Abstract

A model for brightness and hue perception of self-luminous stimuli surrounded by a self-luminous achromatic background has been developed based on a series of visual experiments. In the model, only the absolute spectral radiance values of the stimulus and background are used as input. Normalized cone excitations are calculated using the 10° Commission Internationale de l’Éclairage (CIE) 2006 cone fundamentals. A von Kries chromatic adaptation transform applied in the CIE 2006 cone space is adopted, and luminance compression and adaptation due to the self-luminous background are included by using a Michaelis–Menten function. Model parameters are determined by fitting the model to the experimental visual data obtained for brightness, hue, and the amount of color versus neutral. The model is validated with additional experimental data. An absolute brightness scale expressed in “bright” is proposed.

© 2018 Optical Society of America

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References

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2018 (1)

S. Hermans, K. A. G. Smet, and P. Hanselaer, “Brightness model for neutral self-luminous stimuli and backgrounds,” LEUKOS 14, 231–244 (2018).
[Crossref]

2017 (2)

2015 (2)

M. Withouck, K. A. G. Smet, W. R. Ryckaert, and P. Hanselaer, “Experimental driven modelling of the color appearance of unrelated self-luminous stimuli: CAM15u,” Opt. Express 23, 12045–12065 (2015).
[Crossref]

D. Xing, A. Ouni, S. Chen, H. Sahmoud, J. Gordon, and R. Shapley, “Brightness-color interactions in human early visual cortex,” J. Neurosci. 35, 2226–2232 (2015).
[Crossref]

2014 (1)

2013 (1)

2012 (1)

C. Fu, C. Li, G. Cui, M. R. Luo, R. W. G. Hunt, and M. R. Pointer, “An investigation of colour appearance for unrelated colours under photopic and mesopic vision,” Color Res. Appl. 37, 238–254 (2012).
[Crossref]

2009 (1)

M. H. Kim, T. Weyrich, and J. Kautz, “Modeling human color perception under extended luminance levels,” ACM Trans. Graph. 28, 27 (2009).
[Crossref]

2007 (1)

2002 (1)

S. L. Guth, “ATD01 model for color appearances, color differences, and chromatic adaptation,” Proc. SPIE 4421, 303–307 (2002).
[Crossref]

2000 (1)

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

1999 (1)

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

1997 (1)

Y. Nayatani, “Simple estimation methods for the Helmholtz–Kohlrausch effect,” Color Res. Appl. 22, 385–401 (1997).
[Crossref]

1994 (1)

H.-W. Bodmann and M. La Toison, “Predicted brightness-luminance phenomena,” Light. Res. Technol. 26, 135–143 (1994).
[Crossref]

1993 (2)

L. E. Arend and B. Spehar, “Lightness, brightness, and brightness contrast: 2. Reflectance variation,” Percept. Psychophys. 54, 457–468 (1993).
[Crossref]

R. Carter, “Gray scale and achromatic color difference,” J. Opt. Soc. Am. A 10, 1380–1391 (1993).
[Crossref]

1983 (1)

J. M. Valeton and D. van Norren, “Light adaptation of primate cones: an analysis based on extracellular data,” Vision Res. 23, 1539–1547 (1983).
[Crossref]

1969 (1)

P. Whittle and P. D. C. Challands, “The effect of background luminance on the brightness of flashes,” Vision Res. 9, 1095–1110 (1969).
[Crossref]

1967 (1)

1959 (1)

1957 (1)

L. M. Hurvich and D. Jameson, “An opponent-process theory of color vision,” Psychol. Rev. 64, 384–404 (1957).
[Crossref]

1949 (1)

R. M. Hanes, “The construction of subjective brightness scales from fractionation data: a validation,” J. Exp. Psychol. 39, 719–728 (1949).
[Crossref]

1913 (1)

V. L. Michaelis, M. L. Maud Menten, R. S. Goody, and K. A. Johnson, “Die Kinetik der Invertinwirkung,” Biochem. Z. 49, 333–369 (1913) [Bul. Math. Biophys. 13(4), 303 (1951)].

1902 (1)

J. von Kries, “Theoretische Studien über die Umstimmung des Sehorgans,” Festschr. Albrecht-Ludwigs-Univ. Freiburg 32, 143–158 (1902).

Arend, L. E.

L. E. Arend and B. Spehar, “Lightness, brightness, and brightness contrast: 2. Reflectance variation,” Percept. Psychophys. 54, 457–468 (1993).
[Crossref]

Bartleson, C. J.

Bodmann, H.-W.

H.-W. Bodmann and M. La Toison, “Predicted brightness-luminance phenomena,” Light. Res. Technol. 26, 135–143 (1994).
[Crossref]

Breneman, E. J.

Carter, R.

Challands, P. D. C.

P. Whittle and P. D. C. Challands, “The effect of background luminance on the brightness of flashes,” Vision Res. 9, 1095–1110 (1969).
[Crossref]

Chen, S.

D. Xing, A. Ouni, S. Chen, H. Sahmoud, J. Gordon, and R. Shapley, “Brightness-color interactions in human early visual cortex,” J. Neurosci. 35, 2226–2232 (2015).
[Crossref]

Cui, G.

C. Fu, C. Li, G. Cui, M. R. Luo, R. W. G. Hunt, and M. R. Pointer, “An investigation of colour appearance for unrelated colours under photopic and mesopic vision,” Color Res. Appl. 37, 238–254 (2012).
[Crossref]

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]

C. Li, Z. Li, Z. Wang, Y. Xu, M. R. Luo, G. Cui, M. Melgosa, and M. Pointer, “A revision of CIECAM02 and its CAT and UCS,” in Color Imaging Conference (2016), Vol. 1, pp. 208–212.

Deconinck, G.

Fach, C.

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

Fairchild, M. D.

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

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in Color Imaging Conference (2002), pp. 23–27.

Fechner, G.

G. Fechner, Elemente der Psychophysik, 1st ed. (Breitkopf und Härtel, 1860), Vol. 1.

Fernandez-Maloigne, C.

C. Fernandez-Maloigne, Advanced Color Image Processing and Analysis (Springer International Publishing, 2013).

Fu, C.

C. Fu, C. Li, G. Cui, M. R. Luo, R. W. G. Hunt, and M. R. Pointer, “An investigation of colour appearance for unrelated colours under photopic and mesopic vision,” Color Res. Appl. 37, 238–254 (2012).
[Crossref]

García, P. A.

Goody, R. S.

V. L. Michaelis, M. L. Maud Menten, R. S. Goody, and K. A. Johnson, “Die Kinetik der Invertinwirkung,” Biochem. Z. 49, 333–369 (1913) [Bul. Math. Biophys. 13(4), 303 (1951)].

Gordon, J.

D. Xing, A. Ouni, S. Chen, H. Sahmoud, J. Gordon, and R. Shapley, “Brightness-color interactions in human early visual cortex,” J. Neurosci. 35, 2226–2232 (2015).
[Crossref]

Guth, S. L.

S. L. Guth, “ATD01 model for color appearances, color differences, and chromatic adaptation,” Proc. SPIE 4421, 303–307 (2002).
[Crossref]

Hanes, R. M.

R. M. Hanes, “The construction of subjective brightness scales from fractionation data: a validation,” J. Exp. Psychol. 39, 719–728 (1949).
[Crossref]

Hanselaer, P.

Hermans, S.

S. Hermans, K. A. G. Smet, and P. Hanselaer, “Brightness model for neutral self-luminous stimuli and backgrounds,” LEUKOS 14, 231–244 (2018).
[Crossref]

Huertas, R.

Hunt, R. W. G.

C. Fu, C. Li, G. Cui, M. R. Luo, R. W. G. Hunt, and M. R. Pointer, “An investigation of colour appearance for unrelated colours under photopic and mesopic vision,” Color Res. Appl. 37, 238–254 (2012).
[Crossref]

R. W. G. Hunt and M. Pointer, Measuring Colour, 4th ed. (Wiley, 2011).

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in Color Imaging Conference (2002), pp. 23–27.

Hurvich, L. M.

Jameson, D.

Johnson, K. A.

V. L. Michaelis, M. L. Maud Menten, R. S. Goody, and K. A. Johnson, “Die Kinetik der Invertinwirkung,” Biochem. Z. 49, 333–369 (1913) [Bul. Math. Biophys. 13(4), 303 (1951)].

Kautz, J.

M. H. Kim, T. Weyrich, and J. Kautz, “Modeling human color perception under extended luminance levels,” ACM Trans. Graph. 28, 27 (2009).
[Crossref]

Kim, M. H.

M. H. Kim, T. Weyrich, and J. Kautz, “Modeling human color perception under extended luminance levels,” ACM Trans. Graph. 28, 27 (2009).
[Crossref]

Kingdom, F.

F. Kingdom and N. Prins, Psychophysics: A Practical Introduction, 2nd ed. (Elsevier, 2013).

Koenderink, J.

La Toison, M.

H.-W. Bodmann and M. La Toison, “Predicted brightness-luminance phenomena,” Light. Res. Technol. 26, 135–143 (1994).
[Crossref]

Li, C.

C. Fu, C. Li, G. Cui, M. R. Luo, R. W. G. Hunt, and M. R. Pointer, “An investigation of colour appearance for unrelated colours under photopic and mesopic vision,” Color Res. Appl. 37, 238–254 (2012).
[Crossref]

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in Color Imaging Conference (2002), pp. 23–27.

C. Li, Z. Li, Z. Wang, Y. Xu, M. R. Luo, G. Cui, M. Melgosa, and M. Pointer, “A revision of CIECAM02 and its CAT and UCS,” in Color Imaging Conference (2016), Vol. 1, pp. 208–212.

M. R. Luo and C. Li, “CIECAM02 and its recent developments,” in Advanced Color Image Processing and Analysis (Springer, 2013).

Li, Z.

C. Li, Z. Li, Z. Wang, Y. Xu, M. R. Luo, G. Cui, M. Melgosa, and M. Pointer, “A revision of CIECAM02 and its CAT and UCS,” in Color Imaging Conference (2016), Vol. 1, pp. 208–212.

Luo, M. R.

K. A. G. Smet, Q. Zhai, M. R. Luo, and P. Hanselaer, “Study of chromatic adaptation using memory color matches, Part I: neutral illuminants,” Opt. Express 25, 7732–7748 (2017).
[Crossref]

K. A. Smet, Q. Zhai, M. R. Luo, and P. Hanselaer, “Study of chromatic adaptation using memory color matches, Part II: colored illuminants,” Opt. Express 25, 8350–8365 (2017).
[Crossref]

C. Fu, C. Li, G. Cui, M. R. Luo, R. W. G. Hunt, and M. R. Pointer, “An investigation of colour appearance for unrelated colours under photopic and mesopic vision,” Color Res. Appl. 37, 238–254 (2012).
[Crossref]

C. Li, Z. Li, Z. Wang, Y. Xu, M. R. Luo, G. Cui, M. Melgosa, and M. Pointer, “A revision of CIECAM02 and its CAT and UCS,” in Color Imaging Conference (2016), Vol. 1, pp. 208–212.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in Color Imaging Conference (2002), pp. 23–27.

M. R. Luo and C. Li, “CIECAM02 and its recent developments,” in Advanced Color Image Processing and Analysis (Springer, 2013).

Marks, L. E.

S. S. Stevens, G. Stevens, and L. E. Marks, Psychophysics: Introduction to Its Perceptual, Neural, and Social Prospects (Wiley, 1986).

Maud Menten, M. L.

V. L. Michaelis, M. L. Maud Menten, R. S. Goody, and K. A. Johnson, “Die Kinetik der Invertinwirkung,” Biochem. Z. 49, 333–369 (1913) [Bul. Math. Biophys. 13(4), 303 (1951)].

Melgosa, M.

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]

C. Li, Z. Li, Z. Wang, Y. Xu, M. R. Luo, G. Cui, M. Melgosa, and M. Pointer, “A revision of CIECAM02 and its CAT and UCS,” in Color Imaging Conference (2016), Vol. 1, pp. 208–212.

Michaelis, V. L.

V. L. Michaelis, M. L. Maud Menten, R. S. Goody, and K. A. Johnson, “Die Kinetik der Invertinwirkung,” Biochem. Z. 49, 333–369 (1913) [Bul. Math. Biophys. 13(4), 303 (1951)].

Moroney, N.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in Color Imaging Conference (2002), pp. 23–27.

Nayatani, Y.

Y. Nayatani, “Simple estimation methods for the Helmholtz–Kohlrausch effect,” Color Res. Appl. 22, 385–401 (1997).
[Crossref]

Newman, T.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in Color Imaging Conference (2002), pp. 23–27.

Ouni, A.

D. Xing, A. Ouni, S. Chen, H. Sahmoud, J. Gordon, and R. Shapley, “Brightness-color interactions in human early visual cortex,” J. Neurosci. 35, 2226–2232 (2015).
[Crossref]

Pointer, M.

R. W. G. Hunt and M. Pointer, Measuring Colour, 4th ed. (Wiley, 2011).

C. Li, Z. Li, Z. Wang, Y. Xu, M. R. Luo, G. Cui, M. Melgosa, and M. Pointer, “A revision of CIECAM02 and its CAT and UCS,” in Color Imaging Conference (2016), Vol. 1, pp. 208–212.

Pointer, M. R.

M. Withouck, K. A. G. Smet, W. R. Ryckaert, M. R. Pointer, G. Deconinck, J. Koenderink, and P. Hanselaer, “Brightness perception of unrelated self-luminous colors,” J. Opt. Soc. Am. A 30, 1248–1255 (2013).
[Crossref]

C. Fu, C. Li, G. Cui, M. R. Luo, R. W. G. Hunt, and M. R. Pointer, “An investigation of colour appearance for unrelated colours under photopic and mesopic vision,” Color Res. Appl. 37, 238–254 (2012).
[Crossref]

Prins, N.

F. Kingdom and N. Prins, Psychophysics: A Practical Introduction, 2nd ed. (Elsevier, 2013).

Ryckaert, W. R.

Sahmoud, H.

D. Xing, A. Ouni, S. Chen, H. Sahmoud, J. Gordon, and R. Shapley, “Brightness-color interactions in human early visual cortex,” J. Neurosci. 35, 2226–2232 (2015).
[Crossref]

Shapley, R.

D. Xing, A. Ouni, S. Chen, H. Sahmoud, J. Gordon, and R. Shapley, “Brightness-color interactions in human early visual cortex,” J. Neurosci. 35, 2226–2232 (2015).
[Crossref]

Sharpe, L. T.

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

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

Smet, K. A.

Smet, K. A. G.

Spehar, B.

L. E. Arend and B. Spehar, “Lightness, brightness, and brightness contrast: 2. Reflectance variation,” Percept. Psychophys. 54, 457–468 (1993).
[Crossref]

Stevens, G.

S. S. Stevens, G. Stevens, and L. E. Marks, Psychophysics: Introduction to Its Perceptual, Neural, and Social Prospects (Wiley, 1986).

Stevens, S. S.

S. S. Stevens, G. Stevens, and L. E. Marks, Psychophysics: Introduction to Its Perceptual, Neural, and Social Prospects (Wiley, 1986).

S. S. Stevens, “The psychophysics of sensory function,” in Sensory Communication (Wiley, 1962), pp. 934–940.

Stockman, A.

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

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

Valeton, J. M.

J. M. Valeton and D. van Norren, “Light adaptation of primate cones: an analysis based on extracellular data,” Vision Res. 23, 1539–1547 (1983).
[Crossref]

van Norren, D.

J. M. Valeton and D. van Norren, “Light adaptation of primate cones: an analysis based on extracellular data,” Vision Res. 23, 1539–1547 (1983).
[Crossref]

von Kries, J.

J. von Kries, “Theoretische Studien über die Umstimmung des Sehorgans,” Festschr. Albrecht-Ludwigs-Univ. Freiburg 32, 143–158 (1902).

Wang, Z.

C. Li, Z. Li, Z. Wang, Y. Xu, M. R. Luo, G. Cui, M. Melgosa, and M. Pointer, “A revision of CIECAM02 and its CAT and UCS,” in Color Imaging Conference (2016), Vol. 1, pp. 208–212.

Weyrich, T.

M. H. Kim, T. Weyrich, and J. Kautz, “Modeling human color perception under extended luminance levels,” ACM Trans. Graph. 28, 27 (2009).
[Crossref]

Whittle, P.

P. Whittle and P. D. C. Challands, “The effect of background luminance on the brightness of flashes,” Vision Res. 9, 1095–1110 (1969).
[Crossref]

Withouck, M.

Xing, D.

D. Xing, A. Ouni, S. Chen, H. Sahmoud, J. Gordon, and R. Shapley, “Brightness-color interactions in human early visual cortex,” J. Neurosci. 35, 2226–2232 (2015).
[Crossref]

Xu, Y.

C. Li, Z. Li, Z. Wang, Y. Xu, M. R. Luo, G. Cui, M. Melgosa, and M. Pointer, “A revision of CIECAM02 and its CAT and UCS,” in Color Imaging Conference (2016), Vol. 1, pp. 208–212.

Zhai, Q.

ACM Trans. Graph. (1)

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Figures (11)

Fig. 1.
Fig. 1. (left) Experimental room and (right) the central colored stimulus and part of the neutral low luminous background.
Fig. 2.
Fig. 2. Chromaticity coordinates of all stimuli plotted in the CIE 1976 u 10 , v 10 chromaticity diagram. (+) Saturated red, green, and blue; (triangle) less saturated red, green, and blue; and (circle) white stimuli (see Figs. 5 and 6).
Fig. 3.
Fig. 3. Graphical observer response sheet.
Fig. 4.
Fig. 4. Rescaled brightness values of the average observer as a function of the predicted brightness value of the achromatic brightness model together with all proportionality factors a ; R 2 and STRESS values for each background independently. Error bars are standard errors and boxplots are included for L b = 0    cd / m 2 .
Fig. 5.
Fig. 5. Brightness perception of the average observer as a function of the luminance level of the stimulus ( L B = 50    cd / m 2 ). (+) Most saturated red, green, and blue; (triangle) less saturated red, green, and blue; and (circle) white stimuli (see Fig. 2).
Fig. 6.
Fig. 6. Brightness perception as a function of the background luminance for all three different luminance levels of stimuli. (+) Saturated red, green, and blue; (triangle) less saturated red, green, and blue; and (circle) white stimuli (see Fig. 2). Full lines indicate the predicted brightness values by CAM18sl for the most saturated and white stimuli.
Fig. 7.
Fig. 7. Amount of neutral of the average observer as a function of the CIE1976 u 10 , v 10 saturation. Background luminance level, L B = 50    cd / m 2 . Error bars are standard errors.
Fig. 8.
Fig. 8. Hue quadrature of the average observer as a function of the predicted hue quadrature by the model. Error bars are standard errors.
Fig. 9.
Fig. 9. Brightness perception of the average observer as a function of the predicted brightness value by CAM18sl. Error bars are standard errors and RMSE, R 2 , and STRESS values for each background separately.
Fig. 10.
Fig. 10. Amount of neutral for the average observer as a function of the CAM18sl saturation, S.
Fig. 11.
Fig. 11. Average observer amount of white against the predicted amount of white of the CAM18sl model.

Tables (2)

Tables Icon

Table 1. Unique Hue Data for Calculating the Unique Hue Quadrature ( H i )

Tables Icon

Table 2. R 2 , RMSE, and Average STRESS Values of the Brightness, Hue Quadrature, and Amount of Neutral Perception of the Test Experiment and of the Validation Experiment

Equations (20)

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STRESS = 1 n i = 1 n j = 1 k ( A i , j f B i , j ) 2 j = 1 k ( f B i , j ) 2 , with    f = j = 1 k ( A i , j ) 2 j = 1 k A i , j B i , j .
ρ = 676.7 390 830 L e , λ ( λ ) l ¯ 10 ( λ ) d λ , γ = 794.0 390 830 L e , λ ( λ ) m ¯ 10 ( λ ) d λ , β = 1461.5 390 830 L e , λ ( λ ) s ¯ 10 ( λ ) d λ .
[ ρ c γ c β c ] = [ ρ w r / ρ B 0 0 0 γ w r / γ B 0 0 0 β w r / β B ] [ ρ γ β ] .
ρ c , a = ρ c 0.58 ρ c 0.58 + ( 291.20 + 71.8 α w r 0.78 ) 0.58 , γ c , a = γ c 0.58 γ c 0.58 + ( 291.20 + 71.8 α w r 0.78 ) 0.58 , β c , a = β c 0.58 β c 0.58 + ( 291.20 + 71.8 α w r 0.78 ) 0.58 .
a = c a ( ρ c , a 12 11 γ c , a + β c , a 11 ) , b = c b ( ρ c , a + γ c , a 2 β c , a ) .
h = 180 π tan 1 ( b / a ) .
H = H i + 100 h h i h i + 1 h i ,
M = c M ( a 2 + b 2 ) .
A = ( 2 ρ c , a + γ c , a + 1 20 β c , a ) , Q = c A ( A + c 1 M c HK 2 ) .
S = M Q .
W = 1 1 + 2.29 S 2.09 .
Q = 0.937 ( ( 2 ρ c , a + γ c , a + 1 20 β c , a ) + 0.0024 M 1.09 ) .
ρ = 676.7 390 830 L e , λ ( λ ) l ¯ 10 ( λ ) d λ , γ = 794.0 390 830 L e , λ ( λ ) m ¯ 10 ( λ ) d λ , β = 1461.5 390 830 L e , λ ( λ ) s ¯ 10 ( λ ) d λ .
[ ρ c γ c β c ] = [ ρ w r / ρ B 0 0 0 γ w r / γ B 0 0 0 β w r / β B ] [ ρ γ β ] ,
ρ c , a = ρ c 0.58 ρ c 0.58 + ( 291.20 + 71.8 α w r 0.78 ) 0.58 , γ c , a = γ c 0.58 γ c 0.58 + ( 291.20 + 71.8 α w r 0.78 ) 0.58 , β c , a = β c 0.58 β c 0.58 + ( 291.20 + 71.8 α w r 0.78 ) 0.58 ,
a = 0.63 ( ρ c , a 12 11 γ c , a + β c , a 11 ) , b = 0.12 ( ρ c , a + γ c , a 2 β c , a ) .
M = 3260 ( a 2 + b 2 ) ;
Q = 0.937 ( ( 2 ρ c , a + γ c , a + 1 20 β c , a ) + 0.0024 M 1.09 ) ;
S = M Q ;
W = 1 1 + 2.29 S 2.09 .

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