1. Introduction

Accurate models for color difference thresholds are necessary for many technologies and applications, for example: displays [1], medical decision-making [2,3], medical research [4], perceptual image processing and video communication [5], and color quality control in many industries [6,7]. Since the 1930s [8] considerable time and effort has been spend to the development of color difference metrics by academia and industries. Multiple color difference formulas were recommended by international standardisation bodies [9] [10] (pp.825-830). In many cases an approximately uniform color space was formulated with an Euclidean distance as a measure for small color differences. However, additional weighting functions were required to fit experimental color difference data sets [11] (pp.96-105). The color difference formula CIEDE2000 [12] is an example of this approach.

Experiments confirm that the color space is a three-dimensional space, but not Euclidean and a more general Riemannian space is required [13–17]. The small distance between two neighbouring points in color space is given by a positive definite differential form, called the line element. Describing small color differences with a line element has a long history which started at the end of the 19th century [18]. Several line elements for color space, derived from theoretical considerations and experimental data, were developed [19]. The Friele (1961) [20] and Friele (1965) [21] line elements showed good agreement with experimental color matching data, and were amended and optimized by MacAdam [22] and Chickering [23] with the intention to be widely applied. However, this approach was almost abandoned since the end of the 1970s. [In the following parts of the paper the Friele (1961) [20] and the Friele (1965) [21] line elements are denoted as respectively F61 LE and F65 LE.]

In the meantime psychophysical color discrimination research has made important progress and this opened up renewed interest to tackle the problem of the prediction of visibility thresholds of small color differences with a line element. In the present paper we describe the development of a new line element for color space suitable for test fields of $\approx 2^{\circ }$ angular diameter and a luminance in the range of about 1 cd m$^{-2}$ to 400 cd m$^{-2}$. This new line element, denoted as ’new LE’, is based on the earlier work of Friele and on psychophysical models which were mainly developed since the early 1980s. We have validated the existing and the new LE against data sets available in the literature, starting with MacAdam’s [24,25] famous ellipses published in 1942 for a single observer PGN (we will refer to these as MA42). Additional measurements were published by Brown-MacAdam in 1949 [26,27] (abbreviated as BM49) for two observers (-WRJB and -DLM). Similar ’color-matching ellipses’ were published by Wyszecki and Fielder in 1971 [28] (abbreviation WF71, with observers -GF, -AR and -GW) as well as larger ’color-difference matches’ also in 1971 [29] (abbreviation WF71d, with the same observers). Finally we used the well-known RIT-DuPont dataset published in 1991 [30–32] which is an average over 50 observers (abbreviated as RD91). Additional details about these sets can be found in the top halve of Table 2. Note that only dataset RD91 covers a relatively large luminance range, whereas the range for the BM49 data sets is moderate and the other sets only consider a single luminance level. Also the chromaticity ranges are quite different as can be seen in Figs. 6, 7 and 10.

In the next section a brief overview of the relevant psychophysical models is given and the discrimination threshold models along the cardinal directions are worked out. Section 3 describes the development of the new LE for color space. In section 4 the agreement between color discrimination ellipsoids calculated with the new LE and experimental color discrimination ellipsoids reported in literature is presented. Also in this section the impact of new off-diagonal elements of the metric tensor and the von Kries transformation are discussed, and a comparison of the new LE with several color difference formulas and the F65 LE is made.

2. Discrimination threshold models

2.1 Psychophysical models

Color vision is modeled as a series of stages, the first and second stage are important for color discrimination. In the first stage incident photons can be absorbed by the long-wavelength, middle-wavelength and short-wavelength sensitive cone photoreceptors, which results in three signals with magnitude $L$, $M$ and $S$ respectively. In the present study the spectral sensitivities of the foveal cone photoreceptors as determined by Smith-Pokorny [33] and the chromaticities as defined by MacLeod-Boynton [34,35] were used. The Smith-Pokorny spectral sensitivities are linear transformations of the Judd modified color matching functions [10] (pp.331, 615) [36]. The MacLeod-Boynton chromaticities are respectively $l=\frac {L}{Y}$, $m=\frac {M}{Y}$ and $s=\frac {S}{Y}$, with $Y=L+M$ the retinal illuminance [td] (troland). The $(l,m,s)$ chromaticities are determined by Eq. (1) with $(x_{\text {J}},y_{\text {J}},z_{\text {J}})$ the chromaticities according to the color matching functions modified by Judd.:

In the second stage the cone contrast signals $\frac {dL}{L}$, $\frac {dM}{M}$ and $\frac {dS}{S}$ are encoded into an achromatic signal, a red-green color opponent signal and a blue-yellow color opponent signal. For discrimination of color differences there are three independent directions, i.e. the achromatic direction $(dl=0$ and $ds=0)$, the red-green $(dY=0$ and $ds=0)$ and the blue-yellow $(dY=0$ and $dl=0)$ color opponent directions. These directions are called ’cardinal directions’ [37] (pp.11.1 and 11.31-33). Important are the changes of the operating properties of the visual system caused by changes in the visual field, called visual adaptation. Ekroll and Faul [38] considered two distinct visual adaptation effects caused by backgrounds: (i) von Kries adaptation [39] (pp.168-171) [40] [11] (pp.21-23) and (ii) crispening effects [41,42]. The effect of backgrounds on the discrimination thresholds along the cardinal directions was examined in several psychophysical experiments. For the achromatic thresholds we used the experimental results of Blackwell [43] and Whittle [44], and for the discrimination thresholds along the color opponent directions we used the experimental results of Yeh et al. [45], Krauskopf-Gegenfurtner [46], Miyahara et al. [47], Smith et al. [48] and Rovamo et al. [49].

2.2 Discrimination threshold along the achromatic cardinal direction

Blackwell [43] measured the achromatic threshold contrast for obs ervers adapted to a $10^\circ \times 10^\circ$ background and a circular test field of about $2^{\circ }$ angular diameter with the same luminance as the background. In the retinal illuminance range from $0.08$ td to $2\,\,10^4$ td, covering both the Rose-deVries and the Weber zone, Eq. (2) was fitted to the data reported by Blackwell [43] (Table VIII p.642). The result is shown in Fig. 3(a).

To take into account the effect of the background luminance on the contrast threshold, we used the results of the brightness scaling experiment by Whittle [44]. In this experiment 25 disks with diameter $2.1^\circ$ were shown in a spiral arrangement with a gap of $1^\circ$, the first (bright) and the last (dark) disk had a fixed luminance. The task was to adjust the brightness levels of the other 23 disks to a series of equal appearing steps in grayness. Whittle found that the threshold contrast has a sharp minimum when the luminance of the test field approximates the luminance of the background. This is called the crispening effect [41], and a model for this effect was devised [44]. It was also found that the crispening effect almost disappears by inserting a thin black annulus between the background and the circular test field, or when the background and the test field have a hue difference. For the threshold contrast Eq. (3) was derived from Whittle’s model:

The crispening effect is shown in Fig. 2(b). From Eq. (3) and Eq. (5) it is clear that the measurements by Whittle were done in the Weber zone, because for $Y=Y_a$ the threshold contrast $\frac {dY}{Y}$ is independent of $Y$. Equation (3) was simplified by setting $a_p=|a_n|$, i.e. it was assumed that the small difference between $a_p$ and $|a_n|$ is negligible. Generalization of the achromatic threshold contrast for both the Rose-deVries and the Weber zone is done by replacing the luminance-independent factor in Eq. (3) by Eq. (2). Therefore we propose that the achromatic threshold contrast $\frac {dY(Y,Y_a)}{Y}$ (for $dl=0$ and $ds=0$) is equal to:

 

Fig. 1. Luminance levels of two neighbouring disks with positive and negative contrast.

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Fig. 2. (a) Threshold elevation $\Theta (Y,Y_a)$ as a function of $\frac {Y}{Y_a}$ for $b=1$. (b) Crispening effect: normalized luminance difference threshold as a function of $\frac {Y-Y_a}{Y_a}$.

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The justification for the simplification $Y_d=0$ in Eq. (4) is as follows. For very low luminance levels the threshold becomes constant [50]. Often this is modeled by a so-called ’dark light’ constant [51]. To take this into account we should modify the square-root factor in Eq. (6) and therefore the dark light constant introduced by Whittle in the threshold elevation Eq. (4) becomes superfluous. However, considering the luminance range of the test data there is really no need to introduce a dark light constant. As already mentioned, among the considered data sets only the RD91 data set covers a relatively large luminance range. The model for the achromatic threshold contrast was therefore applied on the RD91 data set. In this case the observation of the sample pairs was on a background with a different hue, therefore Eq. (6) with $b=1$ was fitted to the RD91 threshold contrast data points with $Y_a=379$ td as per the measurement conditions, with model parameter $Y_A=100$ td and free model parameters $\kappa _0$ and $k_{\text {sc}}$. The results were $\kappa _0=4.23\, 10^{-2}$ and $k_{\text {sc}}=0.14$. The corresponding achromatic threshold contrast $\psi _A(Y)$ is shown in Fig. 3(b).

 

Fig. 3. (a) Achromatic threshold contrast as reported by Blackwell [43] (Table VIII p.642) (red squares) and Eq. (2) with with $a_w=0.00283$ and $Y_A=23.2$ td (blue line). (b) Achromatic threshold contrast of the RD91 data set (red squares) and model $\psi _A(Y)$ (blue line).

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2.3 Discrimination thresholds along the red-green cardinal direction

Yeh et al. (1993) [45], Miyahara et al. (1993) [47], Krauskopf-Gegenfurtner (1992) [46], and Rovamo et al. (2001) [49] measured discrimination thresholds along the red-green cardinal direction. In this direction the S-cone excitation is constant and the variations of the L-cone and M-cone excitations are opposite with equal magnitude to keep the luminance constant. These experiments used different measurement apparatuses and pychophysical methods. For the first three experiments the stimuli were aperiodic and were observed on various backgrounds (white, yellow, red and dark). Each of these experiments showed that, for a steady adapting background, the discrimination threshold as a function of the L-cone excitation has a V-shape with a minimum for equal $l$-chromaticity of the stimulus and the background (crispening effect). The $l$-chromaticities of the stimulus and the adapting background are denoted by $l$ and $l_a$ respectively. For the experiments of Yeh et al. the background was dark, and they found the minimum threshold for $l\approx 2/3$, this corresponds with the $l$-chromaticity of equal-energy white (EEW) $l_E\approx 2/3$; this was also found by Boynton-Kambe (1980) [51]. Rovamo et al. measured the $Y$-dependence of the contrast sensitivity of a red-green chromatic grating in the retinal illuminance range from 2.4 td to 2400 td ($\approx$ 0.1 cd m$^{-2}$ to 500 cd m$^{-2}$), whereas the measurements of Yeh et al. were in a more limited range between 2.9 td and 290 td. The unmodulated test signal was white $(l\approx 2/3)$ and the background had the same color $(l=l_a)$. Rovamo et al. approximated the measured threshold contrast $\frac {dL}{L}$ by:

Based on the above mentioned experimental results we propose that the red-green threshold contrast $\frac {dl}{l}$ (for $dY=0$ and $ds=0$) is equal to:

Eq. (8) was fitted to the data points of the red-green threshold contrast as derived from the measurements by MA42 with $l_a=0.656$ and $Y=237$ td as per the measurement conditions, with model parameter $Y_T=100$ td and free model parameters $\kappa _1$ and $\kappa _2$. The results are $\kappa _1=6.05\,\,10^{-4}$ and $\kappa _2=5.46\,\,10^{-3}$. The corresponding red-green threshold contrast $\psi _T(l)$ is shown in Fig. 4(a).

 

Fig. 4. (a) Red-green threshold contrast of the MA42 data set (red squares) and $\psi _T(l)$ (blue line). (b) Blue-yellow threshold contrast of the WF71-AR data set (red squares) and $\psi _D(s)$ (blue line).

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2.4 Discrimination thresholds along the blue-yellow cardinal direction

In the same series of experiments as in paragraph 2.3, Yeh et al. (1993) [45], Miyahara et al. (1993) [47], Krauskopf-Gegenfurtner (1992) [46], and Rovamo et al. (2001) [49] measured discrimination thresholds along the blue-yellow cardinal direction on various backgrounds. In the blue-yellow cardinal direction the S-cone excitation varies while the L-cone excitation and the luminance are constant. The findings were discrimination threshold curves with V-shapes with minima for equal $s$-chromaticity of the stimulus and the background (crispening effect). The $s$-chromaticities of the stimulus and the background are denoted as $s$ and $s_a$ respectively. On a dark background the minimum threshold occurred at $s=0$. Rovamo et al. measured the $Y$-dependence of the contrast sensitivity for blue-yellow gratings in the retinal illuminance range from 2.4 td to 2400 td, whereas the measurements of Yeh et al. were limited to the range from 2.9 td to 290 td. The unmodulated test signal was white with $s$-chromaticity $s\approx 0.01608 \triangleq s_E$ and the background had the same color $(s=s_a)$. The $s$-chromaticity of EEW is denoted by $s_E$. They approximated the measured threshold contrast $\frac {dS}{S}$ by:

Based on the above mentioned experimental results we propose that the blue-yellow threshold contrast $\frac {ds}{s}$ (for $dY=0$ and $dl=0$) is equal to:

Eq. (10) was fitted to the data points of the blue-yellow threshold contrast as derived from the measurements by WF71-AR with $s_a=0.01608$ and $Y=137.9$ td as per the measurement conditions, model parameter $Y_D=100$ td and free model parameters $\kappa _3$ and $\kappa _4$. The results are $\kappa _3=6.8\,\,10^{-3}$ and $\kappa _4=3.2\,\,10^{-4}$. The corresponding blue-yellow threshold contrast $\psi _D(s)$ is shown in Fig. 4(b).

3. Constructing a new line element

3.1 Friele’s line elements

During the 1960s Ludwig F.C. Friele, researcher at the Fibre Research Institute TNO in The Netherlands, developed the F61 LE and the F65 LE for the human color space. Both are based on the qualitative 3-stage Müller theory [10] (pp.634-639). Friele assumed that color discrimination occurs at the level of an achromatic ($A$) ’process’, a red-green ($T$) ’process’ and a blue-yellow ($D$) ’process’. For small color differences Friele’s line elements are expressed as:

3.2 Line element in MacLeod-Boynton contrast space

The interpretation of the metric tensor elements and threshold ellipsoids is fairly straightforward in the $(\frac {dY}{Y},\frac {dl}{l},\frac {ds}{s})$ space, which we call the MacLeod-Boynton contrast space. In this space the line element is written as:

The diagonal elements $g_{ii}$ are immediately given by the threshold contrasts given previously in Eqs. (6), (8) and (10):

We note that $g_{22}\gg g_{11}+g_{33}$ for (nearly) all color centers. The transformation from the MacLeod-Boynton contrast space to the cone contrast space is given by:

For the graphical representation and analysis of the threshold ellipsoids and ellipses we use a hybrid space called the MacLeod-Boynton hybrid space. In this space the line element is, according to Eq. (14), expressed as:

Following Brown-MacAdam [26], a scale factor $k=0.2\, \log _{10}(e)\approx 0.0869$ for the achromatic contrast $\frac {dY}{Y}$ was introduced to obtain threshold ellipsoids with sizes of equal order of magnitude in the achromatic direction and the chromaticity directions. Without this scaling the threshold ellipses would be excessively stretched in the $\frac {dY}{Y}$ direction and consequently the possible tilting of the ellipsoids would be hidden.

The determination of the off-diagonal elements $g_{ij}\,\, (i\neq j)$ in Eq. (14) was based on an analysis of the F61 LE and F65 LE and the results of threshold experiments. As a first step of this analysis the metric tensor of the Friele line elements was determined in the MacLeod-Boynton contrast space and is denoted as $\mathsf {F_{MLB}}$. From Eq. (13) and Eq. (16) it follows that:

In the hybrid space (17) this tensor can be written as:

For $\alpha =0.5$ (cf. F61 LE) $Q^2$ diverges if $l$ approaches unity and therefore condition (20) will eventually fail for large values of $l$. For $\alpha =\frac {l^2}{l^2+m^2}$ (cf. F65 LE) $Q^2<1$ and condition  20) is usually met since for almost all data points the right-hand side of (20) is larger than unity. The principle axes of threshold ellipsoids are parallel to the coordinate axes if each off-diagonal element of the metric tensor is zero, if not then the ellipsoids are rotated with respect to the coordinate axes. Therefore, in a next step of the analysis, the orientations of threshold ellipsoids with metric tensor (19) were compared with the orientation of the experimental threshold ellipsoids of RD91 and BM49-WRJB. Threshold ellipses in the respective cross sections $dl=0$, $\frac {dY}{Y}=0$ and $ds=0$ were considered.

The threshold ellipses in the cross section $dl=0$ are according to the metric tensor (19) perfectly parallel to the coordinate axes since $g_{13}=0$. The RD91 and BM49-WRJB threshold ellipses in the cross section $dl=0$ are shown in respectively Fig. 8(a) and Fig. 8(b). These figures show threshold ellipses with principal axes nearly parallel to the coordinate axes, and consequently this confirms that $g_{13}$ is zero.

 

Fig. 5. Threshold ellipses of the RD91 data set (black solid line) and calculated threshold ellipses according to the metric tensor (19) for $\alpha =0.5$ (red solid line) and $\alpha =\frac {l^2}{l^2+m^2}$ (green dotted line) in (a) Cross section $\frac {dY}{Y}=0$ (the ellipses are 2$\times$ enlarged) and (b) Cross section $ds=0$ (the ellipses are 3$\times$ enlarged) $Y_{\text {max}}$ [td] is the highest retinal illuminance of the test field in the RD91 data set. In (a) the gamut boundary of the chromaticities according the standard Rec.2020 [53] (gray dashed line) and the spectrum locus (red dotted line) are depicted.

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Fig. 6. Threshold ellipses (2$\times$ enlarged) in the cross section $\frac {dY}{Y}=0$ of the MacLeod-Boynton hybrid space of the RD91 data set (black solid line) and the new LE with respectively optimised parameters (red solid line) and generic parameters (blue solid line). The insets show the threshold ellipses for color 3 (medium gray), 10 (black) and 18 (light gray). Also the gamut boundary of the chromaticities according the standard Rec.2020 [53] (gray dashed line) and the spectrum locus (red dotted line) are depicted.

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Fig. 7. Threshold ellipses (7$\times$ enlarged) in the MacLeod-Boynton chromaticity plane of the MA42 data set (black solid line) and the new LE with respectively optimised parameters (red solid line) and generic parameters (blue solid line). Also the gamut boundary of the chromaticities according the standard Rec.2020 [53] (gray dashed line) and the spectrum locus (red dotted line) are depicted.

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Fig. 8. Cross section $dl=0$ of the threshold ellipsoids in the MacLeod-Boynton hybrid space for (a) RD91 (3$\times$ enlarged). (b) BM49-WRJB (10$\times$ enlarged). At each color center the measured ellipse (black solid line) and the ellipses according to the new LE with respectively the optimized parameter values (red solid line) and generic parameter values (blue solid line) are shown. $Y_{\text {max}}$ [td] is the highest retinal illuminance of the test field in the considered data set.

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For the analysis in the cross sections $\frac {dY}{Y}=0$ and $ds=0$ the threshold ellipsoids were calculated according to the RD91 measurement conditions and metric tensor Eq. (19) for respectively $\alpha =0.5$ and $\alpha =\frac {l^2}{l^2+m^2}$. The diagonal elements $g_{ii}$ are according to Eqs. (15). The results are shown in Figs. 5(a) and 5(b). In the cross section $\frac {dY}{Y}=0$ the calculated threshold ellipses have, for $\alpha =0.5$ and $\alpha =\frac {l^2}{l^2+m^2}$, principal axes parallel to the coordinate axes, whereas the measured threshold ellipses show for increasing $s$-chromaticity an increasing clockwise rotation. This points to an $s$-dependency of $g_{23}$. Notice that the MA42 data set in the MacLeod-Boynton chromaticity plane [Fig. 7] also shows a clockwise rotation that increases with increasing $s$-chromaticity. In the cross section $ds=0$ the calculated threshold ellipses have, for $\alpha =0.5$ and $\alpha =\frac {l^2}{l^2+m^2}$, principal axes parallel to the coordinate axes, whereas the measured threshold ellipses of the RD91 data set [Fig. 9(a)] show for increasing $|l-l_a|$ an increasing counterclockwise rotation. As shown in Fig. 9(b) the BM49-WRJB data set shows for $l>0.8$ similar rotations. However, in this case there is in the complete range of $l$-chromaticities more variability of the orientation of the threshold ellipses, especially for $l$<0.6. This is probably caused by the data noise.

 

Fig. 9. Cross section $ds=0$ of the threshold ellipsoids in the MacLeod-Boynton hybrid space for (a) RD91 (3$\times$ enlarged). (b) BM49-WRJB (10$\times$ enlarged). At each color center the measured ellipse (black solid line) and the ellipses according to the new LE with respectively the optimized parameter values (red solid line) and generic parameter values (blue solid line) are shown. $Y_{\text {max}}$ [td] is the highest retinal illuminance of the test field in the considered data set.

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From this analysis it is clear that the off-diagonal elements $g_{12}=Q \, g_{11}$ and $g_{23}=-Q\, g_{33}$ of the F61 LE and the F65 LE are not in accordance with the measurement data. Notice that if $\alpha =l$ then $Q=0$ and consequently the threshold ellipsoids are perfectly parallel to the coordinate axes, moreover condition (20) is always met. Other equations for $g_{12}$ and $g_{23}$ are required to obtain threshold ellipsoids with orientations which are more closely in line with the measurements. Positive definiteness of the metric tensors $\mathsf {G_{MLB}}$ and $\mathsf {G_H}$ requires that:

In order to comply with the observed orientation dependencies in the the cross sections $\frac {dY}{Y}=0$ and $ds=0$ we propose the following off-diagonal elements:

Eventually, the new LE is given by:

Table 1. Values of the model parameters $Y_A$, $Y_T$, $Y_D$ and $k_{\text {sc}}$.

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3.3 Determining the model parameters

The model parameters $Y_A$, $Y_T$, $Y_D$, $k_{\text {sc}}$, $\kappa _i\, (i=0,1,2,3,4), k_{12} \text { and } k_{23}$ were determined by comparison of the new LE with the measurement by MA42, BM49, WF71, WF71d, and RD91. These experiments and data sets are well-documented and are assumed to be reliable. [The parameters $\kappa _i\, (i=0,1,2,3,4)$ are hereafter denoted as $\kappa _{0\cdot \cdot 4}$.]

First the parameters $Y_A$, $Y_T$ and $Y_D$ and $k_{\text {sc}}$ were considered. Rovamo et al. [49] found for red-green and blue-yellow chromatic gratings an average value of 160 td for the critical retinal illuminances $Y_T$ and $Y_D$. Our fitting of Eq. (2) to the achromatic discrimination data of Blackwell [43] resulted in 23 td for the critical retinal illuminance $Y_A$. The complete model of the new LE was applied to the above mentioned data sets and we found a good fit for values of $Y_A$, $Y_T$ and $Y_D$ around 100 td. Moreover the model is not very sensitive for changes of $Y_A$, $Y_T$ and $Y_D$ around 100 td. Our fitting of Eq. (6) to the achromatic threshold of the RD91 experiments with $Y_A=100$ td results in $k_{\text {sc}}\approx 0.15$. Therefore for each data set we fixed the values as given in Table 1.

In a next step the model parameters $\kappa _{0\cdot \cdot 4},\, k_{12} \text { and } k_{23}$ were considered. An algorithm was developed for the computation of these model parameters. The algorithm was based on the global minimisation of the dissimilarity between pairs of threshold ellipsoids for a specific data set. A pair consists of a measured threshold ellipsoid and a threshold ellipsoid calculated according to the new LE (25), both at the same color center. To quantify the dissimilarity between pairs of ellipsoids we developed a specific algorithm, which is described in the Appendix. The algorithm for the global minimisation of the dissimilarity is described in the next paragraph.

A data set of measured threshold ellipsoids at $N$ color centers was selected, and it was assumed that the complete experiment was conducted in the same conditions and with the same psychophysical method. The threshold ellipsoids were expressed in the MacLeod-Boynton hybrid space $(k\frac {dY}{Y},dl,ds)$. A transformation might be necessary because usually the measured threshold ellipsoids are specified in the CIE 1931 color space. Moreover, the various experiments were conducted with adaptation to different neutral backgrouds (denoted with superscript $a$). This difference was eliminated by a transformation of the color coordinates of the test fields, according to the von Kries rule and assuming complete adaptation, to a common adaptive condition (denoted with superscript $b$). EEW was chosen as the common adaptive condition. The equations for the von Kries rule and the transformation between the color spaces $(k\frac {dY^a}{Y^a},dx^a,dy^a)$ and $(k\frac {dY^b}{Y^b},dl^b,ds^b)$ are given in Supplement 1. For each color center $j=1, 2,\ldots,N$ the tensor elements according to Eq. (25) were calculated with suitable initial estimates for the parameters $\kappa _{0\cdot \cdot 4}$, $k_{12} \text { and } k_{23}$. Therefore, for each color center a pair of ellipsoids, i.e. a measured threshold ellipsoid and a calculated threshold ellipsoid based on the new LE, was available. However, so far the latter is based on initial estimates of the parameters $\kappa _{0\cdot \cdot 4}$, $k_{12} \text { and } k_{23}$. In a next step, for each pair the dissimilarity measure $d_j$ (cf. Appendix) was calculated as well as the root-mean-square error $d_{\text {rms}}=\sqrt {\frac {1}{N}\sum _{j=1}^N d_j^2}$ of the complete data set. To obtain the final parameter values, the minimisation of $d_{\text {rms}}$ with free variables $\kappa _{0\cdot \cdot 4}$, $k_{12} \text { and } k_{23}$ was executed. For this global minimisation the MATLAB solver fmincon was used.

The new LE and algorithm were applied to the data sets MA42, BM49, WF71, WF71d and RD91. For each data set the parameters $\kappa _{0\cdot \cdot 4}$, $k_{12} \text { and } k_{23}$ were calculated for an optimized fit between the measured and calculated threshold ellipsoids. The results are given in Table 2. Notice that the dissimilarity values $d_{\text {rms}}$ are rather small as well as the differences of the $d_{\text {rms}}$ values between the measurements. This suggests an overall good correlation between the measured and the calculated threshold ellipsoids. A qualitative impression of the correlation can be verified in the Figs. 6 to 10.

 

Fig. 10. Threshold ellipses in the CIE 1931 $(x,y)$ chromaticity plane for: (a) MA42 (7$\times$ enlarged). (b) BM49-WRJB (7$\times$ enlarged). (c) WF71-AR (7$\times$ enlarged). (d) WF71d-AR (3$\times$ enlarged). In each case the measured threshold ellipses (black solid line) and the ellipses according to the new LE with respectively optimized parameter values (red solid line) and generic parameter values (blue solid line) are depicted. Also the gamut boundary of the chromaticities according the standard Rec.2020 [53] (gray dashed line) and the spectrum locus (black dashed line) are depicted.

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Table 2. Main parameters of the 10 experiments and the corresponding values of the optimised model parameters $\kappa _{0\cdot \cdot 4}$, $k_{12}$, $k_{23}$ (for $d\sigma =1$). The second last row gives the values of $d_{\text {rms}}$ for the optimized values of the model parameters (as described in paragraph 3.3). The last row gives the values of $d^{\ast }_{\text {rms}}$ for the generic values of the model parameters (as described in section 4). Symbols and abbreviations: $\dagger$: length boundary [deg] of juxtaposed subfields of test field, $\ddagger$: diameter of background [deg] and color of background, W: white, m/b: monocularly or binocularly viewing, n/a: natural pupil or artificial pupil, $N$: number of color centers, $\leftarrow$: same number or symbol as on the left side of the arrow, $\S$: number of observers, $\star$: average age of 50 observers with standard deviation of 7.7 years [31].

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3.4 Experiments: main characteristics and differences

With the exception of the RD91 experiments, all of the considered measurements were color matching experiments conducted with specifically constructed colorimeters. For the color matching experiments by MA42, BM49 and WF71 the test field consisted of two juxtaposed fields and the task was to match an adjustable color to a fixed color of the juxtaposed comparison field. These experiments measured the visibility threshold which is expressed as the standard deviation (SD) of the color matching errors of the repeated color matches [10] (pp.306-323). The WF71d color difference matching experiments differed from the other color matching experiments because the test field consisted of three juxtaposed fields, one with an adjustable color and two with predefined colors with a color difference $\Delta E_{\text {CIE}}=6.7$ (CIE 1964 color difference formula [10] (p.828)), which is substantially larger than the visibility threshold. The task was to match the color difference between each of the three fields. For the RD91 measurements the observers viewed, under a constant daylight simulator, painted sample pairs next to a fixed near-neutral anchor pair with a color difference $\Delta E_{ab}^{\ast }=1$ (CIELAB color difference formula [54]). This color difference was considered to be a representative sample of an acceptable industrial color tolerance which is substantially larger than visibility thresholds measured under laboratory conditions. The task was to judge whether the color difference of the sample pair was larger or smaller than the color difference of the anchor pair (method of constant stimuli). The corresponding threshold ellipsoids were derived by Melgosa et al. [32]. The MA42 measurements were conducted at constant luminance, whereas in each of the other experiments the color discrimination was determined for combined chromaticity and luminance differences.

It should be noted that the considered experiments differed also in various other aspects, such as: viewing with natural pupil or artificial pupil, binocularly or monocularly viewing, dark or white background, size and shape of the test field and background, luminance of test field and background, range and number of color centers, number and age of observers, duration of the observation and the psychophysical method. In addition there are intra- en inter-observer differences. Consequently, the differences between the experiments had an impact on the calculated values of the model parameters $\kappa _{0\cdot \cdot 4},\, k_{12} \text { and } k_{23}$ [Table 2].

3.5 Generic parameters metric tensor elements

So far the parameters $\kappa _{0\cdot \cdot 4},\, k_{12} \text { and } k_{23}$ were determined for an optimized fit between the measured and calculated threshold ellipsoids of the 10 experiments separately. We were wondering if an acceptable fit between measured and calculated threshold ellipsoids could be realized with a set of generic values for the model parameters. To verify this, it was necessary to eliminate first the differences between the experiments which have a predictable impact on the value of the model parameters.

Monocularly viewing, in the MA42 and BM49 experiments, and binocularly viewing, in the other experiments, is such a difference. Indeed, it was found by Campbell and Green [55] that the threshold contrast of sinusoidal gratings is about a factor $\sqrt {2}$ lower for binocularly viewing compared with monocularly viewing. Assuming that this holds true for the threshold contrasts $\psi _A$, $\psi _T$ and $\psi _D$, the parameters $\kappa _{0\cdot \cdot 4}$ for MA42 and BM49 were normalized for binoculary viewing.

Another difference that needed attention was the size of the test fields. Brown (1952) [56] found that the sensitivity to color differences is $\approx 2 \times$ better for circular split fields of $12^\circ$ diameter compared with split fields of $2^\circ$ diameter. However, it was not clear whether the area of the test field or the boundary length between the two subfields was the root cause of the difference. This aspect was investigated by Lamar et al. (1947) [57,58]. They determined the threshold contrast for a series of achromatic rectangular stimuli with varying areas and varying length/width ratios ${a}/{b}$. The rectangular stimuli were located in the central region of a circular background of $30^\circ$ diameter with uniform luminance. For stimuli with a width $b\geq 3$ arcmin, they found that the threshold contrast was independent of the total area of the stimuli and only dependent on the length of its perimeter. The contrast threshold decreased with increasing perimeter length. They stated: ’contrast is not judged over the area of a target, but across its boundary’ [57]. Therefore, we assumed that the boundary length of juxtaposed subfields or sample pairs is the spatial parameter that influences the threshold contrast. Because the variation of the boundary lengths in the considered experiments is small, we assumed that a normalization for this parameter was not necessary.

The magnitude of the color differences is an important difference. For the WF71d color difference matches, the color difference between the pairs of test colors $\Delta E_{\text {CIE}}=6.7$ (CIE 1964 color difference formula [10] (p.828)), this corresponds to $6.7 \times 0.7=4.7$ SD’s [59] (p.468). The observers were asked to adjust the third field so that the color difference between each of the three colors was equal. This was a more difficult task compared to color matching of two fields, and consequently the errors of the matching were larger. This is reflected in the values of the parameters $\kappa _{0\cdot \cdot 4}$ which are about $2\times$ larger compared with the $\kappa _{0\cdot \cdot 4}$ for SD color matching experiments and binocularly viewing. For RD91 the color difference of the mid-gray anchor pair was $\Delta E^*_{ab}=1.02$ [31] (CIELAB color difference formula [54]), this value corresponds to an acceptable industrial color difference and it is assumed that this corresponds to about 5 SD’s [60] (p.319). This is reflected in the values of the $\kappa _{0\cdot \cdot 4}$ for these measurements.

After normalization of $\kappa _{0\cdot \cdot 4}$ for the MA42 and BM49 experiments to binocularly viewing with a factor $1/\sqrt {2}$, and for the WF71d color difference matching experiments with a factor $1/2$, it was assumed that the respective $\kappa _{0\cdot \cdot 4}$ for the 9 matching experiments are on a par with each other. The $\kappa _{0\cdot \cdot 4}$ of the RD91 experiments were normalized to color differences corresponding to SD’s with a factor 1/5. All the normalized values are given in Table 3, as well as the averages $\mu$ and standard deviations $\sigma$ of the 9 matching experiments.

Table 3. Normalized values of the parameters $\kappa _{0\cdot \cdot 4}$ of the 9 matching experiments: WF71, MA42, BM49, WF71d, as well as the averages $\mu$ and standard deviations $\sigma$ of the 9 matching experiments. The rightmost column shows the normalized values of the parameters $\kappa _{0\cdot \cdot 4}$ of the RD91 experiments [$\S$ number of observers].

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Table 3 shows that the averages of $\kappa _0$, $\kappa _1$ and $\kappa _3$ for the 9 matching experiments are close to the corresponding normalized values of the RD91 experiments, whereas the averages of $\kappa _2$ and $\kappa _4$ for the 9 matching experiments have a larger relative difference with the corresponding values of the RD91 experiments but the relative differences are still within reasonable limits. Therefore, we proposed for the generic values of the model parameters $\kappa _{0\cdot \cdot 4}$ the rounded averages of the 9 matching experiments as given in Table 4a. Notice that only one significant digit is specified. The reason for this is explained in the next paragraph. As already discussed, the color differences of the RD91 experiments correspond to acceptable industrial color differences (about 5 SDs). The model fits significantly better to the RD91 threshold ellipsoids with the proposed Eqs. (23) and (24) with parameters $k_{12}=56.6$ and $k_{23}=30.5$ (Table 2). The same Eqs. (23) and (24) were applied for the SD color difference experiments. However in these cases there was not a clear pattern in the calculated values of $k_{12}$ and $k_{23}$, but lower values compared to the case of RD91 data set seem appropriate (Table 2). Notice that the metric tensor of the F65 LE has also off-diagonal elements which depend on the size of the color differences. Based on the available data the proposed generic values for $k_{12}$ and $k_{23}$ are as given in Table 4b.

Table 4. Generic values of the model parameters.

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4. Results and discussion

The new LE for SD color difference ellipsoids and binocularly viewing consists of Eqs. (25), (6), (8) and (10), and the generic parameter values given in Tables 1, 4a and 4b. By means of the new LE and generic parameter values the threshold ellipsoids were calculated for the experiments by WF71, MA42, BM49, WF71d and RD91. The differences of the experiments as discussed in paragraph 3.5 were taken into account. For each of the 10 experiments the dissimilarity between the measured and the calculated threshold ellipsoids for each color center, denoted $d^{\ast }_j$, and the root-mean-square (RMS) dissimilarity $d^{\ast }_{\text {rms}}=\sqrt {\frac {1}{N}\sum _{j=1}^N d_j^{\ast \, 2}}$ for the complete data set corresponding with each experiment and observer were calculated. The $d^{\ast }_{\text {rms}}$ values are given in the last row of Table 2. The change from optimized values for the parameters $\kappa _{0\cdot \cdot 4}$, $k_{12}$ and $k_{23}$ to generic parameter values results in an increase of $\langle d_{\text {rms}} \rangle = 0.368$ to $\langle d_{\text {rms}}^{\ast } \rangle =0.406$. With $\langle d_{\text {rms}} \rangle =\sqrt {\frac {1}{\sum _{i=1}^{10} N_i} \sum _{i=1}^{10}{N_i\, d_{\text {rms,}i}^2}}$ which is a weighted RMS of the RMS dissimilarities of each data set with as weights the number of samples of each data set. $\langle d_{\text {rms}}^{\ast }\rangle$ was calculated similarly. Changing the generic parameter values of $\kappa _{0\cdot \cdot 4}$ to rounded averages of the 9 matching experiments with 2 significant digits decreases $\langle d_{\text {rms}}^{\ast } \rangle$ with about 0.001, therefore one significant digit is sufficient.

A qualitative impression of the correlation between the visually perceived (measured) and the calculated color differences with the new LE and generic parameters is shown in the Figs. 6 to 10.

Chromaticity discrimination improves with increasing luminance of the test field [61]. The RD91 data set has three sample colors with very similar chromaticities and different retinal illuminance levels, i.e. color 10 (’black’, $Y=61$ td, $x=0.314$, $y=0.335$), color 3 (’medium gray’, $Y=952$ td, $x=0.315$, $y=0.335$) and color 18 (’light gray’, $Y=2193$ td, $x=0.315$, $y=0.331$) [32]. The effect of the retinal illuminance level on the chromaticity discrimination are for these three colors shown in the insets of Fig. 6. There is a substantial increase of the sensitivity to chromaticity variations when the luminance of the test field increases from 61 td (’black’) to 952 td (’medium gray’). However, the sensitivity to chromaticity variation is nearly stable between 952 td (’medium gray’) and 2193 td (’light gray’). The insets of Fig. 6 show that the predictions of the chromaticity discrimination are in good agreement with the measurements. We evaluated the impact of various off-diagonal elements $g_{12}$ and $g_{23}$ on the predictability of the measured threshold ellipses/ellipsoids. For this we calculated the dissimilarity between the measured threshold ellipses/ellipsoids and those calculated with the new LE, but with the off-diagonal elements $g_{12}$ and $g_{23}$ as shown in the top row of Table 5. This was done for the RD91, MA42 and BM49 data sets. In each case the generic values of the model parameters were used. The results are shown in Table 5. For the RD91 data set it is clear that the dissimilarity significantly decreases when the off-diagonal elements of the F65 LE are replaced by the proposed off-diagonal elements (23) and (24). The greatest effect is due to off-diagonal element (24). For the MA42 data set only $g_{23}$ is relevant. In this case the decrease of the dissimilarity $d_{\text {rms}}^{\ast }$ is rather small. However, this is due to the small proportion of threshold ellipses for which the new expression (24) is significant. Indeed, when we consider the small number of measured threshold ellipses with principal axes that are not parallel with the coordinate axes, then a significant decrease of the dissimilarity is found when the off-diagonal element $g_{23}$ of the F65 LE is replaced by Eq. (24). For example, for the color centers 1, 2 and 3 (Fig. 7) the $d^{\ast }$ values decrease from 0.727, 0.578 and 0.448 (with $g_{23}$ according to the F65 LE) to 0.395, 0.481 and 0.419 (with $g_{23}$ according to Eq. (24)) respectively. The BM49 data sets cover the largest range of $l$-chromaticities, however, only small improvements of the predictability of the BM49 data sets were found when the off-diagonal elements of the F65 LE were replaced by Eqs. (23) and (24). This is probably caused by the larger statistical spread of the BM49 data sets. We also applied the new LE to the detection threshold measurements by Chaparro et al. (1995) [62]. In these experiments the eye was adapted to a $6.2^{\circ }$ diameter background. A test flash of 200 ms duration and $2.2^{\circ }$ diameter was in the center of the background field. In the absence of the test flash a uniform $6.2^{\circ }$ diameter disk was visible i.e. the color of the background and the test field were equal. One set of experiments was conducted with narrowband background light of 400 td with spectral centroids of respectively 525 nm (green), 579 nm (yellow) and 610 nm (red). For stimuli with wavelength $>520$ nm the achromatic mechanism and red-green mechanism are isolated; only the $L$ and $M$ signals need to be considered [63]. The test flashes could increase or decrease the L-cone and M-cone excitations relative to the background level. One of the findings of Chaparro et al. [62] was that for each of the adapting backgrounds the detection contours are straight lines with a slope of 45$^\circ$ in the $(\frac {dL}{L},\frac {dM}{M})$ plane. This experiment as well as other similar experiments [64] showed that the sensitivity for chromatic variations is substantially higher than for achromatic variations. The metric tensors of the new LE with generic parameter values and a test field with the same color as the background $(l=l_a)$ were determined in the cone contrast space for monochromatic light of respectively 525 nm, 579 nm and 610 nm. Notice that, in accordance with Eq. (23), $g_{12}$ is zero because $l=l_a$. The threshold ellipses calculated with the new LE are shown in Fig. 11. The principal axes of the threshold ellipses have an orientation of 45$^\circ$ with respect to the coordinate axes, in accordance with the detection thresholds found by Chaparro et al. We evaluated the impact of the von Kries transformation on the predictability of the measured threshold ellipses/ellipsoids. For this we calculated the dissimilarity between the measured threshold ellipses with those calculated according to the new LE with and without the von Kries adaptation. This was done for the WF71-AR, WF71d-AR, MA42 and RD91 data sets. In each case the optimized and generic values of the model parameters were used. From the results shown in the Fig. 12 (circle markers) we see slightly but consistently higher dissimilarities without the von Kries transformation.

 

Fig. 11. Threshold ellipses in the cone contrast plane $(\frac {dL}{L},\frac {dM}{M})$ predicted with the new LE for monochromatic light of respectively 525 nm, 579 nm and 610 nm. The principal axes of the threshold ellipses have an orientation of 45$^\circ$ with respect to the coordinate axes, this is in agreement with the findings of Chaparro et al. (1995) [62].

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Fig. 12. Dissimilarity between the measured and calculated threshold ellipses/ellipsoids for the data sets WF71-AR, MA42, WF71d-AR and RD91 according to: (a) CIELAB, CIEDE2000, $\Delta$ICtCp, F65 LE and the new LE+ (’+’ refers to the optimised sizes $d\sigma$) [square markers], (b) the new LE with and without the von Kries transformation, and with optimised (denoted as O) and generic model parameters (denoted as G) [circle markers].

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Table 5. Dissimilarity metrics of the data sets RD91, MA42 and BM49 for the various off-diagonal elements $g_{12}$ and $g_{23}$. The off-diagonal elements are shown in the top row. The off-diagonal elements $g_{12}=Q\,g_{11}$ and $g_{23}=-Q\,g_{33}$ are according to the F65 LE.

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We also made a comparison of the new LE with existing color difference formulas and with the F65 LE. Therefore we compared the threshold ellipsoids/ellipses as predicted by the new LE for the data sets MA42, WF71-AR, WF71d- AR and RD91 with those predicted by the color difference formulas CIE 1976 $(L^*a^*b^*)$ (referred to as ’CIELAB’) [54], CIEDE2000 [11] (pp.100-105) [12,65,66], $\Delta \text {I}\text {Ct} \text {Cp}$ [67,68] and the F65 LE [21]. The color difference formulas CIELAB and CIEDE2000 are based on the CIELAB color space [11] (pp.74-75), consequently the color coordinates $(X_n,Y_n,Z_n)$ of a white reference must be chosen. For RD91 we have $Y_n=\frac {2000}{\pi }$ cd m$^{-2}$ and the chromaticites of the reference white were those of the illuminant D65 ($10^{\circ }$ standard observer) [32]. For the other data sets $Y_n$ was chosen to have a lightness $L^*=70$ for the test field, and the chromaticities of the reference white were those of the backgrounds used in the respective measurements. The ’parametric factors’ $k_L$, $k_C$ and $k_H$ of CIEDE2000 were set equal to 1. The color difference equations were converted to the differential form and the metric tensors expressed in the initial color spaces were transformed to the $(\frac {dY}{Y},dx,dy)$ space. The differential form of CIEDE2000 is nontrivial; we followed in this case the solution proposed by Pant and Farup [66]. This allowed to calculate the $d_{\text {rms}}$ value for each data set and each color difference formula. However, attention must be given to the different threshold criteria and experimental differences. Therefore, in each case the size of the line element $d\sigma$ was chosen to minimize $d_{\text {rms}}$. The results are shown in Fig. 12 (square markers) and in Table 6.

Table 6. Comparison between the color difference equations CIELAB, CIEDE2000, $\Delta$ICtCp, and the F65 LE and the new LE+.

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We estimated the overall performance of each color difference metric by calculating the weighted RMS dissimilarity $\langle d_{\text {rms}} \rangle$ for the four data sets (MA42, WF71-AR, WF71d-AR and RD91). $\langle d_{\text {rms}} \rangle$ is calculated similarly as above and listed in the last row of Table 6. From this table it is clear that the CIELAB threshold ellipses/ellipsoids have the highest $\langle d_{\text {rms}} \rangle$, which means they deviate most from the experimental results. CIEDE2000, $\Delta$ICtCp and F65 LE have an intermediate $\langle d_{\text {rms}} \rangle$. The color difference formula CIEDE2000 performs well for the RD91 data set, probably because this data set was one of the the data sets used for its development [69]. $\Delta$ICtCp and F65 LE have both the lowest and highest $d_{\text {rms}}$ for respectively the MA42 and the RD91 data set (large color differences). Notice that for the F65 LE specific parameters depending on the size of the color difference must be chosen cf. [21] (p.1319). We used the parameter values $l=0.25$ and $f=0$ (’threshold differences’) for MA42, WF71 and WF71d, and $l=0.1$ and $f=3/4$ (’color differences of commercial importance’) for RD91. The new LE+ (’+’ refers to the optimised sizes $d\sigma$) has the lowest $\langle d_{\text {rms}} \rangle$. Also notice that the optimal sizes $d\sigma$ follow the same trend among the four data sets, and that the optimal sizes $d\sigma$ of the new LE are close to the inverse of the normalization factors discussed in paragraph 3.5.In Supplement 1 the results are given of the validation of the new LE with generic parameters against the data set of Romero et al. (1993) [70]. Notice that the data set for this additional validation was not used to determine the model parameters.

5. Conclusions

We developed a new line element (LE) for the human color space based on the Friele line elements. For this development we used existing data sets for color discrimination with luminance levels of the test fields in the range of 1 cd m$^{-2}$ to 400 cd m$^{-2}$. The test fields were split fields with a boundary length of about $2^{\circ }$ visual angle and without a gap. Since the metric tensor of the line element $[g_{ij}]$ is almost diagonal in the MacLeod-Boynton contrast space, we expressed the new LE in this space, whereas Friele’s line elements were expressed in the cone contrast space. New equations for the threshold contrasts along the three cardinal directions were derived from psychophysical experiments reported in the literature. Visual adaptation to the background light has a strong influence on the color discrimination sensitivity. For the chromatic channels this crispening effect was incorporated in the diagonal elements of the new metric tensor, whereas Friele’s line elements did not include the crispening effect. No luminance crispening was observed in the data sets considered and therefore was also not included in the calculations, although this could easily be done if appropriate. Generic values for the model parameters, corresponding to standard deviation color difference thresholds and binocularly viewing, were determined. We found adequate agreement between the new LE and measurements reported in the literature. The new LE was compared with several color difference formulas and with the Friele (1965) LE. From this comparison a better overall predictability of threshold ellipses/ellipsoids of the investigated data sets was found with the new LE. Whereas in Friele’s LE’s the off-diagonal elements $g_{12}$ and $g_{23}$ are related to respectively the diagonal elements $g_{11}$ and $g_{33}$, in the new LE these off-diagonal elements are defined independently. They are only significant for high $s$ chromaticities and, high and low $l$ chromaticities. We found a great impact of the new expression for $g_{23}$ on the predictability of threshold ellipses/ellipsoids for the RIT-DuPont (1991) and for the MacAdam (1942) data sets. Also the new expression for $g_{12}$ improved to some extent the predictability of the RIT-DuPont (1991) threshold ellipsoids. On the other hand the new expressions for $g_{12}$ and $g_{23}$ had little effect on the predictability of the ellipsoids for the Brown-MacAdam (1949) data set, despite the large $l$-chromaticity range of this data set. This is probably due to the bigger statistical spread of the experimental data. In our model the background is transformed to the reference equal energy white background by a simple von Kries transformation. Although the effect of this transformation is rather small, because the experimental backgrounds are close to this reference, the transformation yields consistently better predictions. It remains to be investigated whether this can be extended to colored backgrounds.