Abstract

The Optical Society of America’s Uniform Color Scales (OSA-UCS) is one of the color spaces that most closely approximate a “true” uniform color space. Different techniques have been used to convert OSA-UCS-based color specification parameters, L, j, and g, to the CIE tristimulus values, X, Y, and Z. However, none of these methods provides a direct method of inverting OSA-UCS to CIEXYZ values. Thus, numerical algorithms, such as the Newton–Raphson method, have been employed to obtain the transformations. The relative low accuracy and long computation time of this method makes it undesirable for practical applications. An artificial neural network (ANN) was employed to convert OSA-UCS to CIEXYZ. Its performance was compared with that of numerical methods. After optimization, ANN gave a better performance with a mean error (ΔEXYZ) of 1.0×104 and a conversion time of less than 1 s for 1891 samples.

© 2013 Optical Society of America

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References

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  21. Disclosure: While we have made attempts to ensure the code is safe and risk free, the resources are offered “as is” with no warranty. No liability for any damage to your system or material will be accepted.

2012 (1)

N. Moroney, H. S. Fairman, and P. Chong, “An inverse to the Optical Society of America-Uniform Color System,” Color Res. Appl. 37, 106–108 (2012).
[CrossRef]

2009 (1)

2008 (2)

2006 (2)

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[CrossRef]

R. Huertas, M. Melgosa, and C. Oleari, “Performance of a color-difference formula based on OSA-UCS space using small-medium color differences,” J. Opt. Soc. Am. A 23, 2077–2084 (2006).
[CrossRef]

2004 (1)

2002 (3)

M. Kobayasi and K. Yosiki, “Effective conversion algorithm from OSA-UCS to CIEXYZ,” Proc. SPIE 4421, 4 (2002).
[CrossRef]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[CrossRef]

R. G. Kuehni, “CIEDE2000, milestone or final answer?” Color Res. Appl. 27, 126–127 (2002).
[CrossRef]

2001 (1)

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]

1998 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

1978 (1)

1974 (1)

Beale, M. H.

M. H. Beale, M. T. Hagan, and H. B. Demuth, Neural Network Toolbox User’s Guide (The MathWorks, 2011).

Berns, R. S.

Cárdenas, L.

R. Shamey, D. Hinks, M. Melgosa, R. Luo, G. Cui, R. Huertas, L. Cárdenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two ncsu experimental datasets,” in The 5th European Conference on Colour in Graphics, Imaging, and Vision and the 12th International Symposium on Multispectral Colour Science, Joensuu, Finland (2010), pp. 423–428.

Chong, P.

N. Moroney, H. S. Fairman, and P. Chong, “An inverse to the Optical Society of America-Uniform Color System,” Color Res. Appl. 37, 106–108 (2012).
[CrossRef]

Cui, G.

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[CrossRef]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[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. Shamey, D. Hinks, M. Melgosa, R. Luo, G. Cui, R. Huertas, L. Cárdenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two ncsu experimental datasets,” in The 5th European Conference on Colour in Graphics, Imaging, and Vision and the 12th International Symposium on Multispectral Colour Science, Joensuu, Finland (2010), pp. 423–428.

Demuth, H. B.

M. H. Beale, M. T. Hagan, and H. B. Demuth, Neural Network Toolbox User’s Guide (The MathWorks, 2011).

Fairman, H. S.

N. Moroney, H. S. Fairman, and P. Chong, “An inverse to the Optical Society of America-Uniform Color System,” Color Res. Appl. 37, 106–108 (2012).
[CrossRef]

Hagan, M. T.

M. H. Beale, M. T. Hagan, and H. B. Demuth, Neural Network Toolbox User’s Guide (The MathWorks, 2011).

Haykin, S.

S. Haykin, Neural Networks and Learning Machines, 3rd ed. (Prentice-Hall, 2009).

Hinks, D.

R. Shamey, D. Hinks, M. Melgosa, R. Luo, G. Cui, R. Huertas, L. Cárdenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two ncsu experimental datasets,” in The 5th European Conference on Colour in Graphics, Imaging, and Vision and the 12th International Symposium on Multispectral Colour Science, Joensuu, Finland (2010), pp. 423–428.

Huertas, R.

Kobayasi, M.

M. Kobayasi and K. Yosiki, “Effective conversion algorithm from OSA-UCS to CIEXYZ,” Proc. SPIE 4421, 4 (2002).
[CrossRef]

Kuehni, R. G.

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

R. G. Kuehni, “CIEDE2000, milestone or final answer?” Color Res. Appl. 27, 126–127 (2002).
[CrossRef]

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Lee, S. G.

R. Shamey, D. Hinks, M. Melgosa, R. Luo, G. Cui, R. Huertas, L. Cárdenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two ncsu experimental datasets,” in The 5th European Conference on Colour in Graphics, Imaging, and Vision and the 12th International Symposium on Multispectral Colour Science, Joensuu, Finland (2010), pp. 423–428.

Li, C.

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[CrossRef]

Luo, M. R.

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[CrossRef]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[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]

Luo, R.

R. Shamey, D. Hinks, M. Melgosa, R. Luo, G. Cui, R. Huertas, L. Cárdenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two ncsu experimental datasets,” in The 5th European Conference on Colour in Graphics, Imaging, and Vision and the 12th International Symposium on Multispectral Colour Science, Joensuu, Finland (2010), pp. 423–428.

MacAdam, D. L.

Melgosa, M.

Moroney, N.

N. Moroney, H. S. Fairman, and P. Chong, “An inverse to the Optical Society of America-Uniform Color System,” Color Res. Appl. 37, 106–108 (2012).
[CrossRef]

Oleari, C.

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Rigg, B.

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[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]

Roesler, G.

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[CrossRef]

Shamey, R.

R. Shamey, D. Hinks, M. Melgosa, R. Luo, G. Cui, R. Huertas, L. Cárdenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two ncsu experimental datasets,” in The 5th European Conference on Colour in Graphics, Imaging, and Vision and the 12th International Symposium on Multispectral Colour Science, Joensuu, Finland (2010), pp. 423–428.

Witt, K.

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[CrossRef]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Yosiki, K.

M. Kobayasi and K. Yosiki, “Effective conversion algorithm from OSA-UCS to CIEXYZ,” Proc. SPIE 4421, 4 (2002).
[CrossRef]

Color Res. Appl. (6)

R. G. Kuehni, “CIEDE2000, milestone or final answer?” Color Res. Appl. 27, 126–127 (2002).
[CrossRef]

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

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[CrossRef]

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[CrossRef]

N. Moroney, H. S. Fairman, and P. Chong, “An inverse to the Optical Society of America-Uniform Color System,” Color Res. Appl. 37, 106–108 (2012).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Proc. SPIE (1)

M. Kobayasi and K. Yosiki, “Effective conversion algorithm from OSA-UCS to CIEXYZ,” Proc. SPIE 4421, 4 (2002).
[CrossRef]

SIAM J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Other (7)

Disclosure: While we have made attempts to ensure the code is safe and risk free, the resources are offered “as is” with no warranty. No liability for any damage to your system or material will be accepted.

S. Haykin, Neural Networks and Learning Machines, 3rd ed. (Prentice-Hall, 2009).

M. H. Beale, M. T. Hagan, and H. B. Demuth, Neural Network Toolbox User’s Guide (The MathWorks, 2011).

http://www.mathworks.com/help/techdoc/ref/fzero.html .

http://www.mathworks.com/help/matlab/ref/fminsearch.html .

Maltab and Neural Network Toolbox Release 2011b (The MathWorks, Inc.).

R. Shamey, D. Hinks, M. Melgosa, R. Luo, G. Cui, R. Huertas, L. Cárdenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two ncsu experimental datasets,” in The 5th European Conference on Colour in Graphics, Imaging, and Vision and the 12th International Symposium on Multispectral Colour Science, Joensuu, Finland (2010), pp. 423–428.

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Figures (7)

Fig. 1.
Fig. 1.

Architecture of an MLP neural network.

Fig. 2.
Fig. 2.

Distribution of the dataset in the Ljg space.

Fig. 3.
Fig. 3.

Distribution of the dataset in the XYZ space.

Fig. 4.
Fig. 4.

Distribution of all data in the Ljg space (blue circles, training samples; red dots, testing samples).

Fig. 5.
Fig. 5.

Distribution of ΔX, ΔY, and ΔZ of the Kobayasi 3 (K3) and MLP methods for all samples.

Fig. 6.
Fig. 6.

Distribution of the ΔX, ΔY, and ΔZ simulated, Kobayasi 3 (K3), and MLP methods for a sample with Ljg value [0 2 0].

Fig. 7.
Fig. 7.

Distribution of OSA color samples in an xy chromaticity diagram.

Tables (4)

Tables Icon

Table 1. Results of the Conversion of L, j, g to X, Y, and Z

Tables Icon

Table 2. Results of Approximation of Y0 in Kobayasi’s Method

Tables Icon

Table 3. Results of Approximation of X, Y, and Z Using Different MLP Architectures

Tables Icon

Table 4. Distributions of ΔEXYZ and ΔELjg

Equations (23)

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

ϕ([X,Y,Z])=[L,j,g].
x=XX+Y+Z,y=YX+Y+Z,
Y0=(4.4934x2+4.3034y24.27xy1.3744x2.5643y+1.8103)Y,
Λ=5.9[Y01/32/3+0.042(Y030)1/3],
C=Λ5.9(Y01/32/3),
R=0.799X+0.4194Y0.1648Z,G=0.4493X+1.3265Y+0.0927Z,B=0.1149X+0.3394Y+0.717Z,
L=(Λ14.4)21/2,j=C(1.7R1/3+8G1/39.7B1/3),g=C(13.7R1/3+17.7G1/34B1/3).
X=1.0626ω30.4121(ω+a)3+0.2975(ω+b)3,Y=0.3599ω3+0.6401(ω+a)30.0000(ω+b)3,Z=0.0001ω30.3690(ω+a)3+1.4424(ω+b)3,
Y0=ψ(ω).
(21/2L+14.45.9t+23)30.0423(t330)=0.
(L0,j0,g0)=ϕ(Xi,Yi,Zi),(L1,j1,g1)=ϕ(Xi+ΔX,Yi,Zi),(L2,j2,g2)=ϕ(Xi,Yi+ΔY,Zi),(L3,j3,g3)=ϕ(Xi,Yi,Zi+ΔZ).
[Xi+1Yi+1Zi+1]=[(L1L0)/ΔX(L2L0)/ΔY(L3L0)/ΔZ(j1j0)/ΔX(j2j0)/ΔY(j3j0)/ΔZ(g1g0)/ΔX(g2g0)/ΔY(g3g0)/ΔZ]1×[LL0jj0gg0]+[XiYiZi].
Lk=f1(Xk,Yk,Zk),jk=f2(Xk,Yk,Zk),gk=f3(Xk,Yk,Zk),
F(Xi,Yi,Zi)=[f1(Xi,Yi,Zi)L0]2+[f2(Xi,Yi,Zi)j0]2+[f3(Xi,Yi,Zi)g0]2,
τ={x(i),d(i)}i=1N,
F(x,Γ)=Φ(WMΦ(W1x+b(1))+b(M)),
ξav(Γ)=12Ni=1N[d(i)F(x(i),Γ)]2.
Ω={[Li,ji,gi],[Xi,Yi,Zi]}i=1N=1891.
ΔEXYZi=OiDi=ΔXi2+ΔYi2+ΔZi2.
ΔELjgi=ΔLi2+Δji2+Δgi2.
F(x,Γ)=Ψ(W3×30O(W30×303Φ(W30×302Φ(W30×31x+b30×1(1))+b30×1(2))+b30×1(3))+b30×1(O)),
Φ(x)=21+e(2x)1,
Ψ(x)=x.

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