Abstract

We propose a method to improve the prediction performance of existing color-difference formulas with additional visual data. The formula is treated as the mean function of a Gaussian process, which is trained with experimentally determined color-discrimination data. Color-difference predictions are calculated using Gaussian process regression (GPR) considering the uncertainty of the visual data. The prediction accuracy of the CIE94 formula is significantly improved with the GPR approach for the Leeds and the Witt datasets. By upgrading CIE94 with GPR we achieve a significantly lower STRESS value of 26.58 compared with that for CIEDE2000 (27.49) on a combined dataset. The method could serve to improve the prediction performance of existing color-difference equations around particular color centers without changing the equations themselves.

© 2010 Optical Society of America

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References

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  1. Committee of the Society of Dyers and Colorists, “BS 6923: Method for calculation of small colour differences,” Tech. rep. (British Standards Institution, London, 1988).
  2. CIE Publication No. 116, “Industrial colour-difference evaluation,” Tech. rep. (Central Bureau of the CIE, Vienna, 1995).
  3. CIE Publication No. 142, “Improvement to industrial colour-difference evaluation,” Tech. rep. (Central Bureau of the CIE, Vienna, 2001).
  4. R. S. Berns, Billmeyer and Saltzman’s Principles of Color Technology, 3rd ed. (Wiley, 2000).
  5. M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. 26, 340–350 (2001).
    [CrossRef]
  6. M. Melgosa, R. Huertas, and R. S. Berns, “Relative significance of the terms in the CIEDE2000 and CIE94 color-difference formulas,” J. Opt. Soc. Am. A 21, 2269–2275 (2004).
    [CrossRef]
  7. R. G. Kuehni, “Variability in estimation of suprathreshold small color differences,” Color Res. Appl. 34, 367–374 (2009).
    [CrossRef]
  8. M. Melgosa, R. Huertas, and R. S. Berns, “Performance of recent advanced color-difference formulas using the standardized residual sum of squares index,” J. Opt. Soc. Am. A 25, 1828–1834 (2008).
    [CrossRef]
  9. S. Shen and R. S. Berns, “Evaluating color difference equation performance incorporating visual uncertainty,” Color Res. Appl. 34, 375–390 (2009).
    [CrossRef]
  10. R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, “Visual determination of suprathreshold color-difference tolerances using probit analysis,” Color Res. Appl. 16, 297–316 (1991).
    [CrossRef]
  11. M. R. Luo and B. Rigg, “Chromaticity-discrimination ellipses for surface colours,” Color Res. Appl. 11, 25–42 (1986).
    [CrossRef]
  12. D. H. Kim and J. H. Nobbs, “New weighting functions for the weighted CIELAB colour difference formula,” in Proceedings of l'Association Internationale de la Couleur (AIC) (AIC, 1997), pp. 446–449.
  13. K. Witt, “Geometric relations between scales of small colour differences,” Color Res. Appl. 24, 78–92 (1999).
    [CrossRef]
  14. R. Kuehni, “Color difference formulas: An unsatisfactory state of affairs,” Color Res. Appl. 33, 324–326 (2008).
    [CrossRef]
  15. I. Lissner and P. Urban, “Improving color-difference formulas using visual data,” in Proceedings of the 5th European Conference on Colour in Graphics, Imaging, and Vision (CGIV) (IS&T, 2010).
  16. CIE Publication No. 101, “Parametric effects in colour-difference evaluation,” Tech. rep. (Central Bureau of the CIE, Vienna, 1993).
  17. C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (The MIT Press, 2006).
  18. L. Silberstein and D. L. MacAdam, “The distribution of color matchings around a color center,” J. Opt. Soc. Am. 35, 32–39 (1945).
    [CrossRef]
  19. W. R. J. Brown, “Statistics of color-matching data,” J. Opt. Soc. Am. 42, 252–256 (1952).
    [CrossRef]
  20. W. R. J. Brown, W. G. Howe, J. E. Jackson, and R. H. Morris, “Multivariate normality of the color-matching process,” J. Opt. Soc. Am. 46, 46–49 (1956).
    [CrossRef]
  21. K. V. Mardia and R. J. Marshall, “Maximum likelihood estimation of models for residual covariance in spatial regression,” Biometrika 71, 135–146 (1984).
    [CrossRef]
  22. M. Kuss, “Gaussian process models for robust regression, classification, and reinforcement learning,” Ph.D. thesis (Technische Universität Darmstadt, 2006).
  23. C. E. Rasmussen, “The Gaussian processes web site,” http://www.gaussianprocess.org.
  24. S.-S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl. 24, 331–343 (1999).
    [CrossRef]
  25. P. A. García, R. Huertas, M. Melgosa, and G. Cui, “Measurement of the relationship between perceived and computed color differences,” J. Opt. Soc. Am. A 24, 1823–1829 (2007).
    [CrossRef]
  26. S. Shen, “Color difference formula and uniform color space modeling and evaluation,” Master's thesis (Rochester Institute of Technology, 2009).

2010 (1)

I. Lissner and P. Urban, “Improving color-difference formulas using visual data,” in Proceedings of the 5th European Conference on Colour in Graphics, Imaging, and Vision (CGIV) (IS&T, 2010).

2009 (3)

S. Shen and R. S. Berns, “Evaluating color difference equation performance incorporating visual uncertainty,” Color Res. Appl. 34, 375–390 (2009).
[CrossRef]

R. G. Kuehni, “Variability in estimation of suprathreshold small color differences,” Color Res. Appl. 34, 367–374 (2009).
[CrossRef]

S. Shen, “Color difference formula and uniform color space modeling and evaluation,” Master's thesis (Rochester Institute of Technology, 2009).

2008 (2)

2007 (1)

2006 (2)

M. Kuss, “Gaussian process models for robust regression, classification, and reinforcement learning,” Ph.D. thesis (Technische Universität Darmstadt, 2006).

C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (The MIT Press, 2006).

2004 (1)

2001 (2)

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

CIE Publication No. 142, “Improvement to industrial colour-difference evaluation,” Tech. rep. (Central Bureau of the CIE, Vienna, 2001).

2000 (1)

R. S. Berns, Billmeyer and Saltzman’s Principles of Color Technology, 3rd ed. (Wiley, 2000).

1999 (2)

K. Witt, “Geometric relations between scales of small colour differences,” Color Res. Appl. 24, 78–92 (1999).
[CrossRef]

S.-S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl. 24, 331–343 (1999).
[CrossRef]

1997 (1)

D. H. Kim and J. H. Nobbs, “New weighting functions for the weighted CIELAB colour difference formula,” in Proceedings of l'Association Internationale de la Couleur (AIC) (AIC, 1997), pp. 446–449.

1995 (1)

CIE Publication No. 116, “Industrial colour-difference evaluation,” Tech. rep. (Central Bureau of the CIE, Vienna, 1995).

1993 (1)

CIE Publication No. 101, “Parametric effects in colour-difference evaluation,” Tech. rep. (Central Bureau of the CIE, Vienna, 1993).

1991 (1)

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, “Visual determination of suprathreshold color-difference tolerances using probit analysis,” Color Res. Appl. 16, 297–316 (1991).
[CrossRef]

1988 (1)

Committee of the Society of Dyers and Colorists, “BS 6923: Method for calculation of small colour differences,” Tech. rep. (British Standards Institution, London, 1988).

1986 (1)

M. R. Luo and B. Rigg, “Chromaticity-discrimination ellipses for surface colours,” Color Res. Appl. 11, 25–42 (1986).
[CrossRef]

1984 (1)

K. V. Mardia and R. J. Marshall, “Maximum likelihood estimation of models for residual covariance in spatial regression,” Biometrika 71, 135–146 (1984).
[CrossRef]

1956 (1)

1952 (1)

1945 (1)

Alman, D. H.

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, “Visual determination of suprathreshold color-difference tolerances using probit analysis,” Color Res. Appl. 16, 297–316 (1991).
[CrossRef]

Balonon-Rosen, M. R.

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, “Visual determination of suprathreshold color-difference tolerances using probit analysis,” Color Res. Appl. 16, 297–316 (1991).
[CrossRef]

Berns, R. S.

S. Shen and R. S. Berns, “Evaluating color difference equation performance incorporating visual uncertainty,” Color Res. Appl. 34, 375–390 (2009).
[CrossRef]

M. Melgosa, R. Huertas, and R. S. Berns, “Performance of recent advanced color-difference formulas using the standardized residual sum of squares index,” J. Opt. Soc. Am. A 25, 1828–1834 (2008).
[CrossRef]

M. Melgosa, R. Huertas, and R. S. Berns, “Relative significance of the terms in the CIEDE2000 and CIE94 color-difference formulas,” J. Opt. Soc. Am. A 21, 2269–2275 (2004).
[CrossRef]

R. S. Berns, Billmeyer and Saltzman’s Principles of Color Technology, 3rd ed. (Wiley, 2000).

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, “Visual determination of suprathreshold color-difference tolerances using probit analysis,” Color Res. Appl. 16, 297–316 (1991).
[CrossRef]

Brown, W. R. J.

Cui, G.

P. A. García, R. Huertas, M. Melgosa, and G. Cui, “Measurement of the relationship between perceived and computed color differences,” J. Opt. Soc. Am. A 24, 1823–1829 (2007).
[CrossRef]

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

García, P. A.

Guan, S.-S.

S.-S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl. 24, 331–343 (1999).
[CrossRef]

Howe, W. G.

Huertas, R.

Jackson, J. E.

Kim, D. H.

D. H. Kim and J. H. Nobbs, “New weighting functions for the weighted CIELAB colour difference formula,” in Proceedings of l'Association Internationale de la Couleur (AIC) (AIC, 1997), pp. 446–449.

Kuehni, R.

R. Kuehni, “Color difference formulas: An unsatisfactory state of affairs,” Color Res. Appl. 33, 324–326 (2008).
[CrossRef]

Kuehni, R. G.

R. G. Kuehni, “Variability in estimation of suprathreshold small color differences,” Color Res. Appl. 34, 367–374 (2009).
[CrossRef]

Kuss, M.

M. Kuss, “Gaussian process models for robust regression, classification, and reinforcement learning,” Ph.D. thesis (Technische Universität Darmstadt, 2006).

Lissner, I.

I. Lissner and P. Urban, “Improving color-difference formulas using visual data,” in Proceedings of the 5th European Conference on Colour in Graphics, Imaging, and Vision (CGIV) (IS&T, 2010).

Luo, M. R.

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

S.-S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl. 24, 331–343 (1999).
[CrossRef]

M. R. Luo and B. Rigg, “Chromaticity-discrimination ellipses for surface colours,” Color Res. Appl. 11, 25–42 (1986).
[CrossRef]

MacAdam, D. L.

Mardia, K. V.

K. V. Mardia and R. J. Marshall, “Maximum likelihood estimation of models for residual covariance in spatial regression,” Biometrika 71, 135–146 (1984).
[CrossRef]

Marshall, R. J.

K. V. Mardia and R. J. Marshall, “Maximum likelihood estimation of models for residual covariance in spatial regression,” Biometrika 71, 135–146 (1984).
[CrossRef]

Melgosa, M.

Morris, R. H.

Nobbs, J. H.

D. H. Kim and J. H. Nobbs, “New weighting functions for the weighted CIELAB colour difference formula,” in Proceedings of l'Association Internationale de la Couleur (AIC) (AIC, 1997), pp. 446–449.

Rasmussen, C. E.

C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (The MIT Press, 2006).

C. E. Rasmussen, “The Gaussian processes web site,” http://www.gaussianprocess.org.

Reniff, L.

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, “Visual determination of suprathreshold color-difference tolerances using probit analysis,” Color Res. Appl. 16, 297–316 (1991).
[CrossRef]

Rigg, B.

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

M. R. Luo and B. Rigg, “Chromaticity-discrimination ellipses for surface colours,” Color Res. Appl. 11, 25–42 (1986).
[CrossRef]

Shen, S.

S. Shen and R. S. Berns, “Evaluating color difference equation performance incorporating visual uncertainty,” Color Res. Appl. 34, 375–390 (2009).
[CrossRef]

S. Shen, “Color difference formula and uniform color space modeling and evaluation,” Master's thesis (Rochester Institute of Technology, 2009).

Silberstein, L.

Snyder, G. D.

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, “Visual determination of suprathreshold color-difference tolerances using probit analysis,” Color Res. Appl. 16, 297–316 (1991).
[CrossRef]

Urban, P.

I. Lissner and P. Urban, “Improving color-difference formulas using visual data,” in Proceedings of the 5th European Conference on Colour in Graphics, Imaging, and Vision (CGIV) (IS&T, 2010).

Williams, C. K. I.

C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (The MIT Press, 2006).

Witt, K.

K. Witt, “Geometric relations between scales of small colour differences,” Color Res. Appl. 24, 78–92 (1999).
[CrossRef]

Biometrika (1)

K. V. Mardia and R. J. Marshall, “Maximum likelihood estimation of models for residual covariance in spatial regression,” Biometrika 71, 135–146 (1984).
[CrossRef]

Color Res. Appl. (8)

S.-S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl. 24, 331–343 (1999).
[CrossRef]

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

R. G. Kuehni, “Variability in estimation of suprathreshold small color differences,” Color Res. Appl. 34, 367–374 (2009).
[CrossRef]

S. Shen and R. S. Berns, “Evaluating color difference equation performance incorporating visual uncertainty,” Color Res. Appl. 34, 375–390 (2009).
[CrossRef]

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, “Visual determination of suprathreshold color-difference tolerances using probit analysis,” Color Res. Appl. 16, 297–316 (1991).
[CrossRef]

M. R. Luo and B. Rigg, “Chromaticity-discrimination ellipses for surface colours,” Color Res. Appl. 11, 25–42 (1986).
[CrossRef]

K. Witt, “Geometric relations between scales of small colour differences,” Color Res. Appl. 24, 78–92 (1999).
[CrossRef]

R. Kuehni, “Color difference formulas: An unsatisfactory state of affairs,” Color Res. Appl. 33, 324–326 (2008).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Other (11)

Committee of the Society of Dyers and Colorists, “BS 6923: Method for calculation of small colour differences,” Tech. rep. (British Standards Institution, London, 1988).

CIE Publication No. 116, “Industrial colour-difference evaluation,” Tech. rep. (Central Bureau of the CIE, Vienna, 1995).

CIE Publication No. 142, “Improvement to industrial colour-difference evaluation,” Tech. rep. (Central Bureau of the CIE, Vienna, 2001).

R. S. Berns, Billmeyer and Saltzman’s Principles of Color Technology, 3rd ed. (Wiley, 2000).

I. Lissner and P. Urban, “Improving color-difference formulas using visual data,” in Proceedings of the 5th European Conference on Colour in Graphics, Imaging, and Vision (CGIV) (IS&T, 2010).

CIE Publication No. 101, “Parametric effects in colour-difference evaluation,” Tech. rep. (Central Bureau of the CIE, Vienna, 1993).

C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (The MIT Press, 2006).

D. H. Kim and J. H. Nobbs, “New weighting functions for the weighted CIELAB colour difference formula,” in Proceedings of l'Association Internationale de la Couleur (AIC) (AIC, 1997), pp. 446–449.

S. Shen, “Color difference formula and uniform color space modeling and evaluation,” Master's thesis (Rochester Institute of Technology, 2009).

M. Kuss, “Gaussian process models for robust regression, classification, and reinforcement learning,” Ph.D. thesis (Technische Universität Darmstadt, 2006).

C. E. Rasmussen, “The Gaussian processes web site,” http://www.gaussianprocess.org.

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Figures (9)

Fig. 1
Fig. 1

Relation between color pair distance ϕ x , x ́ and Euclidean length difference ρ x , x ́ evaluated on the RIT-DuPont dataset. One bar represents the mean length difference for all color pair distances within the corresponding distance range.

Fig. 2
Fig. 2

Relation between angle arccos ( α x , x ́ ) and Euclidean length difference ρ x , x ́ evaluated on the RIT-DuPont dataset. The length differences were calculated separately for each RIT-DuPont color center and then averaged over all color centers. One bar represents the mean length difference for all angles within the corresponding angle range. The function 1 α x , x ́ 2 (normalized) is shown as a black curve to illustrate the agreement of our model with the visual data.

Fig. 3
Fig. 3

Components of the covariance function. (a) Distance and angle components ϕ x , x ́ and α x , x ́ . (b) Length-difference component ρ x , x ́ .

Fig. 4
Fig. 4

Damping function f ( δ , m ) . For m 2 < δ < m 2 the damping function has no effect on the correction term δ. For δ and δ , the damped correction term converges against m and m , respectively.

Fig. 5
Fig. 5

Damping function f ( δ , m * ) for varying m . The allowed range of values of the correction term δ increases with increasing mean function value m . For δ and δ the damped correction term converges against m and m , respectively.

Fig. 6
Fig. 6

CIE Gray color center at ( L , a , b ) = ( 59.3 , 0.78 , 1.05 ) with selected RIT-DuPont ± T 50 color-difference vectors ( Δ V = 1.02 ) , CIE94 iso-distance contour (nearly circular ellipse, blue online) with Δ E 94 * = 1 , GPR iso-distance contour (ellipse, red online) with Δ GPR = 1 , and 95% confidence interval ( ± 2 var ( v * ) ) of the predictions (shaded gray ellipse) computed according to Eq. (17). Note that due to our damping function f this variance is only an approximation. The Gaussian process uses CIE94 as its mean function and was trained with the RIT-DuPont dataset.

Fig. 7
Fig. 7

Negative log marginal likelihood under varying noise. There is a distinct minimum at σ ϵ 2 = 0.0080 ( σ ϵ = 0.0894 ) with a negative log marginal likelihood of 233.4 . The remaining hyperparameters l 1 , l 2 , and l 3 are set to their optimal values determined by hyperparameter optimization (Section 3). The Gaussian process uses CIE94 as its mean function and was trained with the Witt dataset.

Fig. 8
Fig. 8

(a) Leeds visual data versus CIE94 predictions. (b) Leeds visual data versus computed GPR CIE 94 predictions (trained with the Leeds data). (c) Witt visual data versus CIE94 predictions. (d) Witt visual data versus GPR CIE 94 predictions (trained with the Witt data).

Fig. 9
Fig. 9

(a) Filled CIE94 iso-distance contours ( Δ E 94 * = 1 ) . (b) Filled GPR CIE 94 iso-distance contours ( Δ GPR CIE 94 = 1 ) at L * = 0 , (c) L * = 40 , (d) L * = 60 . Blue contours [online in (b)-(d)] indicate good agreement between CIE94 and GPR predictions; pink contours [online in (c) hue angles around 45° and 270°, (d) hue angles around 180° and ellipses close to the gray axis] indicate high disagreement. The Gaussian process was trained with the Witt data. Note that divergence from the mean function (pink contours) is possible only close to visual data.

Tables (4)

Tables Icon

Table 1 Overview of the Visual Data Used in the Experiments a

Tables Icon

Table 2 Optimized Hyperparameters (Using Marginal Likelihood Optimization) for All Training Sets Employed in the Experiments a

Tables Icon

Table 3 STRESS Values of Color-Difference Measures Applied to Five Sets of Visual Data a

Tables Icon

Table 4 PF/3 Values of Color-Difference Measures Applied to Five Sets of Visual Data a

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

x v ( x ) ,
p ( v | X ) = N ( m , K ) ,
K = [ k ( x 1 , x 1 ) k ( x 1 , x n ) k ( x n , x 1 ) k ( x n , x n ) ] ,
p ( v , v | x , X ) = N ( [ m m ] , [ K k k T k ] ) ,
p ( v | v , x , X ) = N ( E ( v ) , var ( v ) ) ,
E ( v ) = m + k T K 1 ( v m ) ,
var ( v ) = k k T K 1 k .
v ( x ) = v ( x ) + ϵ .
E ( v ) = m + k T ( K + σ ϵ 2 I ) 1 ( v m ) ,
var ( v ) = k k T ( K + σ ϵ 2 I ) 1 k .
m ( x ) = Δ E 94 * ( x ) ,
ϕ x , x ́ = ( x 1 + x 2 2 ) ( x ́ 1 + x ́ 2 2 ) 2 .
α x , x ́ = x d T x ́ d x d 2 x ́ d 2 = cos { ( x d , x ́ d ) } .
ρ x , x ́ = abs ( x d 2 x ́ d 2 ) .
k ( x , x ́ ) = exp [ ϕ x , x ́ 2 2 l 1 2 ] exp [ ρ x , x ́ 2 2 l 2 2 ] α x , x ́ 2 l 3 ,
var ( v ) = k k T ( K + σ ϵ 2 I ) 1 k ,
f ( δ , m ) = { m 2 exp ( 1 + 2 δ m ) m , if δ < m 2 m 2 exp ( 1 2 δ m ) + m , if δ > m 2 δ , else . }
log p ( v | X , θ ) = log N ( m , K + σ ϵ 2 I ) = 1 2 ( v m ) T ( K + σ ϵ 2 I ) 1 ( v m ) 1 2 log | K + σ ϵ 2 I | n 2 log 2 π .

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