Abstract

CIELAB is based on the CIE 1931 color matching functions. If we are given a new set of color matching functions, how do we define a CIELAB-like uniform color space for the new functions? This problem arises because the CIE recommended its physiological cone fundamentals in 2006 and is considering a new set of color matching functions based on them. In fact, the same problem exists for many practical applications in digital imaging. Typical solutions involve using illuminant-dependent color correction matrices to transform the device-dependent color space into the CIE XYZ color space. This conversion process suffers information loss unless the two sets of color matching functions are linear combinations of each other. In this paper, we propose a design process that allows us to develop a CIELAB-like color space using the native sensor fundamentals. The basic idea is to choose the daylight locus as the yellow–blue opponent color process. We call this class of color space DLAB. We describe the design procedures and compare the resulting Munsell color uniformity under CIELAB (L*,a*,b*) and DLAB (L+,a+,b+).

© 2014 Optical Society of America

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References

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2012 (3)

2009 (1)

R. T. Eskew, “Higher order color mechanisms: a critical review,” Vis. Res. 49, 2686–2704 (2009).
[CrossRef]

2006 (2)

J. Mollon, “Monge,” Vis. Neurosci. 23, 297–309 (2006).
[CrossRef]

J. J. Vos, “From lower to higher colour metrics: a historical account,” Clin. Exp. Optom. 89, 348–360 (2006).
[CrossRef]

2004 (1)

E. P. Hornstein, J. Verweij, and J. L. Schnapf, “Electrical coupling between red and green cones in primate retina,” Nat. Neurosci. 7, 745–750 (2004).
[CrossRef]

2001 (1)

2000 (1)

A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and the long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vis. Res. 40, 1711–1737 (2000).
[CrossRef]

1999 (2)

A. Stockman, L. T. Sharpe, and C. Fach, “The spectral sensitivity of human short-wavelength sensitive cones derived from thresholds and color matches,” Vis. Res. 39, 2901–2927 (1999).
[CrossRef]

R. G. Kuehni, “Towards an improved uniform color space,” Color Res. Appl. 24, 253–265 (1999).
[CrossRef]

1995 (1)

E. J. Chichilnisky and B. A. Wandell, “Photoreceptor sensitivity changes explain color appearance shifts induced by large uniform backgrounds in dichoptic matching,” Vis. Res. 35, 239–254 (1995).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

1970 (1)

D. B. Judd, “Ideal color space,” Color Eng. 8, 37–52 (1970).

1964 (1)

1957 (2)

1944 (1)

D. L. MacAdam, “On the geometry of color space,” J. Franklin Inst. 238, 195–210 (1944).
[CrossRef]

1943 (1)

1942 (1)

1933 (1)

1930 (1)

D. B. Judd, “Reduction of data on mixture of color stimuli,” Bur. Stand. J. Res. 4, 515–548 (1930).
[CrossRef]

Adams, E. Q.

Berns, R. S.

M. W. Derhak and R. S. Berns, “Analysis and correction of the Joensuu Munsell glossy spectral database,” in Proceedings of the 20th Color and Imaging Conference (2012), pp. 191–194.

Brainard, D. H.

D. H. Brainard and B. A. Wandell, “Asymmetric color matching: how color appearance depends on the illuminant,” J. Opt. Soc. Am. A 9, 1433–1448 (1992).
[CrossRef]

A. Stockman and D. H. Brainard, “Color vision mechanisms,” in The OSA Handbook of Optics, M. Bass, ed., 3rd ed. (McGraw-Hill, 2010), pp. 11.1–11.104.

Chichilnisky, E. J.

E. J. Chichilnisky and B. A. Wandell, “Photoreceptor sensitivity changes explain color appearance shifts induced by large uniform backgrounds in dichoptic matching,” Vis. Res. 35, 239–254 (1995).
[CrossRef]

Conway, B. R.

Derhak, M. W.

M. W. Derhak and R. S. Berns, “Analysis and correction of the Joensuu Munsell glossy spectral database,” in Proceedings of the 20th Color and Imaging Conference (2012), pp. 191–194.

Drew, M. S.

Ebner, F. F.

F. F. Ebner, “Derivation and modeling of hue uniformity and development of the IPT color space,” Ph.D. dissertation (Rochester Institute of Technology, 1998).

Eskew, R. T.

R. T. Eskew, “Higher order color mechanisms: a critical review,” Vis. Res. 49, 2686–2704 (2009).
[CrossRef]

R. T. Eskew, J. S. McLellan, and F. Giulianini, “Chromatic detection and discrimination,” in Color Vision: From Genes to Perception, K. R. Gegenfurtner and L. T. Sharpe, eds. (Cambridge University, 1999), pp. 345–368.

Fach, C.

A. Stockman, L. T. Sharpe, and C. Fach, “The spectral sensitivity of human short-wavelength sensitive cones derived from thresholds and color matches,” Vis. Res. 39, 2901–2927 (1999).
[CrossRef]

Fairchild, M. D.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” Proceedings of the IS&T/SID 10th Color Imaging Conference, Scottsdale, 2002, pp. 23–27.

M. D. Fairchild, Color Appearance Models, 3rd ed. (Wiley, 2013).

Finlayson, G. D.

Funt, B. V.

Giulianini, F.

R. T. Eskew, J. S. McLellan, and F. Giulianini, “Chromatic detection and discrimination,” in Color Vision: From Genes to Perception, K. R. Gegenfurtner and L. T. Sharpe, eds. (Cambridge University, 1999), pp. 345–368.

Gu, J.

J. Jiang, D. Liu, J. Gu, and S. Süsstrunk, “What is the space of spectral sensitivity functions for digital color cameras?” IEEE Workshop on the Applications of Computer Vision, Tampa, 2013, pp. 168–179.

Hernández-Andrés, J.

Hornstein, E. P.

E. P. Hornstein, J. Verweij, and J. L. Schnapf, “Electrical coupling between red and green cones in primate retina,” Nat. Neurosci. 7, 745–750 (2004).
[CrossRef]

Hunt, R. W. G.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” Proceedings of the IS&T/SID 10th Color Imaging Conference, Scottsdale, 2002, pp. 23–27.

Jiang, J.

J. Jiang, D. Liu, J. Gu, and S. Süsstrunk, “What is the space of spectral sensitivity functions for digital color cameras?” IEEE Workshop on the Applications of Computer Vision, Tampa, 2013, pp. 168–179.

Judd, D. B.

Kivinen, H.

H. Kivinen, M. Nuutinen, and P. Oittinen, “Comparison of colour difference methods for natural images,” in Proceedings of the 5th European Conference on Colour in Graphics, Imaging, and Vision (2010), pp. 510–515.

Kuehni, R. G.

R. G. Kuehni, “Towards an improved uniform color space,” Color Res. Appl. 24, 253–265 (1999).
[CrossRef]

Kulikowski, J. J.

A. Panorgias, J. J. Kulikowski, N. R. Parry, D. J. McKeefry, and I. J. Murray, “Phases of daylight and the stability of color perception in the near peripheral human retina,” J. Vis. 12(3), 1–11 (2012).
[CrossRef]

Lafer-Sousa, L.

Lafer-Sousa, R.

Lee, H.-C.

H.-C. Lee, Introduction to Color Imaging Science (Cambridge University, 2005).

H.-C. Lee, “A computational model for opponent color encoding,” Advanced Printing of Conference Summaries, SPSE’s 43rd Annual Conference, Rochester, 1990, pp. 178–181.

H.-C. Lee, “A physics-based color encoding model for images of natural scenes,” Proceedings of the Conference on Modern Engineering and Technology, Electro-Optics Session, Taipei, Taiwan, 1992, pp. 25–52.

Lennie, P.

Li, C.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” Proceedings of the IS&T/SID 10th Color Imaging Conference, Scottsdale, 2002, pp. 23–27.

Liu, D.

J. Jiang, D. Liu, J. Gu, and S. Süsstrunk, “What is the space of spectral sensitivity functions for digital color cameras?” IEEE Workshop on the Applications of Computer Vision, Tampa, 2013, pp. 168–179.

Liu, Y. O.

Luo, M. R.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” Proceedings of the IS&T/SID 10th Color Imaging Conference, Scottsdale, 2002, pp. 23–27.

MacAdam, D. L.

McDermott, K. C.

McKeefry, D. J.

A. Panorgias, J. J. Kulikowski, N. R. Parry, D. J. McKeefry, and I. J. Murray, “Phases of daylight and the stability of color perception in the near peripheral human retina,” J. Vis. 12(3), 1–11 (2012).
[CrossRef]

McLellan, J. S.

R. T. Eskew, J. S. McLellan, and F. Giulianini, “Chromatic detection and discrimination,” in Color Vision: From Genes to Perception, K. R. Gegenfurtner and L. T. Sharpe, eds. (Cambridge University, 1999), pp. 345–368.

Mollon, J.

J. Mollon, “Monge,” Vis. Neurosci. 23, 297–309 (2006).
[CrossRef]

Moroney, N.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” Proceedings of the IS&T/SID 10th Color Imaging Conference, Scottsdale, 2002, pp. 23–27.

Murray, I. J.

A. Panorgias, J. J. Kulikowski, N. R. Parry, D. J. McKeefry, and I. J. Murray, “Phases of daylight and the stability of color perception in the near peripheral human retina,” J. Vis. 12(3), 1–11 (2012).
[CrossRef]

Newhall, S.

Newman, T.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” Proceedings of the IS&T/SID 10th Color Imaging Conference, Scottsdale, 2002, pp. 23–27.

Nickerson, D.

Nieves, J. L.

Nuutinen, M.

H. Kivinen, M. Nuutinen, and P. Oittinen, “Comparison of colour difference methods for natural images,” in Proceedings of the 5th European Conference on Colour in Graphics, Imaging, and Vision (2010), pp. 510–515.

Oittinen, P.

H. Kivinen, M. Nuutinen, and P. Oittinen, “Comparison of colour difference methods for natural images,” in Proceedings of the 5th European Conference on Colour in Graphics, Imaging, and Vision (2010), pp. 510–515.

Panorgias, A.

A. Panorgias, J. J. Kulikowski, N. R. Parry, D. J. McKeefry, and I. J. Murray, “Phases of daylight and the stability of color perception in the near peripheral human retina,” J. Vis. 12(3), 1–11 (2012).
[CrossRef]

Parry, N. R.

A. Panorgias, J. J. Kulikowski, N. R. Parry, D. J. McKeefry, and I. J. Murray, “Phases of daylight and the stability of color perception in the near peripheral human retina,” J. Vis. 12(3), 1–11 (2012).
[CrossRef]

Pokorny, J.

Romero, J.

Sanders, C. L.

Schnapf, J. L.

E. P. Hornstein, J. Verweij, and J. L. Schnapf, “Electrical coupling between red and green cones in primate retina,” Nat. Neurosci. 7, 745–750 (2004).
[CrossRef]

Sharpe, L. T.

A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and the long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vis. Res. 40, 1711–1737 (2000).
[CrossRef]

A. Stockman, L. T. Sharpe, and C. Fach, “The spectral sensitivity of human short-wavelength sensitive cones derived from thresholds and color matches,” Vis. Res. 39, 2901–2927 (1999).
[CrossRef]

Shepard, R. N.

R. N. Shepard, “The perceptual organization of colors: an adaptation to regularities of the terrestrial world,” in The Adapted Mind: Evolutionary Psychology and the Generation of Culture, J. H. Barkow, L. Cosmides, and J. Tooby, eds. (Oxford University, 1992), pp. 495–532.

Smith, V. C.

Sproson, W. N.

W. N. Sproson, Colour Science in Television and Display Systems (Adam Hilger, 1982).

Stockman, A.

A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and the long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vis. Res. 40, 1711–1737 (2000).
[CrossRef]

A. Stockman, L. T. Sharpe, and C. Fach, “The spectral sensitivity of human short-wavelength sensitive cones derived from thresholds and color matches,” Vis. Res. 39, 2901–2927 (1999).
[CrossRef]

A. Stockman and D. H. Brainard, “Color vision mechanisms,” in The OSA Handbook of Optics, M. Bass, ed., 3rd ed. (McGraw-Hill, 2010), pp. 11.1–11.104.

Süsstrunk, S.

J. Jiang, D. Liu, J. Gu, and S. Süsstrunk, “What is the space of spectral sensitivity functions for digital color cameras?” IEEE Workshop on the Applications of Computer Vision, Tampa, 2013, pp. 168–179.

Verweij, J.

E. P. Hornstein, J. Verweij, and J. L. Schnapf, “Electrical coupling between red and green cones in primate retina,” Nat. Neurosci. 7, 745–750 (2004).
[CrossRef]

Vos, J. J.

J. J. Vos, “From lower to higher colour metrics: a historical account,” Clin. Exp. Optom. 89, 348–360 (2006).
[CrossRef]

Wandell, B. A.

E. J. Chichilnisky and B. A. Wandell, “Photoreceptor sensitivity changes explain color appearance shifts induced by large uniform backgrounds in dichoptic matching,” Vis. Res. 35, 239–254 (1995).
[CrossRef]

D. H. Brainard and B. A. Wandell, “Asymmetric color matching: how color appearance depends on the illuminant,” J. Opt. Soc. Am. A 9, 1433–1448 (1992).
[CrossRef]

Webster, M. A.

Wiest, M. C.

Wyszecki, G.

Wyszecki, G. W.

Bur. Stand. J. Res. (1)

D. B. Judd, “Reduction of data on mixture of color stimuli,” Bur. Stand. J. Res. 4, 515–548 (1930).
[CrossRef]

Clin. Exp. Optom. (1)

J. J. Vos, “From lower to higher colour metrics: a historical account,” Clin. Exp. Optom. 89, 348–360 (2006).
[CrossRef]

Color Eng. (1)

D. B. Judd, “Ideal color space,” Color Eng. 8, 37–52 (1970).

Color Res. Appl. (1)

R. G. Kuehni, “Towards an improved uniform color space,” Color Res. Appl. 24, 253–265 (1999).
[CrossRef]

J. Franklin Inst. (1)

D. L. MacAdam, “On the geometry of color space,” J. Franklin Inst. 238, 195–210 (1944).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (6)

J. Vis. (1)

A. Panorgias, J. J. Kulikowski, N. R. Parry, D. J. McKeefry, and I. J. Murray, “Phases of daylight and the stability of color perception in the near peripheral human retina,” J. Vis. 12(3), 1–11 (2012).
[CrossRef]

Nat. Neurosci. (1)

E. P. Hornstein, J. Verweij, and J. L. Schnapf, “Electrical coupling between red and green cones in primate retina,” Nat. Neurosci. 7, 745–750 (2004).
[CrossRef]

Vis. Neurosci. (1)

J. Mollon, “Monge,” Vis. Neurosci. 23, 297–309 (2006).
[CrossRef]

Vis. Res. (4)

A. Stockman, L. T. Sharpe, and C. Fach, “The spectral sensitivity of human short-wavelength sensitive cones derived from thresholds and color matches,” Vis. Res. 39, 2901–2927 (1999).
[CrossRef]

A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and the long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vis. Res. 40, 1711–1737 (2000).
[CrossRef]

E. J. Chichilnisky and B. A. Wandell, “Photoreceptor sensitivity changes explain color appearance shifts induced by large uniform backgrounds in dichoptic matching,” Vis. Res. 35, 239–254 (1995).
[CrossRef]

R. T. Eskew, “Higher order color mechanisms: a critical review,” Vis. Res. 49, 2686–2704 (2009).
[CrossRef]

Other (18)

CIE, “A review of chromatic adaptation transforms,” Tech. Rep., (CIE Central Bureau, 2004).

http://www.uef.fi/fi/spectral/munsell-colors-matt-spectrofotometer-measured .

M. W. Derhak and R. S. Berns, “Analysis and correction of the Joensuu Munsell glossy spectral database,” in Proceedings of the 20th Color and Imaging Conference (2012), pp. 191–194.

http://www.rit.edu/cos/colorscience/rc_munsell_renotation.php .

F. F. Ebner, “Derivation and modeling of hue uniformity and development of the IPT color space,” Ph.D. dissertation (Rochester Institute of Technology, 1998).

CIE, “Colorimetry,” 3rd ed., Tech. Rep., (CIE Central Bureau, 2004).

H. Kivinen, M. Nuutinen, and P. Oittinen, “Comparison of colour difference methods for natural images,” in Proceedings of the 5th European Conference on Colour in Graphics, Imaging, and Vision (2010), pp. 510–515.

W. N. Sproson, Colour Science in Television and Display Systems (Adam Hilger, 1982).

CIE, “Fundamental chromaticity diagram with physiological axes—part 1,” Tech. Rep., (CIE Central Bureau, 2006).

H.-C. Lee, “A computational model for opponent color encoding,” Advanced Printing of Conference Summaries, SPSE’s 43rd Annual Conference, Rochester, 1990, pp. 178–181.

H.-C. Lee, “A physics-based color encoding model for images of natural scenes,” Proceedings of the Conference on Modern Engineering and Technology, Electro-Optics Session, Taipei, Taiwan, 1992, pp. 25–52.

A. Stockman and D. H. Brainard, “Color vision mechanisms,” in The OSA Handbook of Optics, M. Bass, ed., 3rd ed. (McGraw-Hill, 2010), pp. 11.1–11.104.

R. T. Eskew, J. S. McLellan, and F. Giulianini, “Chromatic detection and discrimination,” in Color Vision: From Genes to Perception, K. R. Gegenfurtner and L. T. Sharpe, eds. (Cambridge University, 1999), pp. 345–368.

R. N. Shepard, “The perceptual organization of colors: an adaptation to regularities of the terrestrial world,” in The Adapted Mind: Evolutionary Psychology and the Generation of Culture, J. H. Barkow, L. Cosmides, and J. Tooby, eds. (Oxford University, 1992), pp. 495–532.

J. Jiang, D. Liu, J. Gu, and S. Süsstrunk, “What is the space of spectral sensitivity functions for digital color cameras?” IEEE Workshop on the Applications of Computer Vision, Tampa, 2013, pp. 168–179.

H.-C. Lee, Introduction to Color Imaging Science (Cambridge University, 2005).

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” Proceedings of the IS&T/SID 10th Color Imaging Conference, Scottsdale, 2002, pp. 23–27.

M. D. Fairchild, Color Appearance Models, 3rd ed. (Wiley, 2013).

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Figures (8)

Fig. 1.
Fig. 1.

CIE 1931 (x,y) chromaticity diagram. The CIELAB a* and b* axes are straight lines; when the reference white is illuminant C, the b* axis is almost parallel to the CIE daylight locus. It is also clear that the daylight locus is along the major axes of the Munsell chroma contours.

Fig. 2.
Fig. 2.

CIE daylight locus (4000 to 25000 K) is almost a straight line on the intensity-invariant plane of the log human cone responses (CIE 2006 cone fundamentals).

Fig. 3.
Fig. 3.

αlogL+βlogM+γlogS as a function of 1/T for different sets of α+β+γ=0. They are all approximately close to straight lines, as predicted from Wien’s formula, which is described in the text. Note that for a specific α, β, γ, the slope becomes zero (CIE 2006 cone fundamentals).

Fig. 4.
Fig. 4.

Computation of DLAB. The inputs are normalized sensor responses. The outputs are (L+,a+,b+). The coefficients vy1, vy2, and vy3 are computed from the constant Munsell value plane. The coefficients k11, k13, k21, and k22 are computed from the daylight plane and its orthogonal direction. Note that positive and negative terms are grouped separately and then mapped through the nonlinear function. The scaling factors, r and τ, are determined to match Munsell chromas and CIELAB step size. The nonlinear transform, f(·), is the same as the CIELAB cube root function.

Fig. 5.
Fig. 5.

Three perspectives of the 3D distribution of Munsell color data under illuminant C in LMS(SS).

Fig. 6.
Fig. 6.

(a) CIELAB and (b), (c) DLAB comparisons for Munsell value 5, hue, and chroma contours with daylight locus.

Fig. 7.
Fig. 7.

Comparison of the axes of CIELAB and DLAB at different daylight temperatures at 4000, 6500 and 25,000 K. The shaded, thick curve is the daylight locus, yD=3.000xD2+2.870xD0.275. CIELAB axes are dashed lines and DLAB axes are solid lines.

Fig. 8.
Fig. 8.

c0Y(αu+βv+γw) as a function of 1/T. Here, Y=2Y0. (CIE 2006 cone fundamentals).

Tables (2)

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Table 1. Coefficients for Construction of DLAB

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Table 2. Standard Deviation Comparison

Equations (33)

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L*=116f(Y/Yn)16,a*=500[f(X/Xn)f(Y/Yn)],b*=200[f(Y/Yn)f(Z/Zn)],
L=ϕ(λ)l¯(λ)dλ,M=ϕ(λ)m¯(λ)dλ,S=ϕ(λ)s¯(λ)dλ,
logL=logA+logC1logλl5C2λlT,logM=logA+logC1logλm5C2λmT,logS=logA+logC1logλs5C2λsT.
αlogL+βlogM+γlogS=κC2T(αλl+βλm+γλs),
E=RC+,
R=R+E(CC),
nd=[0.6384,0.7600,0.1216]T.
[L1/LnM1/MnS1/SnL2/LnM2/MnS2/SnLt/LnMt/MnSt/Sn]vy=[Y1/YnY2/YnYt/Yn],
L+=116f(vyTq)16.
q=[L/LnM/MnS/Sn]np1LLn+np2MMn+np3SSnνnp[111]=1νnp[np2+np3np2np3np1np1+np3np3np1np2np1+np2]q.
b+=nd×np.
a+=np×b+.
[alinear+blinear+]=[a+Tb+T]q=1νnp[a+Tb+T][np2+np3np2np3np1np1+np3np3np1np2np1+np2]q=[k11k12k13k21k22k23]q=Kq.
ans+=f(k11LLn+k13SSn)f(MMn),bns+=f(k21LLn+k22MMn)f(SSn),
[ans1+bns1+ans2+bns2+ans1021+bns1021+]=UnsΣnsVnsT,
Σns=[σns100σns20000].
r=σns2/σns1.
[a1*b1*a2*b2*a1021*b1021*]=UCΣCVCT,
[ans1+rbns1+ans2+rbns2+ans1021+rbns1021+]=UsΣsVsT.
ΣC=[σC100σC20000],Σs=[σs100σs20000].
τ=σC1+σC2σs1+σs2
a+=τ[f(k11LLn+k13SSn)f(MMn)],b+=rτ[f(k21LLn+k22MMn)f(SSn)].
[l¯HPE(λ)m¯HPE(λ)s¯HPE(λ)]=[0.40640.71850.08210.21311.09760.0430000.5610][x¯(λ)y¯(λ)z¯(λ)].
L+=116f(0.3628LLn+0.6372MMn0.0000SSn)16,a+=937[f(0.8606LLn+0.1394SSn)f(MMn)],b+=183[f(0.4451LLn+0.5549MMn)f(SSn)],
L+=116f(0.6584LLn+0.3702MMn0.0292SSn)16,a+=806[f(0.8400LLn+0.1600SSn)f(MMn)],b+=184[f(0.7340LLn+0.2660MMn)f(SSn)],
X=D(λ)x¯(λ)dλ=d0X0+d1X1+d2X2,Y=D(λ)y¯(λ)dλ=d0Y0+d1Y1+d2Y2,Z=D(λ)z¯(λ)dλ=d0Z0+d1Z1+d2Z2.
[d0d1d2]=[X0X1X2Y0Y1Y2Z0Z1Z2]1[XYZ].
[u0u1u2]=[X0X1X2Y0Y1Y2Z0Z1Z2]1[xyz].
d0=u0Yu0Y0+u1Y1+u2Y2=c0Y,d1=u1Yu0Y0+u1Y1+u2Y2=c1Y,d2=u2Yu0Y0+u1Y1+u2Y2=c2Y.
L=D(λ)l¯(λ)dλ=d0L0+d1L1+d2L2,M=D(λ)m¯(λ)dλ=d0M0+d1M1+d2M2,S=D(λ)s¯(λ)dλ=d0S0+d1S1+d2S2.
LL0=c0Y(1+c1L1c0L0+c2L2c0L0)=c0Y(1+u),MM0=c0Y(1+c1M1c0M0+c2M2c0M0)=c0Y(1+v),SS0=c0Y(1+c1S1c0S0+c2S2c0S0)=c0Y(1+w).
k=αlog(LL0)+βlog(MM0)+γlog(SS0)=αlog(1+u)+βlog(1+v)+γlog(1+w)αu+βv+γw,
α(LL0)+β(MM0)+γ(SS0)=c0Y(αu+βv+γw).

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