Abstract

In a previous article [J. Opt. Soc. Am. A 24, 2196 (2007)] we have modeled skin color using the radiative transfer equation, solved by the auxiliary function method. Three main parameters have been determined as being predominant in the diversity of skin color: the concentrations of melanosomes and of red blood cells and the oxygen saturation of blood. From the reflectance spectrum measured on real Caucasian skin, these parameters are now evaluated by minimizing the standard deviation on the adjusted wavelength range between the experimental spectrum and simulated spectra gathered in a database.

© 2008 Optical Society of America

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References

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  1. M. Shimada, Y. Yamada, M. Itoh, and T. Yatagai, “Melanin and blood concentration in human skin studied by multiple regression analysis: Experiments,” Phys. Med. Biol. 46, 2385-2395 (2001).
    [CrossRef] [PubMed]
  2. G. N. Stamatas, B. Z. Zmudzka, N. Kollias, and J. Z. Beer, “Non-invasive measurements of skin pigmentation in situ,” Pigment Cell Res. 17, 618-626 (2004).
    [CrossRef] [PubMed]
  3. T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
    [CrossRef] [PubMed]
  4. C. Magnain, M. Elias, and J. M. Frigerio, “Skin color modeling using the radiative transfer equation solved by the auxiliary function method,” J. Opt. Soc. Am. A 24, 2196-2205 (2007).
    [CrossRef]
  5. L. A. Goldsmith, Physiology, Biochemistry, and Molecular Biology of the Skin (Oxford U. Press, 1998).
  6. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  7. M. Elias and G. Elias, “New and fast calculation for incoherent multiple scattering,” J. Opt. Soc. Am. A 19, 894-901 (2002).
    [CrossRef]
  8. K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region,” Photochem. Photobiol. 80, 450-455 (2004).
    [PubMed]
  9. E. Angelopoulou, “The reflectance spectrum of human skin,” Tech. Rep. No. MS-CIS-99-29, Department of Computer and Information Science, University of Pennsylvania (1999), pp. 1-14.
  10. S. L. Jacques, http://omlc.ogi.edu/news/jan98/skinoptics.html (1998).
  11. R. R. Anderson and J. A. Parrish, “Optical properties of human skin,” in The Science of Photomedecine (Plenum, 1982), pp. 147-194.
  12. D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102, 1-4 (2004).
    [CrossRef]
  13. J. M. Steinke and A. P. Sheperd, “Comparison of Mie theory and the light scattering of red blood cells,” Appl. Opt. 27, 4027-4033 (1988).
    [CrossRef] [PubMed]
  14. G. Poirier, “Human skin modelling and rendering,” Master's thesis (University of Waterloo, 2004).
  15. S. L. Jacques, http://omlc.ogi.edu/spectra/hemoglobin/index.html (1999).
  16. M. Elias and G. Elias, “Radiative transfer in inhomogeneous stratified scattering media with use of the auxiliary function method,” J. Opt. Soc. Am. A 21, 580-589 (2004).
    [CrossRef]
  17. P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485-1502 (1971).
    [CrossRef] [PubMed]
  18. K. Stamnes, S. Chee Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layer media,” Appl. Opt. 27, 2502-2510 (1988).
    [CrossRef] [PubMed]
  19. M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
    [CrossRef] [PubMed]
  20. C. F. Bohren and D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1983).
  21. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

2007

2005

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

2004

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102, 1-4 (2004).
[CrossRef]

G. Poirier, “Human skin modelling and rendering,” Master's thesis (University of Waterloo, 2004).

M. Elias and G. Elias, “Radiative transfer in inhomogeneous stratified scattering media with use of the auxiliary function method,” J. Opt. Soc. Am. A 21, 580-589 (2004).
[CrossRef]

G. N. Stamatas, B. Z. Zmudzka, N. Kollias, and J. Z. Beer, “Non-invasive measurements of skin pigmentation in situ,” Pigment Cell Res. 17, 618-626 (2004).
[CrossRef] [PubMed]

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region,” Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

2002

2001

M. Shimada, Y. Yamada, M. Itoh, and T. Yatagai, “Melanin and blood concentration in human skin studied by multiple regression analysis: Experiments,” Phys. Med. Biol. 46, 2385-2395 (2001).
[CrossRef] [PubMed]

1999

E. Angelopoulou, “The reflectance spectrum of human skin,” Tech. Rep. No. MS-CIS-99-29, Department of Computer and Information Science, University of Pennsylvania (1999), pp. 1-14.

S. L. Jacques, http://omlc.ogi.edu/spectra/hemoglobin/index.html (1999).

1998

S. L. Jacques, http://omlc.ogi.edu/news/jan98/skinoptics.html (1998).

L. A. Goldsmith, Physiology, Biochemistry, and Molecular Biology of the Skin (Oxford U. Press, 1998).

1989

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

1988

1983

C. F. Bohren and D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1983).

1982

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

R. R. Anderson and J. A. Parrish, “Optical properties of human skin,” in The Science of Photomedecine (Plenum, 1982), pp. 147-194.

1971

1960

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Aalders, M. C. G.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102, 1-4 (2004).
[CrossRef]

Anderson, R. R.

R. R. Anderson and J. A. Parrish, “Optical properties of human skin,” in The Science of Photomedecine (Plenum, 1982), pp. 147-194.

Angelopoulou, E.

E. Angelopoulou, “The reflectance spectrum of human skin,” Tech. Rep. No. MS-CIS-99-29, Department of Computer and Information Science, University of Pennsylvania (1999), pp. 1-14.

Batzer, J.

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

Beer, J. Z.

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

G. N. Stamatas, B. Z. Zmudzka, N. Kollias, and J. Z. Beer, “Non-invasive measurements of skin pigmentation in situ,” Pigment Cell Res. 17, 618-626 (2004).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1983).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Chee Tsay, S.

Coelho, S. G.

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

Elias, G.

Elias, M.

Faber, D. J.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102, 1-4 (2004).
[CrossRef]

Frigerio, J. M.

Goldsmith, L. A.

L. A. Goldsmith, Physiology, Biochemistry, and Molecular Biology of the Skin (Oxford U. Press, 1998).

Hearing, V. J.

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

Hoffman, D. R.

C. F. Bohren and D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1983).

Hooper, B. A.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102, 1-4 (2004).
[CrossRef]

Itoh, M.

M. Shimada, Y. Yamada, M. Itoh, and T. Yatagai, “Melanin and blood concentration in human skin studied by multiple regression analysis: Experiments,” Phys. Med. Biol. 46, 2385-2395 (2001).
[CrossRef] [PubMed]

Jacques, S. L.

S. L. Jacques, http://omlc.ogi.edu/spectra/hemoglobin/index.html (1999).

S. L. Jacques, http://omlc.ogi.edu/news/jan98/skinoptics.html (1998).

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

Jayaweera, K.

Juzenas, P.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region,” Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Kollias, N.

G. N. Stamatas, B. Z. Zmudzka, N. Kollias, and J. Z. Beer, “Non-invasive measurements of skin pigmentation in situ,” Pigment Cell Res. 17, 618-626 (2004).
[CrossRef] [PubMed]

Magnain, C.

Mik, E. G.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102, 1-4 (2004).
[CrossRef]

Miller, S. A.

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

Moan, J.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region,” Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Mudgett, P. S.

Nielsen, K. P.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region,” Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Parrish, J. A.

R. R. Anderson and J. A. Parrish, “Optical properties of human skin,” in The Science of Photomedecine (Plenum, 1982), pp. 147-194.

Poirier, G.

G. Poirier, “Human skin modelling and rendering,” Master's thesis (University of Waterloo, 2004).

Richards, L. W.

Sheperd, A. P.

Shimada, M.

M. Shimada, Y. Yamada, M. Itoh, and T. Yatagai, “Melanin and blood concentration in human skin studied by multiple regression analysis: Experiments,” Phys. Med. Biol. 46, 2385-2395 (2001).
[CrossRef] [PubMed]

Stamatas, G. N.

G. N. Stamatas, B. Z. Zmudzka, N. Kollias, and J. Z. Beer, “Non-invasive measurements of skin pigmentation in situ,” Pigment Cell Res. 17, 618-626 (2004).
[CrossRef] [PubMed]

Stamnes, J. J.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region,” Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Stamnes, K.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region,” Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

K. Stamnes, S. Chee Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layer media,” Appl. Opt. 27, 2502-2510 (1988).
[CrossRef] [PubMed]

Star, W. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

Steinke, J. M.

Sterenborg, H. J. C. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

Tadokoro, T.

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

van Gemert, M. J. C.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102, 1-4 (2004).
[CrossRef]

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

van Leeuwen, T. G.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102, 1-4 (2004).
[CrossRef]

Wiscombe, W.

Wolber, R.

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

Yamada, Y.

M. Shimada, Y. Yamada, M. Itoh, and T. Yatagai, “Melanin and blood concentration in human skin studied by multiple regression analysis: Experiments,” Phys. Med. Biol. 46, 2385-2395 (2001).
[CrossRef] [PubMed]

Yamaguchi, Y.

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

Yatagai, T.

M. Shimada, Y. Yamada, M. Itoh, and T. Yatagai, “Melanin and blood concentration in human skin studied by multiple regression analysis: Experiments,” Phys. Med. Biol. 46, 2385-2395 (2001).
[CrossRef] [PubMed]

Zhao, L.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region,” Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Zmudzka, B. Z.

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

G. N. Stamatas, B. Z. Zmudzka, N. Kollias, and J. Z. Beer, “Non-invasive measurements of skin pigmentation in situ,” Pigment Cell Res. 17, 618-626 (2004).
[CrossRef] [PubMed]

Appl. Opt.

IEEE Trans. Biomed. Eng.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

J. Invest. Dermatol.

T. Tadokoro, Y. Yamaguchi, J. Batzer, S. G. Coelho, B. Z. Zmudzka, S. A. Miller, R. Wolber, J. Z. Beer, and V. J. Hearing, “Mechanisms of skin tanning in different racial/ethnic groups in response to ultraviolet radiation,” J. Invest. Dermatol. 124, 1326--1332 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Photochem. Photobiol.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region,” Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Phys. Med. Biol.

M. Shimada, Y. Yamada, M. Itoh, and T. Yatagai, “Melanin and blood concentration in human skin studied by multiple regression analysis: Experiments,” Phys. Med. Biol. 46, 2385-2395 (2001).
[CrossRef] [PubMed]

Phys. Rev. Lett.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102, 1-4 (2004).
[CrossRef]

Pigment Cell Res.

G. N. Stamatas, B. Z. Zmudzka, N. Kollias, and J. Z. Beer, “Non-invasive measurements of skin pigmentation in situ,” Pigment Cell Res. 17, 618-626 (2004).
[CrossRef] [PubMed]

Other

G. Poirier, “Human skin modelling and rendering,” Master's thesis (University of Waterloo, 2004).

S. L. Jacques, http://omlc.ogi.edu/spectra/hemoglobin/index.html (1999).

C. F. Bohren and D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1983).

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

E. Angelopoulou, “The reflectance spectrum of human skin,” Tech. Rep. No. MS-CIS-99-29, Department of Computer and Information Science, University of Pennsylvania (1999), pp. 1-14.

S. L. Jacques, http://omlc.ogi.edu/news/jan98/skinoptics.html (1998).

R. R. Anderson and J. A. Parrish, “Optical properties of human skin,” in The Science of Photomedecine (Plenum, 1982), pp. 147-194.

L. A. Goldsmith, Physiology, Biochemistry, and Molecular Biology of the Skin (Oxford U. Press, 1998).

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

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Figures (6)

Fig. 1
Fig. 1

Skin structure.

Fig. 2
Fig. 2

Scattering in a complex medium.

Fig. 3
Fig. 3

Influence of the volumic concentration of melanosomes C on the modeled reflectance spectrum of Caucasian skin.

Fig. 4
Fig. 4

Results of the inverse problem for a measurement (solid curve), the minimization of the standard deviation ϵ is realized on 380 780 nm (squares) and 380 600 (triangles).

Fig. 5
Fig. 5

Values of the oxygen saturation S obtained for 11 reflectance spectra for both methods, the minimization of the color change Δ E (solid circles) and the standard deviation ϵ on 380 600 nm (open circles). Unrealistic values for S have been shaded.

Fig. 6
Fig. 6

Four real skin reflectance spectra (solid curves) and their simulated spectra (symbols) calculated with our method.

Tables (4)

Tables Icon

Table 1 Influence of Each Parameter on the Color Change Δ E Calculated between the Two Extreme Values a

Tables Icon

Table 2 Relative Error in Percentages on the Determination of the Parameters C, C S , and S for Both Minimizations of ϵ or Δ E

Tables Icon

Table 3 Parameters Obtained by Minimizing the Standard Deviation Calculated over the Ranges 380 780 nm and 380 600 nm and the Values of ϵ for a Real Skin Spectrum

Tables Icon

Table 4 Determination of the Parameters of the 11 Measured Reflectance Spectra for Both Methods

Equations (5)

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

μ d f ( u , z ) d z = ( k + s ) f ( u , τ ) + s 4 π F ( u , z ) p ( u , u ) + s 4 π 0 4 π f ( u 1 , z ) p ( u , u 1 ) d Ω 1 ,
d f ( u , τ ) d τ = f ( u , τ ) μ + ϖ 4 π μ μ F ( u , τ ) μ p ( u , u ) + ϖ 4 π μ μ 0 4 π f ( u 1 , τ ) μ 1 p ( u , u 1 ) d Ω 1 ,
Δ E = ( L 1 * L 2 * ) 2 + ( a 1 * a 2 * ) 2 + ( b 1 * b 2 * ) 2 .
R ( λ , C , C S , S ) = a + x = C , C S , S ( b x x + c x x 2 ) + 1 2 x = C , C S , S y = C , C S , S x d x , y x . y ,
ϵ = i = 1 i max = 41 ( R unknown ( λ i ) R ref ( λ i ) ) 2 i max 1 ,

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