Abstract

The angle resolved reflectance factor of matte samples is measured with a goniophotometer and simulated using radiative transfer theory. Both measurements and simulations display the same characteristic dependence of the reflectance factor on the observation angle. The angle resolved reflectance spectra are translated to CIELAB color coordinates and the angular color differences are found to be surprisingly large. A chromatic adaptation that is dependent on the observation angle is suggested, in which a nonabsorbing opaque medium is used as the reference white, and the angular color differences are then reduced. Furthermore, the use of an undyed paper as the reference white is evaluated. The angular lightness differences are then reduced further, but the angular differences in chroma are still large. It is suggested that smaller variations in perceived color could be explained by angle dependent chromatic adaptation and a limited sensitivity of the human visual system to changes in chroma.

© 2011 Optical Society of America

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References

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  1. G. Wyszecki and W. S. Stiles, Color Science (Wiley, 2000).
  2. R. W. G. Hunt, Measuring Colour (Fountain, 1998).
  3. International Organization for Standardization, “Paper, board and pulps—measurement of diffuse reflectance factor,” ISO 2469 (International Organization for Standardization, 1994).
  4. Deutsches Institut für Normung e. V., “Colorimetry; spectrophotometric method,” DIN 5033-4 (Deutsches Institut für Normung e. V., 1992).
  5. F. W. Billmeyer Jr. and R. T. Marcus, “Effect of illuminating and viewing geometry on the color coordinates of samples with various surface textures,” Appl. Opt. 8, 763–768 (1969).
    [CrossRef] [PubMed]
  6. E. N. Dalal and K. M. Natale-Hoffman, “The effect of gloss on color,” Color Res. Appl. 24, 369–376 (1999).
    [CrossRef]
  7. W. Ji, M. R. Pointer, R. M. Luo, and J. Dakin, “Gloss as an aspect of the measurement of appearance,” J. Opt. Soc. Am. A 23, 22–33 (2006).
    [CrossRef]
  8. P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 11a, 593–601 (1931).
  9. M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. theory,” J. Opt. Soc. Am. A 27, 1032–1039 (2010).
    [CrossRef]
  10. M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. measurements,” J. Opt. Soc. Am. A 27, 1040–1045 (2010).
    [CrossRef]
  11. M. Mikula, M. Ceppan, and K. Vasko, “Gloss and goniocolorimetry of printed materials,” Color Res. Appl. 28, 335–342(2003).
    [CrossRef]
  12. W. M. Chirdon, W. J. O’Brien, and R. E. Robertson, “Mechanisms of goniochromism relevant to restorative dentistry,” Dent. Mater. 25, 802–809 (2009).
    [CrossRef] [PubMed]
  13. C. Oleari, “Colorimetry in optical coating,” Proc. SPIE 5963, 596305 (2005).
    [CrossRef]
  14. L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
    [CrossRef]
  15. M. Neuman, P. Edström, M. Andersson, L. Coppel, and O. Norberg, “Angular variations of color in turbid media—the influence of bulk scattering on goniochromism in paper,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology , 2010) pp. 407–413.
  16. A. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of Spectralon white reflectance 23 standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt. 50, 2431–2442 (2011).
    [CrossRef] [PubMed]
  17. A. Kienle and F. Foschum, “250 years Lambert surface: does it really exist?” Opt. Express 19, 3881–3889 (2011).
    [CrossRef] [PubMed]
  18. International Organization for Standardization, “Paper—determination of light scattering and absorption coefficients (using Kubelka-Munk theory),” ISO 9416 (International Organization for Standardization, 1998).
  19. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  20. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  21. W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  22. S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32, 559–568 (1993).
    [CrossRef] [PubMed]
  23. N. Joshi, C. Donner, and H. W. Jensen, “Noninvasive measurement of scattering anisotropy in turbid materials by nonnormal incident illumination,” Opt. Lett. 31, 936–938(2006).
    [CrossRef] [PubMed]
  24. P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005).
    [CrossRef]
  25. M. Elias and G. Elias, “New and fast calculation for incoherent multiple scattering,” J. Opt. Soc. Am. A 19, 894–901 (2002).
    [CrossRef]
  26. M. I. Mishchenko, “Poynting-Stokes tensor and radiative transfer in discrete random media: the microphysical paradigm,” Opt. Express 18, 19770–19791 (2010).
    [CrossRef] [PubMed]
  27. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, 1977).
  28. P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” Inverse Probl. Sci. Eng. 16, 927–951 (2008).
    [CrossRef]
  29. M. Oren and S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
    [CrossRef]
  30. L. Simonot, “Photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009).
    [CrossRef] [PubMed]
  31. J.-S. Kim, M.-S. Cho, S. Westland, and M. R. Luo, “Image quality assessment for photographic images,” in Proceedings of AIC Colour 05—10th Congress of the International Colour Association, J.L.Nieves and JavierHernández-Andrés, eds. (AIC, 2005) pp. 1095–1098.

2011

2010

2009

W. M. Chirdon, W. J. O’Brien, and R. E. Robertson, “Mechanisms of goniochromism relevant to restorative dentistry,” Dent. Mater. 25, 802–809 (2009).
[CrossRef] [PubMed]

L. Simonot, “Photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009).
[CrossRef] [PubMed]

2008

P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” Inverse Probl. Sci. Eng. 16, 927–951 (2008).
[CrossRef]

2006

2005

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005).
[CrossRef]

C. Oleari, “Colorimetry in optical coating,” Proc. SPIE 5963, 596305 (2005).
[CrossRef]

2003

M. Mikula, M. Ceppan, and K. Vasko, “Gloss and goniocolorimetry of printed materials,” Color Res. Appl. 28, 335–342(2003).
[CrossRef]

2002

1999

E. N. Dalal and K. M. Natale-Hoffman, “The effect of gloss on color,” Color Res. Appl. 24, 369–376 (1999).
[CrossRef]

1995

M. Oren and S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

1993

1990

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1969

1941

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

1931

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 11a, 593–601 (1931).

Andersson, M.

M. Neuman, P. Edström, M. Andersson, L. Coppel, and O. Norberg, “Angular variations of color in turbid media—the influence of bulk scattering on goniochromism in paper,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology , 2010) pp. 407–413.

Bhandari, A.

Billmeyer, F. W.

Ceppan, M.

M. Mikula, M. Ceppan, and K. Vasko, “Gloss and goniocolorimetry of printed materials,” Color Res. Appl. 28, 335–342(2003).
[CrossRef]

Chandrasekhar, S.

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

Cheong, W.-F.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Chirdon, W. M.

W. M. Chirdon, W. J. O’Brien, and R. E. Robertson, “Mechanisms of goniochromism relevant to restorative dentistry,” Dent. Mater. 25, 802–809 (2009).
[CrossRef] [PubMed]

Cho, M.-S.

J.-S. Kim, M.-S. Cho, S. Westland, and M. R. Luo, “Image quality assessment for photographic images,” in Proceedings of AIC Colour 05—10th Congress of the International Colour Association, J.L.Nieves and JavierHernández-Andrés, eds. (AIC, 2005) pp. 1095–1098.

Coppel, L.

M. Neuman, P. Edström, M. Andersson, L. Coppel, and O. Norberg, “Angular variations of color in turbid media—the influence of bulk scattering on goniochromism in paper,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology , 2010) pp. 407–413.

Dakin, J.

Dalal, E. N.

E. N. Dalal and K. M. Natale-Hoffman, “The effect of gloss on color,” Color Res. Appl. 24, 369–376 (1999).
[CrossRef]

Deutsches Institut für Normung e.V.,

Deutsches Institut für Normung e. V., “Colorimetry; spectrophotometric method,” DIN 5033-4 (Deutsches Institut für Normung e. V., 1992).

Donner, C.

Dupraz, D.

L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
[CrossRef]

Edström, P.

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. theory,” J. Opt. Soc. Am. A 27, 1032–1039 (2010).
[CrossRef]

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. measurements,” J. Opt. Soc. Am. A 27, 1040–1045 (2010).
[CrossRef]

P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” Inverse Probl. Sci. Eng. 16, 927–951 (2008).
[CrossRef]

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005).
[CrossRef]

M. Neuman, P. Edström, M. Andersson, L. Coppel, and O. Norberg, “Angular variations of color in turbid media—the influence of bulk scattering on goniochromism in paper,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology , 2010) pp. 407–413.

Elias, G.

Elias, M.

Foschum, F.

Frette, Ø.

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, 1977).

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hamre, B.

Hébert, M.

L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
[CrossRef]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, 1977).

Hunt, R. W. G.

R. W. G. Hunt, Measuring Colour (Fountain, 1998).

Jensen, H. W.

Ji, W.

Joshi, N.

Kienle, A.

Kildemo, M.

Kim, J.-S.

J.-S. Kim, M.-S. Cho, S. Westland, and M. R. Luo, “Image quality assessment for photographic images,” in Proceedings of AIC Colour 05—10th Congress of the International Colour Association, J.L.Nieves and JavierHernández-Andrés, eds. (AIC, 2005) pp. 1095–1098.

Kubelka, P.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 11a, 593–601 (1931).

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, 1977).

Luo, M. R.

J.-S. Kim, M.-S. Cho, S. Westland, and M. R. Luo, “Image quality assessment for photographic images,” in Proceedings of AIC Colour 05—10th Congress of the International Colour Association, J.L.Nieves and JavierHernández-Andrés, eds. (AIC, 2005) pp. 1095–1098.

Luo, R. M.

Marcus, R. T.

Mikula, M.

M. Mikula, M. Ceppan, and K. Vasko, “Gloss and goniocolorimetry of printed materials,” Color Res. Appl. 28, 335–342(2003).
[CrossRef]

Mishchenko, M. I.

Munk, F.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 11a, 593–601 (1931).

Natale-Hoffman, K. M.

E. N. Dalal and K. M. Natale-Hoffman, “The effect of gloss on color,” Color Res. Appl. 24, 369–376 (1999).
[CrossRef]

Nayar, S. K.

M. Oren and S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

Neuman, M.

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. measurements,” J. Opt. Soc. Am. A 27, 1040–1045 (2010).
[CrossRef]

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. theory,” J. Opt. Soc. Am. A 27, 1032–1039 (2010).
[CrossRef]

M. Neuman, P. Edström, M. Andersson, L. Coppel, and O. Norberg, “Angular variations of color in turbid media—the influence of bulk scattering on goniochromism in paper,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology , 2010) pp. 407–413.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, 1977).

Norberg, O.

M. Neuman, P. Edström, M. Andersson, L. Coppel, and O. Norberg, “Angular variations of color in turbid media—the influence of bulk scattering on goniochromism in paper,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology , 2010) pp. 407–413.

O’Brien, W. J.

W. M. Chirdon, W. J. O’Brien, and R. E. Robertson, “Mechanisms of goniochromism relevant to restorative dentistry,” Dent. Mater. 25, 802–809 (2009).
[CrossRef] [PubMed]

Oleari, C.

C. Oleari, “Colorimetry in optical coating,” Proc. SPIE 5963, 596305 (2005).
[CrossRef]

Oren, M.

M. Oren and S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

Pointer, M. R.

Prahl, S. A.

S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32, 559–568 (1993).
[CrossRef] [PubMed]

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, 1977).

Robertson, R. E.

W. M. Chirdon, W. J. O’Brien, and R. E. Robertson, “Mechanisms of goniochromism relevant to restorative dentistry,” Dent. Mater. 25, 802–809 (2009).
[CrossRef] [PubMed]

Simonot, L.

L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
[CrossRef]

L. Simonot, “Photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009).
[CrossRef] [PubMed]

Stamnes, J. J.

Standardization, International Organization for

International Organization for Standardization, “Paper—determination of light scattering and absorption coefficients (using Kubelka-Munk theory),” ISO 9416 (International Organization for Standardization, 1998).

International Organization for Standardization, “Paper, board and pulps—measurement of diffuse reflectance factor,” ISO 2469 (International Organization for Standardization, 1994).

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science (Wiley, 2000).

van Gemert, M. J. C.

Vasko, K.

M. Mikula, M. Ceppan, and K. Vasko, “Gloss and goniocolorimetry of printed materials,” Color Res. Appl. 28, 335–342(2003).
[CrossRef]

Welch, A. J.

S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32, 559–568 (1993).
[CrossRef] [PubMed]

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Westland, S.

J.-S. Kim, M.-S. Cho, S. Westland, and M. R. Luo, “Image quality assessment for photographic images,” in Proceedings of AIC Colour 05—10th Congress of the International Colour Association, J.L.Nieves and JavierHernández-Andrés, eds. (AIC, 2005) pp. 1095–1098.

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science (Wiley, 2000).

Zhao, L.

Appl. Opt.

Astrophys. J.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Color Res. Appl.

L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
[CrossRef]

E. N. Dalal and K. M. Natale-Hoffman, “The effect of gloss on color,” Color Res. Appl. 24, 369–376 (1999).
[CrossRef]

M. Mikula, M. Ceppan, and K. Vasko, “Gloss and goniocolorimetry of printed materials,” Color Res. Appl. 28, 335–342(2003).
[CrossRef]

Dent. Mater.

W. M. Chirdon, W. J. O’Brien, and R. E. Robertson, “Mechanisms of goniochromism relevant to restorative dentistry,” Dent. Mater. 25, 802–809 (2009).
[CrossRef] [PubMed]

IEEE J. Quantum Electron.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Int. J. Comput. Vis.

M. Oren and S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

Inverse Probl. Sci. Eng.

P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” Inverse Probl. Sci. Eng. 16, 927–951 (2008).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Proc. SPIE

C. Oleari, “Colorimetry in optical coating,” Proc. SPIE 5963, 596305 (2005).
[CrossRef]

SIAM Rev.

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005).
[CrossRef]

Z. Tech. Phys.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 11a, 593–601 (1931).

Other

G. Wyszecki and W. S. Stiles, Color Science (Wiley, 2000).

R. W. G. Hunt, Measuring Colour (Fountain, 1998).

International Organization for Standardization, “Paper, board and pulps—measurement of diffuse reflectance factor,” ISO 2469 (International Organization for Standardization, 1994).

Deutsches Institut für Normung e. V., “Colorimetry; spectrophotometric method,” DIN 5033-4 (Deutsches Institut für Normung e. V., 1992).

J.-S. Kim, M.-S. Cho, S. Westland, and M. R. Luo, “Image quality assessment for photographic images,” in Proceedings of AIC Colour 05—10th Congress of the International Colour Association, J.L.Nieves and JavierHernández-Andrés, eds. (AIC, 2005) pp. 1095–1098.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, 1977).

M. Neuman, P. Edström, M. Andersson, L. Coppel, and O. Norberg, “Angular variations of color in turbid media—the influence of bulk scattering on goniochromism in paper,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology , 2010) pp. 407–413.

International Organization for Standardization, “Paper—determination of light scattering and absorption coefficients (using Kubelka-Munk theory),” ISO 9416 (International Organization for Standardization, 1998).

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

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Figures (6)

Fig. 1
Fig. 1

Measured and simulated angle resolved spectral reflectance factor of samples S1 [(a), (b)] and S4 [(c), (d)]. The reflectance factor depends strongly on polar angle θ in both measurements [(a), (c)] and simulations [(b), (d)].

Fig. 2
Fig. 2

Difference between measured and simulated reflectance factor for all observation angles and wavelengths, here shown for sample S4. The paper samples are matte with low gloss levels, and the difference never exceeds 0.1 reflectance factor unit.

Fig. 3
Fig. 3

Measured (lower) and simulated (upper) L * and C * values for all samples and observation angles with the actual perceived color indicated. The observation angle ranges from polar angle θ = 30 ° (right lower part) to θ = 70 ° (upper left part). It can be seen that the lightness of all samples increases when the observation angle increases and that the chroma decreases with observation angle. The change in chroma increases as the dye amount increases. There is good correspondence between measurements and simulations.

Fig. 4
Fig. 4

Simulated BRDF f r , d of a nonabsorbing opaque medium with g = 0.8 . We see that the BRDF, and thus the reflectance factor, is strongly dependent on observation angle θ r for all angles of incidence θ i . No angle of incidence will give a perfectly diffuse (Lambertian) reflectance.

Fig. 5
Fig. 5

Measured (lower) and simulated (upper) L * and C * values for all samples and observation angles when the nonabsorbing opaque medium is used as the reference white. The observation angle ranges from polar angle θ = 30 ° (right lower part) to θ = 70 ° (upper left part). The increase in lightness when the observation angle increases is now reduced, but the change in chroma is still large.

Fig. 6
Fig. 6

Measured (lower) and simulated (upper) L * and C * values for all samples and observation angles when the undyed paper is used as the reference white. The observation angle ranges from polar angle θ = 30 ° (right lower part) to θ = 70 ° (upper left part). The increase in lightness when the observation angle increases is now further reduced, but the change in chroma remains large.

Tables (4)

Tables Icon

Table 1 Paper Samples (S1–S4) Used in This Work a

Tables Icon

Table 2 Maximum Measured and Simulated Angular Color Difference When the Ideal Diffusor Is Used as the Reference White

Tables Icon

Table 3 Maximum Measured and Simulated Angular Color Difference When the Nonabsorbing Opaque Medium Is Used as the Reference White

Tables Icon

Table 4 Maximum Measured and Simulated Angular Color Difference When the Undyed Paper Is Used as the Reference White

Equations (14)

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

X = K λ P ( λ ) x ¯ ( λ ) d λ , Y = K λ P ( λ ) y ¯ ( λ ) d λ , Z = K λ P ( λ ) z ¯ ( λ ) d λ ,
X = K λ P i ( λ ) R ( λ ) x ¯ ( λ ) d λ , Y = K λ P i ( λ ) R ( λ ) y ¯ ( λ ) d λ , Z = K λ P i ( λ ) R ( λ ) z ¯ ( λ ) d λ .
L * = 116 f ( Y / Y n ) 16 , a * = 500 [ f ( X / X n ) f ( Y / Y n ) ] , b * = 200 [ f ( Y / Y n ) f ( Z / Z n ) ] ,
f ( x ) = { x 1 / 3 , x > 0.008856 7.787 x + 16 / 166 , x 0.008856 .
Δ E ab * = [ ( Δ L * ) 2 + ( Δ a * ) 2 + ( Δ b * ) 2 ] 1 / 2 ,
C * = [ ( a * ) 2 + ( b * ) 2 ] 1 / 2
h = arctan ( b * / a * ) .
d L ( r , θ , φ ) d s = ( σ s + σ a ) L ( r , θ , φ ) + σ s 4 π 4 π p ( cos Θ ) L ( r , θ , φ ) d ω ,
p ( cos Θ ) = 1 g 2 ( 1 + g 2 2 g cos Θ ) 3 / 2 .
d L ( θ i , ϕ i , θ r , ϕ r , E ) = f r ( θ i , ϕ i , θ r , ϕ r ) d E ( θ i , ϕ i ) ,
X ( θ r , ϕ r ) = K λ 2 π f r ( θ i , ϕ i , θ r , ϕ r , λ ) d E ( θ i , ϕ i , λ ) x ¯ ( λ ) d λ , Y ( θ r , ϕ r ) = K λ 2 π f r ( θ i , ϕ i , θ r , ϕ r , λ ) d E ( θ i , ϕ i , λ ) y ¯ ( λ ) d λ , Z ( θ r , ϕ r ) = K λ 2 π f r ( θ i , ϕ i , θ r , ϕ r , λ ) d E ( θ i , ϕ i , λ ) z ¯ ( λ ) d λ ,
X ( θ ) = K f r ( θ , λ ) E ( λ ) x ¯ ( λ ) d λ , Y ( θ ) = K f r ( θ , λ ) E ( λ ) y ¯ ( λ ) d λ , Z ( θ ) = K f r ( θ , λ ) E ( λ ) z ¯ ( λ ) d λ ,
X n ( θ ) = K f r , d ( θ , λ ) E ( λ ) x ¯ ( λ ) d λ , Y n ( θ ) = K f r , d ( θ , λ ) E ( λ ) y ¯ ( λ ) d λ , Z n ( θ ) = K f r , d ( θ , λ ) E ( λ ) z ¯ ( λ ) d λ ,
L * ( θ ) = 116 f ( Y ( θ ) / Y n ( θ ) ) 16 , a * ( θ ) = 500 [ f ( X ( θ ) / X n ( θ ) ) f ( Y ( θ ) / Y n ( θ ) ) ] , b * ( θ ) = 200 [ f ( Y ( θ ) / Y n ( θ ) ) f ( Z ( θ ) / Z n ( θ ) ) ] .

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