## Abstract

A method based on the chromaticity discrimination ellipses is proposed for calculating the color difference between two colors. This method calculates the color difference by counting the number of just noticeable differences between the two colors. The color difference formula CIEDE2000 and the proposed method are compared. It is shown that CIEDE2000 is not suitable for predicting the threshold color difference.

©2012 Optical Society of America

## 1. Introduction

Colorimetry specifies numerically a perceived color. The colorimetric system published by the International Commission on Illumination (CIE) in 1931 is one of the foundations of modern colorimetry [1]. Colorimetry is also concerned with the specification of the color difference between two colors. The CIE 1931 system is not a uniform color space. The perceived color difference between two color points is not linearly proportional to their Euclidian distance in the CIE 1931 system. CIE published two approximately uniform color spaces, CIELAB and CIELUV, in 1976 [1]. CIE recommended the color difference formulas in CIELAB and CIELUV as the Euclidian distance of two color points in the two color spaces. For improving the 1976 color difference formulas, CIE recommended two formulas CIE94 and CIEDE2000 in 1994 and 2000, respectively [2, 3].

A color difference formula is an empirical result derived from its collected data set. The derivation of color difference formula is mainly based on the color difference ellipses measured from psychophysical experiments [1–4]. A color difference ellipse represents the same color difference with respect to the color center of the ellipse for the same lightness. There are two methods for measuring color difference ellipses. The first is the matching method, in which an observer assesses whether or not two colors match [1]. A chromaticity discrimination ellipse can be derived from the threshold color differences measured with the matching method. MacAdam ellipses are the first measured chromaticity discrimination ellipses that showed the non-uniformity of the CIE 1931 system [1]. The second is the gray scale method, in which an observer rates the perceived color difference between two colors in terms of some reference gray scale. A suprathreshold color difference ellipse can be calculated from the rating results. The gray scale method was adopted in the developments of CIE94 and CIEDE2000 [4]. An ideal color difference formula should be able to predict both the threshold and suprathreshold color differences.

The color samples used in the developments of CIE94 and CIEDE2000 are surface colors [4]. The pigments of the color samples are not fluorescent and not highly saturated. It is a trend to develop flat-panel displays of wide color gamut (WCG). Because of the highly saturated primaries of WCG displays, CIE color difference formulas may not be suitable for calculating the color difference between two high saturation colors that can be shown by WCG displays [5]. On the other hand, it was shown that the perceived color difference depends on texture effect [6, 7]. The display color is self-luminous and its color difference characteristics may be different from that of surface color. Recently four chromaticity discrimination ellipses were measured by the use of a liquid crystal display [5]. The measured ellipses agree with MacAdam ellipses but significantly differ from that predicted by CIEDE2000 for red and blue colors.

Color processing is required for the video programs shown on WCG displays for a color rendering intent. For the applications of video color processing, both the predictions of the threshold color difference and suprathreshold color difference are required. The prediction of the threshold color difference also relates to the required bit depth for the avoidance of color banding [8]. This article proposes a color difference metric based on the chromaticity discrimination ellipses for calculating the color difference around the threshold. The color difference between two color points in CIELAB is calculated from the chromaticity discrimination ellipses between them. Numerical examples are shown for comparing the color differences calculated by CIEDE2000 and the proposed method.

## 2. Chromaticity discrimination ellipse

A chromaticity discrimination ellipse can be calculated by fitting a set of assessments of the same lightness to the formula in CIELAB [1]

*g*coefficients,

*g*

_{11},

*g*

_{12}and

*g*

_{22}, are fitting constants; Δ

*a** =

*a**-

*a**

*, Δ*

_{c}*b** =

*b**-

*b**

*; (*

_{c}*a**

*,*

_{c}*b**

*) are the coordinates of the color center. Equation (1) can be transformed to the cardinal coordinates ($\Delta {a}^{\prime}*$, $\Delta {b}^{\prime}*$) by rotating the axis Δ*

_{c}*a** to $\Delta {a}^{\prime}*$ with an angle

*θ*so that the ellipse can be represented as [1]

*a*and

*b*are the lengths of the major and minor axes, respectively, i.e.,

*a> b*;

*θ*is the orientation angle of the ellipse. The

*g*coefficients in Eq. (1) and the characteristic parameters

*a*,

*b*and

*θ*in Eq. (2) are related by

*g*

_{11}= cos

^{2}

*θ/a*sin

^{2}+^{2}

*θ/b*

^{2}, g_{22}= sin

^{2}

*θ/a*cos

^{2}+^{2}

*θ/b*, and

^{2}*g*

_{12}= (1

*/a*1

^{2}-*/b*) sin

^{2}*θ*cos

*θ*.

Figure 1
shows the 25 MacAdam ellipses and their corresponding ellipses predicted by the CIEDE2000 with a unit color difference (Δ*E*_{00} = 1) in CIELAB. The color difference ellipse predicted by the CIEDE2000 is called the CIEDE2000 ellipse for simplicity. The ellipses are barely observed if they are plotted with their actual axis lengths. The MacAdam ellipses and CIEDE2000 ellipses shown in the Fig. 1 are enlarged so that they can be clearly shown. The enlargement factors shown in the caption of Fig. 1 will be explained in the Section 5. We can see the significant difference between the MacAdam ellipses and CIEDE2000 ellipses in red and blue regions. The ellipse axis lengths depend on the parametric effects of the psychophysical experiment [9]. Figure 1 clearly shows that the characteristic parameters *a*, *b* and *θ* of an ellipse significantly depend on the chroma *C** and hue angle *h** defined in CIELAB, in which *C** = (*a**^{2} + *b**^{2})^{1/2} and *h** = tan^{−1}(*b**/*a**)^{1/2}. In this article, the hue angle is in unit of degree except otherwise specified. The empirical formulas of the characteristic parameters can be obtained by the polynomial regression in terms of *C** and *h** from a set of chromaticity discrimination ellipses [5]. The chromaticity discrimination ellipses to be fitted are called the training ellipses.

For satisfying the periodic boundary condition for the hue angle, we modify the polynomial regression given in Ref [5] as

*= a*,

*b*and

*θ*;

*N*is the regression order; α

*are the constants calculated by the regression. The use of higher regression order increases the fitting accuracies for the training ellipses but may result in the less accuracy for the ellipse that is not included in the training ellipses. The reduction of accuracy is called the over-interpolation. The number of α*

_{ijk}*coefficients in Eq. (3) is*

_{ijk}*N*= (

_{t}*N*+ 1)(

*N*+ 2)(

*N*+ 3)/6. The value of

*N*should be much less than the number of the set of training ellipses for avoiding the over-interpolation. The characteristic parameters

_{t}*a*,

*b*and

*θ*of 25 MacAdam ellipses shown in the Fig. 1 are taken as the training ellipses for an example. We take

*N*= 2, which corresponds to

*N*= 10. The ellipses calculated by the regression are called the MacAdam fitting ellipses, which are also shown in the Fig. 1. We can see that the MacAdam fitting ellipses fit the MacAdam ellipses well in the color regions where the density of the distribution of MacAdam ellipses is high. Comparing the MacAdam fitting ellipses with the CIEDE2000 ellipses, we can see the significant improvement of the prediction accuracy of the chromaticity discrimination ellipses with the regression model.

_{t}## 3. Color difference in unit of just noticeable difference

This section presents the color difference metric based on the chromaticity discrimination ellipses. Figure 2
shows the diagram for the calculation procedure of the color difference between the two color points *Q _{s}* and

*Q*. Actually the axis lengths and orientation angles of neighboring chromaticity discrimination ellipses are nearly the same. The differences of the axis lengths and orientation angles shown in the Fig. 2 are exaggerated. The color difference is calculated by counting the number of just noticeable differences between

_{t}*Q*and

_{s}*Q*. We define the color vectors

_{t}**= (**

*Q*_{s}*a**

*,*

_{s}*b**

*) and*

_{s}**= (**

*Q*_{t}*a**

*,*

_{t}*b**

*) corresponding to*

_{t}*Q*and

_{s}*Q*, respectively, and the color difference vector

_{t}**=**

*V***-**

*Q*_{t}**. The unit vector**

*Q*_{s}**=**

*u***/|**

*V***| = (cos**

*V**φ*, sin

*φ*), in which

*φ*is the angle from the Δ

*a** axis to

**. Starting from**

*V**Q*, which is also denoted as

_{s}*P*

_{0}in the Fig. 2, we have the first intersection point (

*P*

_{1}) of the vector

**and the chromaticity discrimination ellipse with the color center at**

*V**P*

_{0}; the second intersection point (

*P*

_{2}) of the vector

**and the chromaticity discrimination ellipse with the color center at**

*V**P*

_{1}, and so on until the last intersection point (

*P*) of the vector

_{n}**and the chromaticity discrimination ellipse with the color center at**

*V**P*

_{n}_{-1}. In the Fig. 2,

*P*

_{n}= P_{2}. The color vector of the

*i*-th intersection point

*P*is denoted as

_{i}**. We have**

*P*_{i}

*P*

_{i+}_{1}=

*P**+*

_{i}*s*

_{i}**, where**

*u**s*is the length between

_{i}*P*and

_{i}*P*

_{i+}_{1};

*s*= (

_{i}*g*

_{11i}cos

^{2}φ +

*g*

_{12i}sin2φ +

*g*

_{22i}sin

^{2}φ)

^{-1/2}. The parameters

*g*

_{11}

*,*

_{i}*g*

_{12}

*and*

_{i}*g*

_{22}

*are the*

_{i}*g*coefficients of the chromaticity discrimination ellipse with the color center at

*P*

_{i}. The color difference calculated from

*Q*to

_{s}*Q*can be expressed as Δ

_{t}*E*(

_{cd}**,**

*Q*_{s}**) =**

*Q*_{t}*n*+ Δ

*s*/

*s*, where

_{n}*n*is the number of intersection points, Δ

*s*= |

**-**

*Q*_{t}**|, and the term Δ**

*P*_{n}*s*/

*s*counts the remaining color difference between

_{n}*P*and

_{n}*Q*. If the color difference is sub-threshold,

_{t}*n*= 0.

The magnitudes of Δ*E _{cd}*(

**,**

*Q*_{1}**) may not be equal to Δ**

*Q*_{2}*E*(

_{cd}**,**

*Q*_{2}**) for the two color points**

*Q*_{1}*Q*

_{1}and

*Q*

_{2}because the

*g*coefficients of intersection points

*P*are slightly different in calculating Δ

_{i}*E*(

_{cd}**,**

*Q*_{1}**) and Δ**

*Q*_{2}*E*(

_{cd}**,**

*Q*_{2}**). Although the difference between Δ**

*Q*_{1}*E*(

_{cd}**,**

*Q*_{1}**) and Δ**

*Q*_{2}*E*(

_{cd}**,**

*Q*_{2}**) is much less than a unit and is negligible, we define the color difference between**

*Q*_{1}*Q*

_{1}and

*Q*

_{2}as Δ

*E*= [Δ

_{jnd}*E*(

_{cd}**,**

*Q*_{1}**) + Δ**

*Q*_{2}*E*(

_{cd}**,**

*Q*_{2}**)]/2 so that the definition is unambiguous. The subscript “**

*Q*_{1}*jnd*” of Δ

*E*emphasizes that its value is in fact the number of just noticeable differences between

_{jnd}*Q*

_{1}and

*Q*

_{2}.

## 4. Discrimination ellipse color difference formula

The calculation method presented in the Section 3 is complicated. From the Eq. (1), we may derive an approximate color difference formula in a similar form as CIEDE2000 for the two color points *Q*_{1} and *Q*_{2} that correspond to the color vector ** Q_{i}** = (

*a**

*,*

_{i}*b**

*) for*

_{i}*i*= 1 and 2. Here Δ

*a** =

*a**

_{2}-

*a**

_{1}, Δ

*b** =

*b**

_{2}-

*b**

_{1}in Eq. (1). The two color vectors can also be represented in the chroma and hue coordinates as

**= (**

*Q*_{i}*C**

*,*

_{i}*h**

*) for*

_{i}*i*= 1 and 2. Under the assumption that the hue angle difference Δ

*h**<< 1, in which Δ

*h** = ︱

*h**

_{2}-

*h**

_{1}︱ and Δ

*h** is in unit of radian instead of degree, we have the color difference formula

*C** = |

*C**

_{2}-

*C**

_{1}|, Δ

*H** =

*C**

*Δ*

_{avg}*h**,

*C**

*= (*

_{avg}*C**

_{1}+

*C**

_{2})/2,

*h**

*= (*

_{avg}*h**

_{1}+

*h**

_{2})/2,

*S*= (

_{C}*g*

_{11}cos

^{2}

*h**

*+*

_{avg}*g*

_{12}sin2

*h**

*+*

_{avg}*g*

_{22}sin

^{2}

*h**

*)*

_{avg}^{-1/2},

*S*= (

_{H}*g*

_{11}sin

^{2}

*h**

*-*

_{avg}*g*

_{12}sin2

*h**

*+*

_{avg}*g*

_{22}cos

^{2}

*h**

*)*

_{avg}^{-1/2}, and

*R*= (

_{T}*g*

_{22}-

*g*

_{11})sin2

*h**

*+ 2*

_{avg}*g*

_{12}cos2

*h**

*. When either one of*

_{avg}*Q*

_{1}and

*Q*

_{2}lies at the origin, its hue angle is taken to be the same as that of the other color point, i.e., if

*C**

_{1}= 0,

*h**

_{1}=

*h**

_{2}; if

*C**

_{2}= 0,

*h**

_{2}=

*h**

_{1}. Because of the discontinuity of the hue angle across 360° to 0

^{o}, if Δ

*h** > π,

*h**

*= mod[180° + (*

_{avg}*h**

_{1}+

*h**

_{2})/2, 360°], Δ

*H** =

*C**

*(2π-Δ*

_{avg}*h**). The

*g*coefficients are calculated from the characteristic parameters

*a*,

*b*and

*θ*. The parameters are calculated by the Eq. (3) with

*C** and

*h** replaced by the average chroma

*C**

*and the average hue angle*

_{avg}*h**

*, respectively. The subscript “*

_{avg}*de*” of Δ

*E*emphasizes that the color difference formula is derived from the discrimination ellipse equation.

_{de}Because the average chroma and average hue angle are used to calculate the characteristic parameters, the color difference between the two color points calculated by Eq. (4) can be regarded as the color difference in unit of the size of the average chromaticity discrimination ellipse between them. We may call Eq. (4) as the discrimination ellipse color difference formula.

## 5. Numerical examples and discussions

It requires a set of training ellipses to obtain the empirical formulas of the characteristic parameters *a*, *b* and *θ* of a chromaticity discrimination ellipse by regression. A chromaticity discrimination ellipse depends on the parametric effects such as environment lighting, adapted white, and other measurement conditions. The dependences have not been experimentally and completely investigated in the state of the art. One may measure the chromaticity discrimination ellipses according to the application condition. As is described in Section 2, we take the 25 MacAdam ellipses shown in the Fig. 1 as the training ellipses, although they do not well sample the color space. Reference [5] measured four chromaticity discrimination ellipses. Their color centers are the same as that of the MacAdam ellipses labeled from 1 to 4 shown in the Fig. 1, which correspond to red, green, blue, and gray colors, respectively. The ratio of axis lengths *a*/*b* and the orientation angle of a measured chromaticity discrimination ellipse are about the same as that of the corresponding MacAdam ellipse. But the measured axis lengths *a* and *b* are longer than that of the corresponding MacAdam ellipses because of higher luminance level of environment lighting for the experiment in the Ref [5]. The increase factors of the measured axis lengths are about 3.1. Due to the calculation procedure of the ellipse parameters, about 68% of the color matchings made by the observer in the psychophysical experiment are expected to fall within the ellipse. The axis lengths increase with the color matching ratio. The desirable color matching ratio depends on applications. For designing a color-banding free display, the use of shorter axis lengths is desirable but may results in the increase of the required signal bit depth. The display performance and cost have to be compromised. In this section, all the 25 MacAdam ellipses with the axis lengths increased by a factor of 3.1 are taken as the training ellipses. If the MacAdam ellipses shown in the Fig. 1 are considered to be the training ellipses, the enlargement factor of the training ellipses plotted in the Fig. 1 is 10/3.1 = 3.226, which is the same as the enlargement factor of the CIEDE2000 ellipses plotted in the same figure.

Figures 3(a)
-3(c) show the color differences Δ*E _{de}* (thind data lines) and Δ

*E*

_{00}(thick data lines) versus the hue angle

*h** for the two colors with (

*C**

_{1},

*h**

_{1}) = (

*C**,

*h**) and (

*C**

_{2},

*h**

_{2}) = (

*C** + Δ

*C**,

*h** + Δ

*h**), where the cases with the chroma

*C** = 20 (red data lines), 40 (green data lines) and 60 (blue data lines) are shown; Δ

*C** and Δ

*h** are the chroma and hue angle differences, respectively. The values of chroma are chosen for the better sampling of MacAdam ellipses over the corresponding color regions. The chroma and hue differences are Δ

*C** = 2.5 and Δ

*h** = 0° for the Fig. 3(a); Δ

*C** = 0 and Δ

*h** = 2.5° for the Fig. 3(b); Δ

*C** = 2.5 and Δ

*h** = 2.5° for the Fig. 3(c). The values of Δ

*C** and Δ

*h** are chosen so that the calculated color differences are around threshold. The maximum relative differences between Δ

*E*and Δ

_{jnd}*E*are 0.051%, 0.055% and 0.076% for the cases shown in the Figs. 3(a)-3(c), respectively. Although the method shown in Section 3 is more accurate to represent the color difference based on the chromaticity discrimination ellipses, the use of the approximate formula, Eq. (4), gives the satisfactory accuracy.

_{de}The color difference decreases as the ellipse axis lengths *a* and *b* increase. For the same hue angle, the major and minor axis lengths of a CIEDE2000 ellipse increase with the chroma; the major axis length of a MacAdam ellipse also increases with the chroma, but its minor axis length decreases as the chroma increases in the blue region around the 270° hue angle. For the same chroma, the axis lengths of both the CIEDE2000 ellipse and MacAdam ellipse change with the hue angle, but the major axis length of a CIEDE2000 ellipse almost does not change with the hue angle outside the blue region and with the chroma larger than about 40.

As the orientation of an ellipse points away from the origin, the color differences due to the chroma difference and hue angle difference increases and decreases, respectively. The orientation of a CIEDE2000 ellipse points toward the origin except for the color region near the origin or the blue region around the 270° hue angle [4]. The orientation of a MacAdam ellipse usually does not point toward the origin as is shown in the Fig. 1.

From the Fig. 3(a) that shows the cases with the small chroma difference, Δ*E*_{00} slightly changes with the hue angle for the case with *C** = 20 and almost does not change with the hue angle for the cases with *C** = 40 and 60. Δ*E _{de}* significantly changes with the hue angle. Δ

*E*

_{00}decreases as the chroma increases. Δ

*E*also increases with the chroma except for the blue region around the 255° hue angle. From the Fig. 1, as the chroma of a MacAdam ellipse increases in this blue region, its minor axis length decreases and its orientation points away from the origin; consequently, Δ

_{de}*E*increases with the chroma in this blue region in spite of the increase of the major axis length. There are two local maxima for Δ

_{de}*E*around the 10° and 255° hue angles. From the Fig. 1, the orientations of the MacAdam ellipses around the two hue angles are almost perpendicular to the directions toward the origin; consequently, the color difference due to the chroma difference is larger around the two hue angles. The two colors cannot be visually discriminated for the case with Δ

_{de}*E*< 1 by definition. For the case with

_{de}*C** = 60 shown in the Fig. 3(a), Δ

*E*shows that the two colors can be discriminated for the hue angle between 339° and 49° in the red region and for the hue angle between 205° and 294° in the blue region, while CIEDE2000 cannot predict the chromaticity discrimination.

_{de}From the Fig. 3(b) that shows the cases with the small hue difference, both Δ*E*_{00} and Δ*E _{de}* increase with the chroma because the Euclidian distance between the two color points on the

*a**

*b** plane increases with the chroma for the same hue angle difference. Both Δ

*E*

_{00}and Δ

*E*significantly change with the hue angle and have the maxima in the blue region around the 290° hue angle. From the Fig. 1, both the angle between the ellipse orientation and the direction toward the origin and the minor axis length of the MacAdam ellipse with

_{de}*h**> 270

^{o}are smaller than that of the MacAdam ellipse with

*h**< 270

^{o}. Therefore, the maximum Δ

*E*of the cases shown in the Fig. 3(b) are around the hue angle larger than 270

_{de}^{o}, while the maximum Δ

*E*of the cases shown in the Fig. 3(a) are around the 255° hue angle.

_{de}The cases with the combined small differences in chroma and hue angle are shown in the Fig. 3(c). The maxima Δ*E _{de}* of the cases shown in the Fig. 3(c) are around the 335° hue angle. There are the local maxima for Δ

*E*

_{00}around the 75° hue angle, but there are the local minima for Δ

*E*around the same hue angle.

_{de}## 6. Conclusion

A method based on the chromaticity discrimination ellipses is proposed for calculating the color difference between two colors. Numerical examples show the different characteristic between the color difference formula CIEDE2000 and the proposed method_{.} The results indicate that CIEDE2000 is not suitable for predicting the threshold color difference. This is reasonable because CIEDE2000 was established with the gray scale method and the data set of suprathreshold color difference, while the chromaticity discrimination ellipses are measured with the matching method and the data set of near threshold color difference. On the other hand, the prediction of the suprathreshold color difference with the proposed method may not be accurate because CIEDE2000 is the more accurate formula for calculating the suprathreshold color difference in the state of the art. It requires further studies to clarify the relation of the suprathreshold color differences calculated by CIEDE2000 and the proposed method. The clarification is helpful for deriving an ideal color difference formula that can be used to accurately calculate both the threshold color difference and suprathreshold color difference. In principle, the proposed method is able to represent more complicated characteristics of the color difference ellipses than CIEDE2000. It can be applied to describe the suprathreshold color difference ellipses measured with the gray scale method for improving the prediction accuracy. Furthermore this method can be extended to the calculation of the color difference based on the color difference ellipsoids [1].

## Acknowledgment

The work was partially supported by National Science Council, R. O. C., under Contract No. 101-2221-E-155-049.

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