_rw, M_w/M_rw, S_w/S_rw) were derived for a range of blackbody radiators, i.e., with color temperatures between 1500 K and 35 000 K. The LMS values were normalized to 1.0 for illuminant D65 and all other LMS-values were thus scaled relative to those of illuminant D65 (all with the same luminance). The resulting ratios are presented in Fig. 3 as a function of the color temperature. The 1/CSRi values were plotted, because the CSRi values of the S cones under low CCT illumination are too large to show the trend in the CSRi values for the L and M cones. In Figs. 3(a) and 3(b), the 1/CSRi values were derived using the CIE 1931 2° CMFs and CIE 1964 10° CMFs, respectively. The tristimulus values XYZ or X₁₀Y₁₀Z₁₀ were converted to cone signals using the HPE matrix. It can be observed that the range of the 1/CSRi values of the Planckian radiators for the S cone is substantially larger than that for the L cones and the M cones. In addition, the red, green, and blue circles in Fig. 3(a) represent the test illuminations of the 20 datasets from the CC7 set except for the Lam & Rigg data for the L, M and S cones, respectively. Similarly, the 1 illumination from the Lam & Rigg data, the 12 illuminations (excluding Binf) from the Smet data, the 5 illuminations from the Wei data and the 12 illuminations from the Ma data (excluding Binf), were plotted in Fig. 3(b) using different marker symbols. Figure 3(b) shows that for both the Smet and Ma data, the data points representing the seven illuminations on or close to the Planckian locus are indeed located on the Planckian curves while the data points representing the other five colorful illuminations deviate from the curves, especially for the S cone. For the 21 test illuminations from the CC7 set, and the 5 illuminations from the Wei data, the data points are located on or close to the Planckian curves. Therefore, the larger range of the 1/CSRi values for the S cones than that for the L- and M-cones was confirmed by the illuminations from the experimental data, indicating that the 1/CSRi values for the L and M cones are not very sensitive to the change in CCT. In addition, as the CSRi values for the L and M cones fluctuate slightly around 1.0, according to Eq. (3), the CSRs values calculated with a von Kries CAT are also close to 1.0, indicating a smaller change in L_c and M_c than S_c under different illuminations. Thus, the relatively large variation in the S-cone signals is also found in “natural” lighting conditions (the ratios for a blackbody radiator are very similar to those for phases of daylight). This suggests that adaptation should mainly be accomplished in the S-cone pathway. This stronger variation in S-cone signals has also been demonstrated by Lucassen & Walraven [35] who showed that a non-linear compression was necessary for the S-cone pathway to obtain a better prediction but to a lesser extend for L- and M-cone pathways.

 

Fig. 3. The 1/CSRi values for the L (red line), M (green line) and S cones (blue line) for blackbody radiators as a function of color temperature. (a). The cone signals were derived from the CIE 1931 2° CMFs. The 21 test illuminations from the CC7 sets except the Lam & Rigg data were plotted as red, green and blue circles for the L, M and S cones, respectively. (b). The cone signals were derived from the CIE 1964 10° CMFs. In this graph, the illuminations from the Lam & Rigg data, the Smet data (excluding Binf), Wei data, and Ma data (excluding Binf) were plotted as asterisks, pentagrams, and circles, respectively.

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2.4 Predicted degrees of adaptation

2.4.1 Standard von Kries CAT

Although the performance of different CAT models for the CC7 and Smet datasets has been compared in Smet et al. [19,20], the D values of the test illuminations in these datasets were not discussed. As for the Wei and Ma data collected later, the level of their agreement with predictions by CAT models have never been compared with those of other datasets. Therefore, this section aims to compare the D value and the prediction error of von Kries CAT among all ten data sources. In addition, this section will further investigate how the discrepancies in the eCSRs values among these data sources (as shown in the previous section) influence the predictive performance of the standard von Kries CAT for these data sets and how the performance changes with illumination color.

The D values for each illumination from these datasets were optimized, henceforth referred to as D_optim, by minimizing the average DEu’v’ or DEu’₁₀v’₁₀ between the predicted corresponding colors and the visual matches made under the reference illumination. The DEu’v’ or DEu’₁₀v’₁₀ coordinates were derived from the adapted tristimulus values X_a Y_a Z_a.

The D_optim values of the four groups of illuminations are plotted against the S-cone CSRi values in Fig. 4. Similar to Fig. 2, the yellow, blue, green, red markers correspond to yellowish, bluish, greenish, reddish illuminations, respectively; and the ten types of marker symbol correspond to the ten data sources. L- or M-cone CSRi were not further analyzed because of the small change in the CSRi value under different illuminations as shown in Fig. 3. For each group of illumination color, a large spread in D_optim values can be observed under similar S-cone CSRi values, which is caused by the internal and external differences in viewing conditions such as a wide range of luminance values (Breneman-C, Wei), and adapting field size (Ma). For example, the D_optim values of the yellowish illuminations in the Breneman data increase from 0.45 to 0.91 when the adapting luminance changes from 15 cd/m² to 11100 cd/m²; the D_optim values of the bluish illuminations in the Wei data range from 0.40 to 1.00 as the adapting luminance increases and the relative luminance of stimulus to illumination decreases; the D_optim values of the reddish illuminations in the Ma data change from 0.30 to 0.60 with the adapting field size ranging from 20° to 80°. As for the CSAJ-C, Kuo & Luo, Lam & Rigg, Helson, LUTCHI, and Braun & Fairchild data sources, the D_optim values for the yellowish illuminations are larger than 0.8, indicating a relatively complete adaptation state.

 

Fig. 4. D_optim values plotted against the S_rw/S_w (S-CSRi) for four groups of illuminations including yellowish illuminations near or close to the Planckian locus (2500 K < CCT < 3500 K), bluish illuminations near or close to the Planckian locus (CCT > 8000 K), greenish illuminations, and reddish illuminations, marked in yellow, blue, green, and red symbols, respectively.

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The minimized prediction error DEu’v’ or DEu’₁₀v’₁₀ for the yellowish, bluish, greenish, and reddish illuminations are summarized in Fig. 5. According to the different CMFs adopted to calculate chromaticity, the 10 data sources are divided into two groups: (1) CC7 (except the Lam & Rigg data) using 2° CMFs, (2) Lam & Rigg, Smet et al., Wei et al. and Ma et al. data sources using 10° CMFs, corresponding to Fig. 5(a), Fig. 5(b), respectively. In each subfigure, the four groups of illuminations (Yellow, Blue, Green, and Red) – represented by yellow, blue, green, and red symbols, respectively – are plotted against S-CSRi. Differently to Fig. 4, the prediction error is averaged over the illuminations with different luminance or adapting field size for each dataset, and the bars refer to the standard deviation of the prediction errors. If there is no standard deviation bar for a data point, it indicates that this dataset only includes one illumination condition, such as the illumination A in CSAJ-C, Kuo & Luo, Lam & Rigg, Helson and LUTCHI data sets. It can be observed that the prediction error tends to increase as the S-CSRi deviates more and more from 1.0, for both yellowish and bluish illuminations. For the Smet, Wei, and Ma data, as shown in Fig. 5(b), the prediction errors of the yellowish illuminations near the Planckian locus are higher than those of the bluish illuminations from the same data source, while the prediction errors for greenish illuminations are fairly small, similar to the bluish illuminations, even though their S-CSRi values are much larger than 1.0. And for other CC7 datasets, as shown in Fig. 5(a), the yellowish illuminations in the Breneman-C data result in a larger DEu’v’ than other datasets in CC7.

 

Fig. 5. Minimized prediction error plotted against the S_rw/S_w (S-CSRi). For a certain illumination from one data source, if there is more than one illumination condition, for example, when there are multiple luminance levels and/or adapting field sizes, both the prediction errors and S_rw/S_w values are averaged over these illumination conditions. Yellowish (2500 K < CCT < 3500 K), bluish illuminations (CCT > 8000 K), greenish, and reddish illuminations are represented by yellow, blue, green, and red symbols, respectively. The illuminations from different data sources are represented by different marker symbols. The black dash line is the piecewise linear function converging at S-CSRi = 1.0 that has been fitted to all the data points in each subfigure. (a). Illuminations from the CC7 sets except the Lam & Rigg data. (b). Illuminations from the Lam & Rigg, the Smet, Wei and Ma CC sets.

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In each subfigure of Fig. 5, the prediction errors were fitted by piecewise linear functions converging at S-CSRi = 1.0 to minimize the root-mean-square-error (RMSE) of each data group. It can be seen that the slope of the line for S_rw/S_w > 1 for the CC7 data excluding the Lam & Rigg data is smaller than that of the Lam & Rigg, Smet, Wei, and Ma data, indicating a slower increment of the prediction error when the illumination becomes less neutral. In addition, as shown in Fig. 5(b), the prediction error DEu’₁₀v’₁₀ for the Smet data is always higher than that for the Wei and Ma data at similar S-CSRi level, probably due to the presence of more corresponding colors per illumination pair in the Smet data (N = 5) than that in the Wei and Ma data (N = 1) which can introduce an extra error from the sample variation in eCSRs. Consistently, the prediction error for the Lam & Rigg data (N = 58) is higher than that for the Wei data (N = 1). From another perspective, the results derived from multiple corresponding colors can be more reliable than from one corresponding color and avoid data overfitting.

For the ten data sources, the root-mean-square (RMS) of prediction errors of two CAT models (full model) widely used in the industry - CAT02, CAT16, and the standard von Kries CAT (with D_optim and HPE sensor) are summarized in Table 2. Note that the CAT16 model adopted here is the corrected version which solves the inconsistency problem of the original CAT16 by excluding the Y_w factor [36]. It can be observed that for almost all the data sources, CAT02 slightly outperforms CAT16 due to its sharpened sensor space. For CC7, the performance of CAT02 and CAT16 is similar to the von Kries CAT, where the DEu’v’ or DEu’₁₀v’₁₀ differences are smaller than 0.0033. While for the Smet, Ma, and Wei data, the von Kries CAT (with D_optim and HPE sensor) substantially outperforms the other two models (CAT02 and CAT16), due to the existence of highly chromatic illuminations in the three data sources.

Table 2. The minimized prediction errors for two CAT models (full model)- CAT02, CAT16, and the standard (Eq. (2)) von Kries CAT with one optimized D, a von Kries CAT with two optimized D values (one for L/ M and one for S cone signals), and the modified von Kries models M1 (Eq. (5)). Note that the M1 model is not described in this section, but it will be introduced later in section 2.4.3. The DEu’v’ or DEu’₁₀v’₁₀ values for each data source are the RMS values overall the datasets. The data sources whose performance are improved and substantially improved (> 0.0033) by M1 are marked in italic and bold, respectively.

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To further investigate the origin of the prediction error, the performance of a von Kries CAT with different optimized D values for L/M and S cones will be tested and compared with the standard von Kries CAT in the next section.

2.4.2 2-D von Kries CAT: one D for the L/M-cones, one D for the S-cone

In the past, some CATs have also treated the S-cones differently from the L and M cones in the chromatic adaptation calculation, such as the CMCCAT97 [37], Bartleson CAT [9,38] and BFD transform [8] by for example including a compression of the S-cone signals. To quantify the prediction error from the standard von Kries CAT with one D for all three cone types (Eq. (2)), another von Kries CAT model with two D factors, one for the L- and M-cones and another for the S cones was tested using these CC data sources. The RMS of prediction errors of the standard, and 2-D von Kries CAT for all these data sources are summarized in Table 2.

For the CSAJ-C, Helson, LUTCHI, Lam & Rigg, Kuo & Luo, and Braun & Fairchild data sources, the optimization with two D values can only improve the performance of the standard von Kries CAT (with a single D value for the three cones) slightly (less than 0.001 in terms of DEu’v’), which is consistent with their similar locations relative to the y = x line for the three cones as shown in Fig. 2. The prediction errors for the standard von Kries CAT are therefore mainly due to the variation of the eCSRs values across different samples (i.e. internal sample inconsistency). For the Breneman-C, Smet, Wei, and Ma data, the RMS of the prediction error of the 2-D optimization is substantially lower than the standard von Kries model with one D value. Note that for the Wei and Ma data, the RMS of DEu’₁₀v’₁₀ of all the illuminations is close to or smaller than 0.001, because there is only one target chromaticity (achromatic matching) in those CC data sets.

The better performance of the 2-D von Kries CAT was also confirmed by the Akaike Information Criterion (AIC) [39] values, which are substantially smaller for the 2-D von Kries CAT (2 free parameters) than for the standard von Kries CAT. Calculation of the relative likelihoods [40] of the two models indicated that the 2-D model is ‘infinitely’ better than the standard one for the Wei and Ma data, and approximately 6 and 33 times better for the Breneman-C and Smet data, respectively. These results suggest that the better performance is not due to the extra free parameter and that the 2-D von Kries CAT is indeed a better, more predictive model than the standard (single D) von Kries CAT.

For the datasets from the Breneman-C and Smet data sources whose prediction errors are reduced by the 2-D optimization, the optimized D values for the L/M-cone and S cone types are shown in Fig. 6, together with the optimized D of the standard von Kries CAT. To investigate the impact of the matching uncertainty (due to the color discrimination threshold) on the degree of adaptation, for each illuminant, the Gaussian process regression bootstrap resampling method was adopted to simulate the matching results for 1000 times with the matching point (from the dataset) as the center, and the JND (just-noticeable-difference) in u’v’ space (0.0033) [41] as the standard deviation in the bivariate Gaussian function. The error bar in Fig. 6 was calculated as the standard deviation of the optimized D values (in the 2-D optimization and the standard von Kries CAT) from the 1000 simulations, and it indicates the variation of the derived optimized D value due to the matching uncertainty. It can be observed that for the illuminations with small S-cone signals – which includes illumination A, B2 K, Yellow, and Green in the Smet data; and illumination A, Yellow and Green in the Breneman-C data – that the optimized D values for the L- and M-cone types are substantially larger than those for the S-cone type. This confirms the analysis in Section 2.3 that for the yellowish and greenish illuminations in the Breneman-C, Smet, Wei, and Ma data, the eCSRs values of the L/M cones are very close to the corresponding CSRi values, while for the S cone, the eCSRs values are considerably smaller than the CSRi value as shown in Fig. 2. As shown in Fig. 6, for all the illuminations in the Smet data, except for the ‘B12K’, ‘Blue’, ‘Red’, and ‘EEW’ illumination conditions, the optimized D values for the L- and M-cones are larger than that for the S cones. Therefore, in the standard von Kries CAT, with a single D for the three cones types, can indeed introduce a large(r) prediction error for the yellowish illuminations, because of the required lower D value for the S-cones.

 

Fig. 6. Summary of the optimized D values using a standard von Kries CAT with one D for all three cones and with one D for the L/M cones and another for the S cones. The error bars indicate the variation of the derived optimized D value due to the matching uncertainty. The two subfigures correspond to the Breneman-C and Smet data sources. In the subfigure of the Breneman data, the labels ‘A’, ‘Y’, ‘G’ in the x-axis represent illuminant A, yellowish illumination, and greenish illumination, respectively, and the number below the label of each illumination indicates its luminance level with the unit cd/m².

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2.4.3 Model M1: modified von Kries CAT with a compressed scaling factor function for the S-cone type

Even though the 2-D von Kries CAT outperforms the standard von Kries CAT, it has two free parameters to be optimized for each dataset. To reduce the prediction errors due to the different degrees of adaptation required for the L/M cone signals and the S cone signals for yellowish illuminations in a von Kries-type CAT that adopts only a single D-value, the following modified von Kries CAT (M1) with a compression of the S cone gain factor is proposed as shown in Eq. (5). As mentioned earlier, some existing CATs–such as the CMCCAT97 [2], the Bartleson CAT [9,38] and the BFD transform [8]–have also treated the S-cones differently from the L and M cones [34] by compressing the stimulus’ S cone excitation using a power function. However, a major drawback in these earlier models is that the power function is not readily invertible. As the modified model (M1) in Eq. (5) applies the compression to the von Kries scaling factor itself, it overcomes the invertibility problem, while remaining a strong von Kries model with an independent linear gain control of all three cones.

The compression factor q (= 0.9) in Eq. (5c) is a constant that has been empirically derived to minimize the RMS of the prediction errors of all 10 CC (i.e. the CC7, Smet, Wei, and Ma) datasets analyzed earlier. The general validity of the derived q-values was analyzed using k-fold cross validation (k = 10). The q factor in M1 optimized for any of the k-1 data sources ranges from 0.68 to 0.97 with a standard deviation of 0.08. The optimized value of the compression factor q in M1 (see Eq. (5c)) is not consistent with that in CMCCAT97 (q = 0.0834) and the Bartleson model (q = 0.45) as the compression exponent in M1 is applied to the von Kries scaling factor of the S cone itself, while the CMCCAT97 and the Bartleson model applies the exponent to the S-cone signal.

In M1, the gain control of the S cone is modeled as a power function with an exponent p that is modulated by the D value and a compressed (1/CSRi) factor (exponent q = 0.90). If D is equal to 1.0, the p factor is equal to 1.0, and M1 will reduce to the standard von Kries model (Eq. (2)). Therefore, M1 meets the boundary condition that S_c = S_rw when D = 1.0 and S = S_w. Note that a similar model was proposed earlier in Ma et al. [42], but it does not meet the boundary condition. If D is smaller than 1.0, the p value is related to S_w/S_rw. Figures 7(a) and 7(b) show the relationship – for D values ranging from 0.0 to 1.0 – between the CSRi and theoretical CSRs predicted by the standard von Kries CAT (Eq. (2)) and M1 (Eq. (5b)), respectively. When D is equal to 1.0, the corresponding curve of M1 is a straight line which is the same as Fig. 7(a). As D becomes smaller, the curve of M1 shown in Fig. 7(b) tends to be flatter, and the CSRs value is less influenced by CSRi when CSRi is larger than 1.0. For the illuminations with S_w smaller than S_rw, such as illuminations A and B2 K, p is smaller than 1.0, leading to a smaller rescaling factor for the S cones than the one in the standard von Kries CAT. While for the illuminations with S_w larger than S_rw such as B12 K and Binf, p is larger than 1.0 and the scaling factor for the S cone will be larger than the one in the standard von Kries CAT.

 

Fig. 7. The relationship between the illumination ratio (CSRi) and stimulus ratio (theoretical CSRs) with different D values varying from 0.0 to 1.0 with steps of 0.2. (a). The von Kries CAT model as shown in Eq. (3). (b). The modified von Kries CAT model (compression only for the S cone) as shown in Eq. (5).

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For each data set, the only free parameter in M1 is the D factor which affects all three cone types. Note that as mentioned earlier, q was optimized over all data sets at the same time and once determined can be considered a constant of the model. Table 2 summarized the RMS prediction error of M1 for all the ten data sources. For the CSAJ-C, Kuo & Luo, Lam & Rigg, Helson, LUTCHI, and Braun & Fairchild data, M1 has a similar performance to the standard von Kries model, and the DEu’v’ difference of the two models is smaller than about a third of a JND [41]. It is consistent with the results mentioned in section 2.4.2, that their prediction errors are mainly due to the sample variation. As M1 does not deal with internal sample inconsistencies, it does not outperform the standard CAT for these data sources. For the Breneman-C, Smet, Wei and Ma data, M1 can however substantially improve the performance compared with the standard von Kries CAT, especially for yellowish illuminations which require a smaller scaling factor for the S-cones than predicted by the standard von Kries model. Note that as an empirical model derived from CC data, M1 is not the only solution to improve the performance of the standard von Kries CAT, but it has the advantage that it remains the form of a von Kries model without including an extra parameter.

3. Discussion

3.1 Discrepancy among the data sources

The analysis shows that different data sources can lead to different conclusions. For example, for most CC7 data sources (CSAJ-C, Kuo & Luo, Lam & Rigg, Helson, LUTCHI, and Braun & Fairchild data), the yellowish illuminations (low CCT) have a similar degree of adaptation to the neutral and bluish ones, while for other data sources (Breneman-C, Smet, Wei, and Ma), the illuminations with a small response in the S cone have a smaller D and a higher prediction error than the neutral and bluish illuminations.

One possible explanation of the discrepancy could be the different experimental methods. As mentioned above, the LUTCHI data [23], and the Kuo & Luo data [22] were derived from magnitude estimation data. For the LUTCHI, and Kuo & Luo data, the chromaticity of the corresponding colors under the reference illumination are derived from their relative position to the iso-hue and iso-colorfulness contours in the CIELAB color space [2]. However, if the illuminant is not close to daylight, CIELAB fails to accurately estimate hue, especially in the blue region [43,44]. The disadvantages of the methods adopted in the derivation of the LUTCHI, and Kuo & Luo data are a low precision due to the relatively large observer estimation error in those types of experiments and the inaccuracies in the color iso-lines in the color diagram [2,45]. The Lam & Rigg data [8] and the Helson data [5] were collected from memory matching experiments and the corresponding color under the reference illumination was interpolated based on the XYZ tristimulus for all the Munsell samples under the reference illumination [2,8]. However, the memory matching method has several drawbacks, such as long training period, complex data analysis, short-term memory distortion, which can result in inaccuracy of the corresponding color data [2]. In Braun & Fairchild’s cross-media matching experiment [24], after viewing the original image (reference stimulus) in the lighting booth for 60 s, the observer was asked to stare at a reproduction on a display for 60 s before starting the matching. However, as suggested by Olkkonen et al. [46], the short-term memory of the original image can be weakened by a short interval (60 s) between the exposure to the original image and the matching task. Both the CSAJ-C [21] and Breneman-C [12] adopted a haploscopic matching method, but Breneman-C overcomes one of the shortcomings of this technique − unnatural viewing conditions. However, binocular rivalry can still lead to a different adaptation state than under normal viewing conditions. The achromatic matching experiments in Ma et al. [28] and Wei et al. [27], and the MCM experiments in Smet et al. [20] share the same principle – long-term memory colors associated with familiar objects are used as an internal reference. From the results of the analysis in our study, similar experiment methods tend to result in similar results, except for the Breneman-C data which was derived using a method similar to the CSAJ-C data, while the results were more similar to the Smet, Wei, and Ma data.

The interaction between the local and global adaptation would be an influential factor. For the CSAJ-C, Kuo & Luo, Lam & Rigg, Helson, LUTCHI, and Ma data, the color perception is dominated by the global adaptation as the stimulus was presented against a uniform adapting field without any adjacent color patch. For the Breneman, Braun & Fairchild, Wei and Smet data, due to the complexity of the adapting field (Breneman, Braun & Fairchild, Wei) or the strong contrast between the stimulus and the background (Smet), the color perception is jointly influenced by both the local adaptation (to the adjacent color) and the global adaptation (to the illumination). However, in the global adaptation group, the predictive performance of the von Kries CAT for the Ma data is substantially different from other five data sources (the CSAJ-C, Kuo & Luo, Lam & Rigg, Helson, LUTCHI), indicating that the local/global adaptation state doesn’t fully account for the discrepancy among data sources.

Another possible reason for discrepancy could be the different sizes of the adapting field. For the CSAJ-C, LUTCHI, Kuo & Luo, Lam & Rigg, Helson, Braun & Fairchild, and Wei data, the visual experiments were conducted in an immersive lighting booth, while the Breneman-C, Smet, and Ma data were collected with an adapting field with a limited size and a dark surround. It has been illustrated in Ma et al. [28] that the degree of adaptation was larger for larger adapting fields, which explains why the yellowish illuminations under the immersive environment have a more complete adaptation state than that under a limited adapting field. However, for the Wei data, even though it was collected in a lighting booth with an immersive adapting field, the degree of adaptation of the yellowish illumination is substantially lower than for data from other lighting booth experiments with similar viewing conditions.

The discrepancy could be explained by the different media of the target object (stimulus). For the CSAJ-C, LUTCHI, Kuo & Luo, Lam & Rigg, Helson data, the visual experiments used real reflective objects as the stimuli which were made by paper or textile. And for these datasets with surface stimuli, the yellowish illuminations (low CCT) result in a similar degree of adaptation as the neutral and bluish ones. While the Breneman-C, Smet, Wei and Ma data where the yellowish illuminations have a lower degree of adaptation and a higher prediction error than the neutral and bluish illuminations were collected with projection or display media. However, for the Braun & Fairchild data, even though the color matching experiment was conducted using a calibrated display, its predictive performance of the von Kries CAT is more similar to the surface stimuli group. One possible explanation is that the adapting field in Braun & Fairchild’s experiment was provided by a complex image which can enhance the degree of adaptation compared to a uniform field [47]. The discrepancies between different stimulus media observed from the ten data sources are consistent with the findings of Zhai et al.’s work [48] where the impact of viewing medium on chromatic adaptation was thoroughly investigated. The medium effect can be due to the metamerism (the spectra of self-luminous stimuli tend to be less ‘smooth’ than that of reflective stimuli) or the human cognitive process, both which deserves further investigation.

3.2 Physiological differences between the L/M- and S-cone

Much physiological evidence indicated the differences between the L/M cone and the S cone in the visual system. Different processing of L/M and S cones has been reported in the literatures as they map to different visual pathways. In post-receptor processing, there are two pathways related to color vision: the koniocellular pathway and parvocellular pathway, corresponding to yellow-blue and red-green opponent, respectively [49,50]. The S-cone signals only map to the koniocellular pathway, while the L- and M-cone signals map to both pathways. Mullen et al. [51] found that the contrast sensitivity of the koniocellular pathway is lower than that of the parvocellular pathway, which is probably caused by the lower discharge rate of koniocellular cells. Medina et al. [52] found that the measured visual reaction time for test stimuli varying along the L - M or S - (L + M) axis cannot be estimated by a single function, indicating the existence of separate adaptation mechanisms for each opponent system. Baudin et al. [53] explored the difference in the kinetics of cone responses between L/M and S cones in the primate retina and found the signal generated by the S cones to be slower than that of the L and M cones. In addition, Baudin et al. also noticed that when the background luminance increases, the change in response kinetics of S cone is around 60% smaller than that of the L and M cones. Several studies have indicated that the chromatic adaptation of the S-cone involves a sensitivity adjustment not only at the photoreceptor, but also at the post-receptoral opponent site [54]. The transient tritanopia found by Pugh and Mollen [55] suggested that the sensitivity of the S-cone was controlled by both the S-cone stimulus and a long-wavelength-dominated mechanism. The combinative euchromatopsia found by Polden and Mollen [56] provided evidence for the sensitivity change at a post-receptoral opponent site. As suggested by these studies, the L/M- and S-cone pathways are processed by separate post-receptoral adaptive mechanisms with spatial and temporal physiological distinctions, supporting the different adaptation states between the L/M and S cone. In addition, to achieve the same adaptation state as the L/M cone, the S cone requires a larger change in the gain control factor due to the large variation in S-cone stimulation (see Fig. 3), but the gain in the S-cone pathway is insufficient to compensate for the reduced stimulation of the S-cone. These reasons likely contribute to its lower degree of adaptation than the L/M cone.

4. Summary and conclusion

The corresponding color pairs collected from 10 different data sources, including the CSAJ-C [21], Kuo & Luo [22], Lam & Rigg [8], Helson [5], LUTCHI [23], Breneman-C [12], Braun & Fairchild [24], Smet [19,20], Wei [27], and Ma [28] data sets, have been analyzed and compared.Firstly, it has been found that for the datasets with more than one sample, the sensor responses under the test illumination are highly correlated with those under the reference illumination, indicating the consistency of the eCSRs (i.e. L_r,c/L, M_r,c/M, S_r,c/S) of different samples. But the variation in the scaling factor (eCSRs) still exists, which can be one of the origins of the prediction errors of the von Kries CAT.

Secondly, for the low CCT illuminants on or near the Planckian locus, substantial discrepancies of the S-cone eCSRs between the ten data sources have been observed. The Breneman-C, Smet, Wei and Ma data have considerably lower eCSRs values for the S cones (despite similar CSRi values) than the other data sources from CC7. In addition, under different Planckian radiators (with different CCT), the CSRi values of the L and M cones are always around 1.0, and their ranges of values are substantially smaller than that of the S cones, indicating a clear difference between L/M and S cone.

Thirdly, it has been found that the optimized degree of adaptation, calculated using the standard von Kries CAT, is lower for the yellowish illuminations from the Breneman-C, Smet, Wei, and Ma data sets than from the other data sources. Additionally, the prediction error increases when the CSRi value of the S cone deviates from 1.0, except for the greenish illuminations. To investigate potential contributions to the prediction error, the von Kries model has been optimized with two D values — one for the L- and M-cones and another for the S cones — to minimize the prediction error of the von Kries CAT for each dataset. The performance of the CSAJ-C, Kuo & Luo, Lam & Rigg, Helson, LUTCHI, Braun & Fairchild data could not be further improved using two D values for the L/M and the S cones, which suggests that their prediction errors are mainly due to sample variation in the eCSRs (i.e. internal sample inconsistencies). However, for the Breneman-C, Smet, Wei, and Ma data, the prediction errors could be improved by incorporating a separate D-factor for the S cone. The latter was found to be substantially smaller than that of the L and M cones for yellowish illuminations. It is probably because the gain of S cone is not sufficient to cover the large variation in S-cone stimulation.

Fourthly, one modified model was proposed by applying a compression only to the rescaling factor of the S-cone. Like the regular von Kries CAT, the modified model also only requires a single D-factor and the difference in the L/M-cones and S-cones is taken care of by the compression function, itself a function of the D-factor. The modified model outperforms the standard von Kries model for the Breneman-C, Smet, Wei, and Ma data. For the other data sets, the modified model has a similar performance. In a conclusion, the modified von Kries CAT is recommended especially for the illuminations with a low excitation of S cone.