## A color difference metric based on the chromaticity discrimination ellipses |

Optics Express, Vol. 20, Issue 24, pp. 26441-26447 (2012)

http://dx.doi.org/10.1364/OE.20.026441

Acrobat PDF (978 KB)

### 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 OSA

## 1. Introduction

4. M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. **26**(5), 340–350 (2001). [CrossRef]

4. M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. **26**(5), 340–350 (2001). [CrossRef]

4. M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. **26**(5), 340–350 (2001). [CrossRef]

6. R.-C. Wu, R. H. Wardman, and M. R. Luo, “A comparison of lightness contrast effects in CRT and surface colours,” Color Res. Appl. **30**(1), 13–20 (2005). [CrossRef]

7. J. H. Xin, H.-L. Shen, and C. C. Lam, “Investigation of texture effect on visual colour difference evaluation,” Color Res. Appl. **30**(5), 341–347 (2005). [CrossRef]

8. T. Fujine, T. Kanda, Y. Yoshida, M. Sugino, M. Teragawa, Y. Yamamoto, and N. Ohta, “Bit depth needed for high image quality TV-evaluation using color distribution index,” J. Disp. Technol. **4**(3), 340–347 (2008). [CrossRef]

## 2. Chromaticity discrimination ellipse

*= 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

*Q*and

_{s}*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.

*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

*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 formulawhere Δ

*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}## 5. Numerical examples and discussions

*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.

*E*(thind data lines) and Δ

_{de}*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}*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.

**26**(5), 340–350 (2001). [CrossRef]

*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*significantly changes with the hue angle. Δ

_{de}*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}*E*

_{00}and Δ

*E*increase with the chroma because the Euclidian distance between the two color points on the

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

_{de}*E*

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

*E*around the same hue angle.

_{de}## 6. Conclusion

_{.}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

## References and links

1. | G. Wyszecki and W. S. Stiles, |

2. | Commission Internationale de l’Eclairage (CIE), |

3. | Commission Internationale de l’Eclairage (CIE), |

4. | M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. |

5. | S. Wen and H.-J. Ma, Investigation of Chromaticity Discrimination Ellipses for Displays,” in |

6. | R.-C. Wu, R. H. Wardman, and M. R. Luo, “A comparison of lightness contrast effects in CRT and surface colours,” Color Res. Appl. |

7. | J. H. Xin, H.-L. Shen, and C. C. Lam, “Investigation of texture effect on visual colour difference evaluation,” Color Res. Appl. |

8. | T. Fujine, T. Kanda, Y. Yoshida, M. Sugino, M. Teragawa, Y. Yamamoto, and N. Ohta, “Bit depth needed for high image quality TV-evaluation using color distribution index,” J. Disp. Technol. |

9. | S.-S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl. |

**OCIS Codes**

(330.1690) Vision, color, and visual optics : Color

(330.1730) Vision, color, and visual optics : Colorimetry

**ToC Category:**

Vision, Color, and Visual Optics

**History**

Original Manuscript: August 9, 2012

Revised Manuscript: October 7, 2012

Manuscript Accepted: November 1, 2012

Published: November 8, 2012

**Virtual Issues**

Vol. 7, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Senfar Wen, "A color difference metric based on the chromaticity discrimination ellipses," Opt. Express **20**, 26441-26447 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26441

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### References

- G. Wyszecki and W. S. Stiles, Color Science, 7th ed. (John Wiley & Sons, 1982).
- Commission Internationale de l’Eclairage (CIE), Industrial colour-difference evaluation, CIE Publication No. 116 (Vienna, 1995).
- Commission Internationale de l’Eclairage (CIE), Improvement to Industrial colour-difference evaluation, CIE Publication No. 114–2001 (Vienna, 2001).
- M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl.26(5), 340–350 (2001). [CrossRef]
- S. Wen and H.-J. Ma, Investigation of Chromaticity Discrimination Ellipses for Displays,” in Society for Information Display 2012 International Symposium Digest of Technical Papers (Society for Information Display, 2012), pp. 725–728.
- R.-C. Wu, R. H. Wardman, and M. R. Luo, “A comparison of lightness contrast effects in CRT and surface colours,” Color Res. Appl.30(1), 13–20 (2005). [CrossRef]
- J. H. Xin, H.-L. Shen, and C. C. Lam, “Investigation of texture effect on visual colour difference evaluation,” Color Res. Appl.30(5), 341–347 (2005). [CrossRef]
- T. Fujine, T. Kanda, Y. Yoshida, M. Sugino, M. Teragawa, Y. Yamamoto, and N. Ohta, “Bit depth needed for high image quality TV-evaluation using color distribution index,” J. Disp. Technol.4(3), 340–347 (2008). [CrossRef]
- S.-S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl.24(5), 331–343 (1999). [CrossRef]

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