Abstract

Widely varying estimates of the number of discernible object colors have been made by using various methods over the past 100 years. To clarify the source of the discrepancies in the previous, inconsistent estimates, the number of discernible object colors is estimated over a wide range of color temperatures and illuminance levels using several chromatic adaptation models, color spaces, and color difference limens. Efficient and accurate models are used to compute optimal-color solids and count the number of discernible colors. A comprehensive simulation reveals limitations in the ability of current color appearance models to estimate the number of discernible colors even if the color solid is smaller than the optimal-color solid. The estimates depend on the color appearance model, color space, and color difference limen used. The fundamental problem lies in the von Kries-type chromatic adaptation transforms, which have an unknown effect on the ranking of the number of discernible colors at different color temperatures.

© 2013 Optical Society of America

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2012

2010

2008

2007

2006

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]

P. D. Pinto, J. M. Linhares, J. A. Carvalhal, and S. M. Nascimento, “Psychophysical estimation of the best illumination for appreciation of Renaissance paintings,” Vis. Neurosci. 23, 669–674 (2006).
[CrossRef]

2001

J. Morovic, P. L. Sun, and P. Morovic, “The gamuts of input and output colour imaging media,” Proc. SPIE 4300, 114–125 (2001).

1998

M. R. Pointer and G. G. Attridge, “The number of discernible colours,” Color Res. Appl. 23, 52–54 (1998).

C. S. McCamy, “On the number of discernible colors,” Color Res. Appl. 23, 337 (1998).

1995

M. D. Fairchild, “Testing colour-appearance models: Guidelines for coordinated research,” Color Res. Appl. 20, 262–267(1995).

1991

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivener, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166–180 (1991).
[CrossRef]

1981

A. Hård and L. Sivik, “NCS–natural color system: a Swedish standard for color notation,” Color Res. Appl. 6, 129–138 (1981).
[CrossRef]

1951

1943

1939

D. B. Judd and K. L. Kelly, “Method of designating colors,” J. Res. Nat. Bur. Stand. 23, 355–366 (1939).
[CrossRef]

1935

1929

N. D. Nyberg, “Zum Aufbau des Farbkörpers im Raume aller Lichtempfindungen,” Z. Phys. 52, 406–419 (1929).
[CrossRef]

1928

S. Rösch, “Die Kennzeichnung der Farben,” Z. Phys. 29, 83–91 (1928).

1927

R. Luther, “Aus dem Gebiete der Farbreiz-Metrik,” Z. Tech. Phys. 8, 540–558 (1927).

1920

E. Schrödinger, “Theorie der Pigmente von größter Leuchtkraft,” Ann. Phys. 367, 603–622 (1920).
[CrossRef]

1916

W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).

Attridge, G. G.

M. R. Pointer and G. G. Attridge, “The number of discernible colours,” Color Res. Appl. 23, 52–54 (1998).

Boring, E. G.

E. G. Boring, H. S. Langfeld, and H. P. Weld, Introduction to Psychology (Wiley, 1939), p. 517.

Capilla, P.

E. Perales, F. Martínez-Verdú, V. Viqueira, M. J. Luque, and P. Capilla, “Computing the number of distinguishable colors under several illuminants and light sources,” in Proceedings of IS&T CGIV 2006: Third European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, 2006), pp. 414–419.

Carreras, J.

Carvalhal, J. A.

P. D. Pinto, J. M. Linhares, J. A. Carvalhal, and S. M. Nascimento, “Psychophysical estimation of the best illumination for appreciation of Renaissance paintings,” Vis. Neurosci. 23, 669–674 (2006).
[CrossRef]

Chapanis, A.

Cheung, V.

J. Morovic, V. Cheung, and P. Morovic, “Why we don’t know how many colors there are,” in Proceedings of IS&T CGIV 2012: Sixth European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, 2012), pp. 49–53.

Cho, M. S.

Chorro, E.

Clarke, A. A.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivener, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166–180 (1991).
[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]

C. Li, M. R. Luo, and G. Cui, “Colour-differences evaluation using colour appearance models,” in Proceedings of IS&T/SID CIC11: Eleventh Color Imaging Conference (Society for Imaging Science and Technology, 2003), pp. 127–131.

de Fez, D.

Fairchild, M. D.

M. D. Fairchild, “Testing colour-appearance models: Guidelines for coordinated research,” Color Res. Appl. 20, 262–267(1995).

Gilabert, E.

Green, P.

P. Green, Color Management: Understanding and Using ICC Profiles, 1st ed. (Wiley, 2010).

Halsey, R. M.

Hård, A.

A. Hård and L. Sivik, “NCS–natural color system: a Swedish standard for color notation,” Color Res. Appl. 6, 129–138 (1981).
[CrossRef]

Hunt, C. E.

Judd, D. B.

Kelly, K. L.

D. B. Judd and K. L. Kelly, “Method of designating colors,” J. Res. Nat. Bur. Stand. 23, 355–366 (1939).
[CrossRef]

Kim, J. S.

Kuehni, R. G.

R. G. Kuehni, Color Space and its Divisions: Color Order from Antiquity to the Present (Wiley, 2003), pp. 202–203.

Langfeld, H. S.

E. G. Boring, H. S. Langfeld, and H. P. Weld, Introduction to Psychology (Wiley, 1939), p. 517.

Li, C.

C. Li, M. R. Luo, M. S. Cho, and J. S. Kim, “Linear programming method for computing the gamut of object color solid,” J. Opt. Soc. Am. A 27, 985–991 (2010).
[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]

C. Li, M. R. Luo, and G. Cui, “Colour-differences evaluation using colour appearance models,” in Proceedings of IS&T/SID CIC11: Eleventh Color Imaging Conference (Society for Imaging Science and Technology, 2003), pp. 127–131.

Linhares, J. M.

P. D. Pinto, J. M. Linhares, and S. M. Nascimento, “Correlated color temperature preferred by observers for illumination of artistic paintings,” J. Opt. Soc. Am. A 25, 623–630(2008).
[CrossRef]

P. D. Pinto, J. M. Linhares, J. A. Carvalhal, and S. M. Nascimento, “Psychophysical estimation of the best illumination for appreciation of Renaissance paintings,” Vis. Neurosci. 23, 669–674 (2006).
[CrossRef]

Luo, M. R.

C. Li, M. R. Luo, M. S. Cho, and J. S. Kim, “Linear programming method for computing the gamut of object color solid,” J. Opt. Soc. Am. A 27, 985–991 (2010).
[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]

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivener, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166–180 (1991).
[CrossRef]

C. Li, M. R. Luo, and G. Cui, “Colour-differences evaluation using colour appearance models,” in Proceedings of IS&T/SID CIC11: Eleventh Color Imaging Conference (Society for Imaging Science and Technology, 2003), pp. 127–131.

Luque, M. J.

E. Perales, F. Martínez-Verdú, V. Viqueira, M. J. Luque, and P. Capilla, “Computing the number of distinguishable colors under several illuminants and light sources,” in Proceedings of IS&T CGIV 2006: Third European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, 2006), pp. 414–419.

Luther, R.

R. Luther, “Aus dem Gebiete der Farbreiz-Metrik,” Z. Tech. Phys. 8, 540–558 (1927).

MacAdam, D. L.

Martínez-Verdú, F.

F. Martínez-Verdú, E. Perales, E. Chorro, D. de Fez, V. Viqueira, and E. Gilabert, “Computation and visualization of the MacAdam limits for any lightness, hue angle, and light source,” J. Opt. Soc. Am. A 24, 1501–1515 (2007).
[CrossRef]

E. Perales, F. Martínez-Verdú, V. Viqueira, M. J. Luque, and P. Capilla, “Computing the number of distinguishable colors under several illuminants and light sources,” in Proceedings of IS&T CGIV 2006: Third European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, 2006), pp. 414–419.

Masaoka, K.

Masuda, O.

McCamy, C. S.

C. S. McCamy, “On the number of discernible colors,” Color Res. Appl. 23, 337 (1998).

Morovic, J.

J. Morovic, P. L. Sun, and P. Morovic, “The gamuts of input and output colour imaging media,” Proc. SPIE 4300, 114–125 (2001).

J. Morovic, V. Cheung, and P. Morovic, “Why we don’t know how many colors there are,” in Proceedings of IS&T CGIV 2012: Sixth European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, 2012), pp. 49–53.

J. Morovic, Hewlett-Packard Ltd., 42 Scythe Way, Colchester, CO3 4SJ, UK (personal communication, 2012).

Morovic, P.

J. Morovic, P. L. Sun, and P. Morovic, “The gamuts of input and output colour imaging media,” Proc. SPIE 4300, 114–125 (2001).

J. Morovic, V. Cheung, and P. Morovic, “Why we don’t know how many colors there are,” in Proceedings of IS&T CGIV 2012: Sixth European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, 2012), pp. 49–53.

Nascimento, S. M.

Newhall, S. M.

Nickerson, D.

Nyberg, N. D.

N. D. Nyberg, “Zum Aufbau des Farbkörpers im Raume aller Lichtempfindungen,” Z. Phys. 52, 406–419 (1929).
[CrossRef]

Ostwald, W.

W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).

Perales, E.

F. Martínez-Verdú, E. Perales, E. Chorro, D. de Fez, V. Viqueira, and E. Gilabert, “Computation and visualization of the MacAdam limits for any lightness, hue angle, and light source,” J. Opt. Soc. Am. A 24, 1501–1515 (2007).
[CrossRef]

E. Perales, Department of Optics, Pharmacology and Anatomy, University of Alicante, Carretera de San Vicente del Raspeig s/n 03690, Alicante, Spain (personal communication, 2012).

E. Perales, F. Martínez-Verdú, V. Viqueira, M. J. Luque, and P. Capilla, “Computing the number of distinguishable colors under several illuminants and light sources,” in Proceedings of IS&T CGIV 2006: Third European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, 2006), pp. 414–419.

Pinto, P. D.

P. D. Pinto, J. M. Linhares, and S. M. Nascimento, “Correlated color temperature preferred by observers for illumination of artistic paintings,” J. Opt. Soc. Am. A 25, 623–630(2008).
[CrossRef]

P. D. Pinto, J. M. Linhares, J. A. Carvalhal, and S. M. Nascimento, “Psychophysical estimation of the best illumination for appreciation of Renaissance paintings,” Vis. Neurosci. 23, 669–674 (2006).
[CrossRef]

Pointer, M. R.

M. R. Pointer and G. G. Attridge, “The number of discernible colours,” Color Res. Appl. 23, 52–54 (1998).

Quintero, J. M.

Rhodes, P. A.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivener, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166–180 (1991).
[CrossRef]

Rösch, S.

S. Rösch, “Die Kennzeichnung der Farben,” Z. Phys. 29, 83–91 (1928).

Schappo, A.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivener, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166–180 (1991).
[CrossRef]

Schrödinger, E.

E. Schrödinger, “Theorie der Pigmente von größter Leuchtkraft,” Ann. Phys. 367, 603–622 (1920).
[CrossRef]

Scrivener, S. A. R.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivener, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166–180 (1991).
[CrossRef]

Sivik, L.

A. Hård and L. Sivik, “NCS–natural color system: a Swedish standard for color notation,” Color Res. Appl. 6, 129–138 (1981).
[CrossRef]

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 2000).

Sun, P. L.

J. Morovic, P. L. Sun, and P. Morovic, “The gamuts of input and output colour imaging media,” Proc. SPIE 4300, 114–125 (2001).

Tait, C. J.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivener, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166–180 (1991).
[CrossRef]

Titchener, E. B.

E. B. Titchener, Outline of Psychology (Macmillan, 1896), p. 48.

E. B. Titchener, Outline of Psychology: New Edition with Additions (Macmillan, 1899), p. 55.

Viqueira, V.

F. Martínez-Verdú, E. Perales, E. Chorro, D. de Fez, V. Viqueira, and E. Gilabert, “Computation and visualization of the MacAdam limits for any lightness, hue angle, and light source,” J. Opt. Soc. Am. A 24, 1501–1515 (2007).
[CrossRef]

E. Perales, F. Martínez-Verdú, V. Viqueira, M. J. Luque, and P. Capilla, “Computing the number of distinguishable colors under several illuminants and light sources,” in Proceedings of IS&T CGIV 2006: Third European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, 2006), pp. 414–419.

von Kries, J.

J. von Kries, Chromatic Adaptation, Festschrift der Albercht-Ludwig-Universität (Fribourg, 1902).

Weld, H. P.

E. G. Boring, H. S. Langfeld, and H. P. Weld, Introduction to Psychology (Wiley, 1939), p. 517.

Wen, S.

S. Wen, “A method for selecting display primaries to match a target color gamut,” J. Soc. Inf. Disp. 15, 1015–1022 (2007).
[CrossRef]

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 2000).

Ann. Phys.

E. Schrödinger, “Theorie der Pigmente von größter Leuchtkraft,” Ann. Phys. 367, 603–622 (1920).
[CrossRef]

Color Res. Appl.

M. R. Pointer and G. G. Attridge, “The number of discernible colours,” Color Res. Appl. 23, 52–54 (1998).

A. Hård and L. Sivik, “NCS–natural color system: a Swedish standard for color notation,” Color Res. Appl. 6, 129–138 (1981).
[CrossRef]

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivener, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166–180 (1991).
[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]

M. D. Fairchild, “Testing colour-appearance models: Guidelines for coordinated research,” Color Res. Appl. 20, 262–267(1995).

C. S. McCamy, “On the number of discernible colors,” Color Res. Appl. 23, 337 (1998).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Res. Nat. Bur. Stand.

D. B. Judd and K. L. Kelly, “Method of designating colors,” J. Res. Nat. Bur. Stand. 23, 355–366 (1939).
[CrossRef]

J. Soc. Inf. Disp.

S. Wen, “A method for selecting display primaries to match a target color gamut,” J. Soc. Inf. Disp. 15, 1015–1022 (2007).
[CrossRef]

Opt. Lett.

Phys. Z.

W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).

Proc. SPIE

J. Morovic, P. L. Sun, and P. Morovic, “The gamuts of input and output colour imaging media,” Proc. SPIE 4300, 114–125 (2001).

Vis. Neurosci.

P. D. Pinto, J. M. Linhares, J. A. Carvalhal, and S. M. Nascimento, “Psychophysical estimation of the best illumination for appreciation of Renaissance paintings,” Vis. Neurosci. 23, 669–674 (2006).
[CrossRef]

Z. Phys.

N. D. Nyberg, “Zum Aufbau des Farbkörpers im Raume aller Lichtempfindungen,” Z. Phys. 52, 406–419 (1929).
[CrossRef]

S. Rösch, “Die Kennzeichnung der Farben,” Z. Phys. 29, 83–91 (1928).

Z. Tech. Phys.

R. Luther, “Aus dem Gebiete der Farbreiz-Metrik,” Z. Tech. Phys. 8, 540–558 (1927).

Other

R. G. Kuehni, Color Space and its Divisions: Color Order from Antiquity to the Present (Wiley, 2003), pp. 202–203.

E. B. Titchener, Outline of Psychology (Macmillan, 1896), p. 48.

E. B. Titchener, Outline of Psychology: New Edition with Additions (Macmillan, 1899), p. 55.

E. G. Boring, H. S. Langfeld, and H. P. Weld, Introduction to Psychology (Wiley, 1939), p. 517.

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry, 3rd ed. (Wiley, 1975).

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 2000).

“Recommended practice for tabulating spectral data for use in colour computations,” CIE 167:2005 (CIE, 2005).

M. D. Fairchild, Color Appearance Models, 2nd ed. (Wiley, 2005).

“Colorimetry,” CIE 15.3:2004 (CIE, 2004).

J. von Kries, Chromatic Adaptation, Festschrift der Albercht-Ludwig-Universität (Fribourg, 1902).

P. Green, Color Management: Understanding and Using ICC Profiles, 1st ed. (Wiley, 2010).

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Figures (15)

Fig. 1.
Fig. 1.

Flowchart for computing an optimal color.

Fig. 2.
Fig. 2.

Two types of spectral reflectance (transmittance) for optimal colors.

Fig. 3.
Fig. 3.

Concatenation of three copies of color-matching functions and illuminant spectrum (top). An optimal color has spectral reflectance R(l) with central wavelength n and half bandwidth hn on the concatenated wavelength scale l (center and bottom).

Fig. 4.
Fig. 4.

Diagram of trapezoidal integration of T(l).

Fig. 5.
Fig. 5.

Diagram of square-packing method. The solid line is a true locus; the dotted and solid squares are squares with different sampling sites.

Fig. 6.
Fig. 6.

Volume of optimal color solids in the CIELAB unit cube (top) and the number of CIE94 color differences (bottom) calculated with and without the von Kries-type CATs (CAT02, HPE, and BFD) embedded in front of CIELAB as a function of the CCT of the light sources (dotted curves, blackbody radiator; solid curves, daylight illuminant).

Fig. 7.
Fig. 7.

Volume of optimal color solids in the CIECAM02 lightness–chroma (top) and lightness–colorfulness (bottom) spaces as a function of the CCT of the light sources with the illuminance of the reference white of 200, 1000, 10,000, and 100,000 lx (dotted curves, blackbody radiator; solid curves, daylight illuminant).

Fig. 8.
Fig. 8.

Volume of optimal color solids in the CIECAM02 lightness–colorfulness space as a function of the CCT of the light sources with the illuminance of the reference white of 1571 lx (dotted curve, blackbody radiator; solid curve, daylight illuminant; squares, illuminant series F and illuminant E; crosses, estimate of Martínez-Verdú et al. [20]).

Fig. 9.
Fig. 9.

Volume of optimal color solids in the CAM02-UCS space as a function of the CCT of the light sources with the illuminance of the reference white of 200, 1000, 10,000, and 100,000 lx (dotted curves, blackbody radiator; solid curves, daylight illuminant).

Fig. 10.
Fig. 10.

Top view of the loci of optimal color solids in the CIELAB color space computed with and without the von Kries-type CATs (CAT02, HPE, and BFD) and in the CIECAM02 lightness–chroma space (J,aC,bC) under blackbody radiators at 2000 and 4000 K and daylight illuminants at 6500 and 10,000 K (1000 lx) for lightness values from 5.5 to 95.5 at intervals of 5 (instead of intervals of 1 to avoid printing too many lines).

Fig. 11.
Fig. 11.

Hue angular volume of optimal color solids in the CIELAB color space computed with and without the von Kries-type CATs (CAT02, HPE, and BFD) and in the CIECAM02 lightness–chroma space (J,aC,bC) under blackbody radiators at 2000 and 4000 K and daylight illuminants at 6500 and 10,000 K (1000 lx).

Fig. 12.
Fig. 12.

Chromaticity distributions of the Munsell matte colors in the CIELAB color space computed with and without the von Kries-type CATs (CAT02, HPE, and BFD) and in the CIECAM02 lightness–chroma space (J,aC,bC) under blackbody radiators at 2000 and 4000 K and daylight illuminants at 6500 and 10,000 K (1000 lx). The marker color represents the Munsell matte colors simulated under illuminant D65.

Fig. 13.
Fig. 13.

Convex-hull volume of Munsell matte colors in the CIELAB color space calculated with and without the von Kries-type CATs (CAT02, HPE, and BFD) and in the CIECAM02 lightness–chroma space (J,aC,bC) as a function of the CCT of the light sources (1000 lx) (dotted curves, blackbody radiator; solid curves, daylight illuminant).

Fig. 14.
Fig. 14.

Chromaticity distributions of the Munsell matte colors in the CIECAM02 lightness–chroma space (J,aC,bC) under blackbody radiators at 2000 and 4000 K and daylight illuminants at 6500 and 10,000 K (1000 lx) with D factors of 1 (complete adaptation), 0.8, and 0.6 (practical lower limit of incomplete adaptation). The marker color represents the Munsell matte colors simulated under illuminant D65.

Fig. 15.
Fig. 15.

Convex-hull volume of Munsell matte colors in the CIECAM02 lightness–chroma space (J,aC,bC) with D factors of 1 (complete adaptation), 0.8, and 0.6 (practical lower limit of incomplete adaptation) as a function of the CCT of the light sources (1000 lx) (dotted curves, blackbody radiator; solid curves, daylight illuminant).

Tables (1)

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Table 1. History of the Estimates of the Number of Discernible Colors

Equations (27)

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Rn(l)={1,lcut-onllcut-off0,otherwise,
Xn=R(l)Tx(l)dl=lcut-onlcut-offTx(l)dl=(Tx(nhn1)·{hn}+Tx(nhn)·(2{hn}))·{hn}/2+(Tx(n+hn+1)·{hn}+Tx(n+hn)·(2{hn}))·{hn}/2+k=nhnn+hnTx(k)Tx(nhn)/2Tx(n+hn)/2.
[XdYdZd]=M1[Lw,d/Lw,s000Mw,d/Mw,s000Sw,d/Sw,s]M[XsYsZs],
MCAT02=[0.73280.42960.16240.70361.69750.00610.00300.01360.9834],
MHPE=[0.389710.688980.078680.229811.183400.046410.000000.000001.00000],
MBFD=[0.89510.26640.16140.75021.71350.03670.03890.06851.0296].
LA=nEw/π,
Lc=(100D/Ls,w+1D)Ls,
Mc=(100D/Ms,w+1D)Ms,
Sc=(100D/Ss,w+1D)Ss,
D=F(1(1/3.6)e(LA+42)/92).
La=L/|L|·400(FL|L|/100)0.4227.13+L/|L|·400(FL|L|/100)0.42+0.1,
FL=0.2k4(5LA)+0.1(1k4)2(5LA)1/3,
k=1/(5LA+1).
M=C·FL1/4.
J=1.7J/(1+0.007J),
M=(1/0.0028)ln(1+0.0028M).
ΔE94*=(ΔL*kLSL)2+(ΔCab*kCSC)2+(ΔHab*kHSH)2,
ΔHab*=ΔEab*2ΔL*2ΔCab*2,
SC=1+k1Cab*(Cab*+ΔCab*),
SH=1+k2Cab*(Cab*+ΔCab*),
ΔE94*=ΔL*2+ΔC94*2+ΔH94*2.
ΔC94*=ΔCab*1+k1(Cab*+ΔCab*)Cab*.
dC94*dCab*=limΔCab*0ΔC94*ΔCab*=11+k1Cab*.
ΔHab*=2Cab*tan(Δhab/2)Cab*Δhab.
ΔH94*Cab*Δhab1+k2Cab*.
ΔA94*=0Cab*ΔH94dC94*0Cab*Cab*Δhab1+k2Cab*dCab*1+k1Cab*=Δhab·k1ln(k2Cab*+1)k2ln(k1Cab*+1)k1k2(k1k2).

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