Abstract

The four-flux model is a method to solve light radiative-transfer problems in planar, possibly multilayer structures. The light fluxes are modeled as two collimated and two diffuse beams propagating forward and backward perpendicularly to the layer stack. In the present contribution, we develop a four-flux model relying on a matrix formalism to determine the reflectance and transmittance factors of stacks of components by knowing those of each individual component. This model is also extended to generate the bidirectional scattering distribution function of the stack by considering an incoming collimated flux in any direction and by taking into account the directionality of the diffuse fluxes exiting from the material at the border components of the stack. The model is applied to opaque Lambertian backgrounds with flat or rough interfaces for which analytical expressions of the BSDF are obtained.

© 2015 Optical Society of America

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References

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2014 (3)

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

2013 (1)

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

2012 (1)

2011 (1)

2009 (2)

N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009).
[Crossref]

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

2008 (1)

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[Crossref]

2007 (1)

M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech. 2007, P02018 (2007).
[Crossref]

2006 (2)

A. B. Murphy, “Modified Kubelka–Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D 39, 3571–3581 (2006).
[Crossref]

L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
[Crossref]

2002 (1)

C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002).
[Crossref]

2001 (3)

M. Elias, L. Simonot, and M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[Crossref]

C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]

C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

1999 (2)

1998 (1)

1997 (2)

1995 (1)

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]

1989 (2)

1988 (1)

1987 (1)

1986 (1)

1984 (1)

1977 (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

1973 (1)

H. Pauli and D. Eitel, “Comparison of different theoretical models of multiple scattering for pigmented media,” Colour 73, 423–426 (1973).

1971 (1)

1967 (1)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[Crossref]

1966 (1)

R. Aronson and D. L. Yarmush, “Transfer-matrix method for gamma-ray and neutron penetration,” J. Math. Phys. 7, 221–237 (1966).
[Crossref]

1954 (1)

1953 (1)

1948 (1)

1942 (2)

J. L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942).
[Crossref]

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Natl. Bur. Stand. 29, 329–332 (1942).

1931 (2)

J. W. Ryde, “The scattering of light by turbid media—Part 1,” Proc. Roy. Soc. (London) A131, 451–464 (1931).

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).

1860 (1)

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1860).
[Crossref]

Ala-Nissila, T.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Aloui, F.

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Amrani, B.

Andraud, C.

Arancibia-Bulnes, C. A.

Aronson, R.

R. Aronson and D. L. Yarmush, “Transfer-matrix method for gamma-ray and neutron penetration,” J. Math. Phys. 7, 221–237 (1966).
[Crossref]

Atkins, J. T.

J. K. Beasley, J. T. Atkins, and F. W. Billmeyer, “Scattering and absorption in turbid media,” in Electromagnetic Scattering, R. L. Rowell and R. S. Stein, eds. (Gordon and Breach, 1967), pp. 765–785.

Bayou, S.

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Beasley, J. K.

J. K. Beasley, J. T. Atkins, and F. W. Billmeyer, “Scattering and absorption in turbid media,” in Electromagnetic Scattering, R. L. Rowell and R. S. Stein, eds. (Gordon and Breach, 1967), pp. 765–785.

Becker, J.-M.

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[Crossref]

Berginc, G.

C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002).
[Crossref]

Billmeyer, F. W.

J. K. Beasley, J. T. Atkins, and F. W. Billmeyer, “Scattering and absorption in turbid media,” in Electromagnetic Scattering, R. L. Rowell and R. S. Stein, eds. (Gordon and Breach, 1967), pp. 765–785.

Bourlier, C.

C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002).
[Crossref]

Briton, J. P.

Chandrasekhar, S.

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

Chauvet, O.

Chee Tsay, S.

Clapper, F. R.

Cook, R. L.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in SIGGRAPH ‘81 Proceedings of the 8th Annual Conference on Computer Graphics and Interactive Techniques (1981), Vol. 15, pp. 307–316.
[Crossref]

d’Eon, E.

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

Dong, N.

N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009).
[Crossref]

Eitel, D.

H. Pauli and D. Eitel, “Comparison of different theoretical models of multiple scattering for pigmented media,” Colour 73, 423–426 (1973).

El Haber, F.

Eldridge, J. I.

L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

Elias, M.

M. Elias, L. Simonot, and M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[Crossref]

Emmel, P.

M. Hébert, R. D. Hersch, and P. Emmel, “Fundamentals of optics and radiometry for color reproduction,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1021–1077.

M. Hébert and P. Emmel, “Two-flux and multiflux matrix model for color surface,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1233–1277.

Forschum, F.

Froyer, G.

Ge, J.

N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009).
[Crossref]

Ginsber, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

Girasole, T.

C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]

C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

Gouesbet, G.

Granqvist, C. G.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Gréhan, G.

C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]

Guo, S. M.

L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

Hébert, M.

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[Crossref]

L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
[Crossref]

M. Hébert and P. Emmel, “Two-flux and multiflux matrix model for color surface,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1233–1277.

M. Hébert, R. D. Hersch, and P. Emmel, “Fundamentals of optics and radiometry for color reproduction,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1021–1077.

Hersch, R. D.

L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
[Crossref]

M. Hébert, R. D. Hersch, and P. Emmel, “Fundamentals of optics and radiometry for color reproduction,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1021–1077.

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Jakob, O.

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

Jakob, W.

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

Jayaweera, K.

Jobic, S.

Judd, D. B.

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Natl. Bur. Stand. 29, 329–332 (1942).

Kienle, A.

Kubelka, P.

Laaksonen, K.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Le Toulouzan, J. N.

Lebaudy, P.

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Lecamp, L.

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Li, H.

B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.

Li, S.-Y.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

Maheu, B.

Marschner, S.

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

Marschner, S. R.

B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.

Menu, M.

M. Elias, L. Simonot, and M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[Crossref]

Molenaar, R.

Mouzali, M.

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Mudgett, P. S.

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).

Murphy, A. B.

A. B. Murphy, “Modified Kubelka–Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D 39, 3571–3581 (2006).
[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]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

Nieminen, R. M.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Niklasson, G. A.

Niklasson, G. A. C.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[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]

Pauli, H.

H. Pauli and D. Eitel, “Comparison of different theoretical models of multiple scattering for pigmented media,” Colour 73, 423–426 (1973).

Puisto, S. R.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Ren, K. F.

Richards, L. W.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

Rocquefelte, X.

Rostedt, N. K. J.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Rozé, C.

C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]

Ruiz-Suarez, J. C.

Ryde, J. W.

J. W. Ryde, “The scattering of light by turbid media—Part 1,” Proc. Roy. Soc. (London) A131, 451–464 (1931).

Saillard, J.

C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002).
[Crossref]

Saunderson, J. L.

Simon, K.

M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech. 2007, P02018 (2007).
[Crossref]

Simonot, L.

Smith, B. G.

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[Crossref]

Stamnes, K.

Stokes, G. G.

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1860).
[Crossref]

Tafforin, A. G.

C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

ten Bosch, J. J.

Torrance, K. E.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in SIGGRAPH ‘81 Proceedings of the 8th Annual Conference on Computer Graphics and Interactive Techniques (1981), Vol. 15, pp. 307–316.
[Crossref]

B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.

Vargas, W. E.

Vöge, M.

M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech. 2007, P02018 (2007).
[Crossref]

Walter, B.

B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.

Wang, L.

L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

Wang, Y. P.

Williams, F. C.

Wiscombe, W.

Yarmush, D. L.

R. Aronson and D. L. Yarmush, “Transfer-matrix method for gamma-ray and neutron penetration,” J. Math. Phys. 7, 221–237 (1966).
[Crossref]

Zhang, Y.

N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009).
[Crossref]

Zheng, S. W.

Zijp, J. R.

ACM Trans. Graph. (1)

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

Acta Mater. (1)

L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

Appl. Opt. (12)

R. Molenaar, J. J. ten Bosch, and J. R. Zijp, “Determination of Kubelka–Munk scattering and absorption coefficient,” Appl. Opt. 38, 2068–2077 (1999).
[Crossref]

P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[Crossref]

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]

B. Maheu, J. N. Le Toulouzan, and G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
[Crossref]

B. Maheu and G. Gouesbet, “Four-flux models to solve the scattering transfer equation. Special cases,” Appl. Opt. 25, 1122–1228 (1986).
[Crossref]

G. A. Niklasson, “Comparison between four flux theory and multiple scattering theory,” Appl. Opt. 26, 4034–4036 (1987).
[Crossref]

B. Maheu, J. P. Briton, and G. Gouesbet, “Four-flux model and a Monte Carlo code: comparisons between two simple and complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
[Crossref]

Y. P. Wang, S. W. Zheng, and K. F. Ren, “Four-flux model with adjusted average crossing parameter to solve the scattering transfer equation,” Appl. Opt. 28, 24–26 (1989).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[Crossref]

W. E. Vargas, “Generalized four-flux radiative transfer model average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 37, 2615–2623 (1998).
[Crossref]

C. A. Arancibia-Bulnes and J. C. Ruiz-Suarez, “Average path-length parameter of diffuse light in scattering media,” Appl. Opt. 38, 1877–1883 (1999).
[Crossref]

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

Atmos. Environ. (1)

C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

Colour (1)

H. Pauli and D. Eitel, “Comparison of different theoretical models of multiple scattering for pigmented media,” Colour 73, 423–426 (1973).

IEEE Trans. Antennas Propag. (2)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[Crossref]

C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002).
[Crossref]

Int. J. Comput. Vis. (1)

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]

J. Math. Phys. (1)

R. Aronson and D. L. Yarmush, “Transfer-matrix method for gamma-ray and neutron penetration,” J. Math. Phys. 7, 221–237 (1966).
[Crossref]

J. Natl. Bur. Stand. (1)

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Natl. Bur. Stand. 29, 329–332 (1942).

J. Opt. A (1)

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[Crossref]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

J. Phys. D (1)

A. B. Murphy, “Modified Kubelka–Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D 39, 3571–3581 (2006).
[Crossref]

J. Res. Natl. Bur. Stand. (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

J. Stat. Mech. (1)

M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech. 2007, P02018 (2007).
[Crossref]

Opt. Commun. (2)

C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]

M. Elias, L. Simonot, and M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[Crossref]

Opt. Express (1)

Polym. J. (1)

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Proc. R. Soc. London (1)

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1860).
[Crossref]

Proc. Roy. Soc. (London) (1)

J. W. Ryde, “The scattering of light by turbid media—Part 1,” Proc. Roy. Soc. (London) A131, 451–464 (1931).

Proc. SPIE (1)

N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009).
[Crossref]

Sol. Energy Mater. Sol. Cells (1)

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Zeitschrift für technische Physik (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).

Other (7)

M. Hébert, R. D. Hersch, and P. Emmel, “Fundamentals of optics and radiometry for color reproduction,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1021–1077.

J. K. Beasley, J. T. Atkins, and F. W. Billmeyer, “Scattering and absorption in turbid media,” in Electromagnetic Scattering, R. L. Rowell and R. S. Stein, eds. (Gordon and Breach, 1967), pp. 765–785.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

M. Hébert and P. Emmel, “Two-flux and multiflux matrix model for color surface,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1233–1277.

B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.

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

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in SIGGRAPH ‘81 Proceedings of the 8th Annual Conference on Computer Graphics and Interactive Techniques (1981), Vol. 15, pp. 307–316.
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. Flux transfers between two components represented by thin arrows. Bold arrows correspond to fluxes.
Fig. 2.
Fig. 2. Useful notations for defining the BSDF.
Fig. 3.
Fig. 3. Flux transfers (a) for a Lambertian component, (b) for a nonscattering component.
Fig. 4.
Fig. 4. (a) Bidirectional reflectance factor r cd ( i , o ) of a nonscattering component (index 1) on a scattering component (index 2). (b) Bidirectional transmittance factor t cd ( i , o ) of a scattering component (index 1) on a nonscattering component (index 2).
Fig. 5.
Fig. 5. (a) Perfectly flat interface on a Lambertian background, (b) micro-facet rough interface on a Lambertian background, (c) distribution of interfaced Lambertian micro-facets.
Fig. 6.
Fig. 6. Volume BRDF in the incident plane, assuming a Beckmann distribution for D σ and the corresponding Smith shadowing masking function for G [47,48], with θ i = 60 ° (backscattering direction at θ o = 60 ° and specular direction at θ o = 60 ° ), n = 1.5 and different roughness parameters σ. (a) Rough interface on a Lambertian background [second term of Eq. (39) with r 10 σ = r 10 ], (b) distribution of interfaced Lambertian facets [second term of Eq. (40)].
Fig. 7.
Fig. 7. Directional-hemispherical reflectance (solid lines) or transmittance (dashed lines) factors of rough interfaces as a function of the incident angle θ i for different roughness parameters σ with n = 1.5 (a) from medium 0 to medium 1, (b) from medium 1 to medium 0.
Fig. 8.
Fig. 8. Bihemispherical factors of a rough interface in terms of the roughness parameter σ with n = 1.5 (a) from medium 0 to medium 1, (b) from medium 1 to medium 0.

Tables (2)

Tables Icon

Table 1. Collimated-to-Diffuse and Diffuse-to-Diffuse Reflectance and Transmittance Factors of a Component According to its Position in the Component Stack a

Tables Icon

Table 2. Expressions of the Different Reflectance or Transmittance Factors in Function of the BRDF or BTDF f , where 2 π f cos θ d ω = θ = 0 π / 2 φ = 0 π / 2 f cos θ sin θ d θ d φ

Equations (44)

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{ J c k 1 = r cc I c k 1 + t cc J c k I c k = t cc I c k 1 + r cc J c k J d k 1 = r cd I c k 1 + t cd J c k + r dd I d k 1 + t dd J d k I d k = t cd I c k 1 + r cd J c k + t dd I d k 1 + r dd J d k .
( r cc 1 0 0 t cc 0 0 0 r cd 0 r dd 1 t cd 0 t dd 0 ) ( I c k 1 J c k 1 I d k 1 J d k 1 ) = ( 0 t cc 0 0 1 r cc 0 0 0 t cd 0 t dd 0 r cd 1 r dd ) ( I c k J c k I d k J d k ) .
( I c k 1 J c k 1 I d k 1 J d k 1 ) = ( M cc 0 2 , 2 M cd M dd ) ( I c k J c k I d k J d k ) ,
M x x = 1 t x x ( 1 r x x r x x t x x t x x r x x r x x ) ,
M cd = 1 t cc t dd ( t cd r cc t cd r cd t cc r cd t dd r dd t cd t cc ( t cd t dd r cd r dd ) r cc ( r cd t dd r dd t cd ) ) .
t x x = 1 M x x ( 1 , 1 ) and r x x = M x x ( 2 , 1 ) M x x ( 1 , 1 ) ,
t cd = M cd ( 1 , 1 ) M cc ( 1 , 1 ) M dd ( 1 , 1 ) and r cd = M cd ( 2 , 1 ) M cc ( 1 , 1 ) M cd ( 1 , 1 ) M dd ( 2 , 1 ) M cc ( 1 , 1 ) M dd ( 1 , 1 ) .
M 1 M 2 = ( M cc 1 M cc 2 0 2 , 2 M cd 1 M cc 2 + M dd 1 M cd 2 M dd 1 M dd 2 ) .
t x x = t x x 1 t x x 2 1 r x x 1 r x x 2 and r x x = r x x 1 + r x x 2 t x x 1 t x x 1 1 r x x 1 r x x 2 .
t cd = t cd 1 t dd 2 1 r dd 1 r dd 2 + t cc 1 t cd 2 1 r cc 1 r cc 2 + t cc 1 t dd 2 ( r cd 1 r cc 2 + r dd 1 r cd 2 ) ( 1 r cc 1 r cc 2 ) ( 1 r dd 1 r dd 2 ) ,
r cd = r cd 1 + t cd 1 r dd 2 t dd 1 1 r dd 1 r dd 2 + t cd 1 r cc 2 t cc 1 1 r cc 1 r cc 2 + t cc 1 t dd 1 ( r cd 2 + r dd 2 r cd 1 r cc 2 ) ( 1 r cc 1 r cc 2 ) ( 1 r dd 1 r dd 2 ) .
t cci t ddi 0 .
f r ( i , o ) = r cd ( i , o ) π and f t ( i , o ) = t cd ( i , o ) π ,
M cc = 1 t cc ( 1 0 0 0 ) .
{ r cd ( i , o ) = r cd ( i ) = r dd t cd ( i , o ) = t cd ( i ) = t dd ( o ) = t dd and { r cd ( i ) = r dd t cd ( i , o ) = t cd ( i ) = t dd ( o ) = t dd .
M cd = ( 1 t cc r dd t dd r dd t cc t dd t dd r dd r dd t dd ) .
{ r cd = r dd = r cd = r dd t cd = t dd = t cd = t dd .
M cd = 0 2 , 2 .
h dd = 1 π θ i = 0 π / 2 φ i = 0 2 π h cc ( i ) cos θ i sin θ i d θ i d φ i = θ i = 0 π / 2 h cc ( i ) sin 2 θ i d θ i ,
t dd ( o ) = t cc ( o ) / n 2 ,
t dd ( o ) = n 2 t cc ( o ) .
r cd ( i , o ) = t cc 1 ( i ) t cc 1 ( o ) n 2 r cd 2 ( i 2 , o 2 ) ( 1 r cc 1 ( i ) r cc 2 ( i ) ) ( 1 r dd 1 r dd 2 ) ,
t cd ( i , o ) = t cc 2 ( o ) n 2 ( t cd 1 ( i , o 1 ) 1 r dd 1 r dd 2 + t cc 1 ( i ) r cd 1 ( i ) r cc 2 ( i ) ( 1 r cc 1 ( i ) r cc 2 ( i ) ) ( 1 r dd 1 r dd 2 ) ) ,
{ n sin ( θ i 1 ) = sin ( θ i ) t cc = T 01 ( i ) = 1 R 01 ( i ) r cc = R 10 ( i 1 ) = R 01 ( i ) t cc = T 10 ( i 1 ) = 1 R 01 ( i ) ,
M cc = 1 T 01 ( i ) ( 1 R 01 ( i ) R 01 ( i ) 1 2 R 01 ( i ) ) .
r dd = r 01 = θ i = 0 π / 2 R 01 ( i ) sin 2 θ i d θ i .
{ t dd = t 01 = 1 r 01 t dd = t 10 = t 01 / n 2 r dd = r 10 = 1 t 10
t dd ( o ) = T 01 ( o ) / n 2 ,
t dd ( o ) = n 2 T 01 ( o ) .
r 01 σ ( i , o ) = π R 01 ( i , h r ) D σ ( h r ) G ( i , o , h r ) 4 ( i · n ) ( o · n ) ,
t 01 σ ( i , o ) = π | i · h t | | o · h t | ( i · n ) ( o · n ) n 1 2 T 01 ( i , h t ) D σ ( h t ) G ( i , o , h t ) ( n 0 ( i · h t ) + n 1 ( o · h t ) ) 2 ,
r 01 σ = θ i = 0 π / 2 r 01 σ ( i ) sin 2 θ i d θ i .
M cc = ( 1 t cc 0 0 0 ) , M dd = ( 1 t dd 0 ρ t dd 0 ) and M cd = ( 0 0 ρ t cc 0 ) .
M cc = ( 1 T 01 ( i ) R 01 ( i ) T 01 ( i ) R 01 ( i ) T 01 ( i ) 1 2 R 01 ( i ) T 01 ( i ) ) , M dd = ( 1 t 01 r 10 t 01 r 01 t 01 t 01 T 01 ( o ) / n 2 r 01 r 10 t 01 ) and M cd = 0 2 , 2 .
M cc = ( 1 t cc T 01 ( i ) 0 R 01 ( i ) t cc T 01 ( i ) 0 ) , M dd = ( 1 ρ r 10 t 01 t dd 0 r 01 + ρ ( t 01 T 01 ( o ) / n 2 r 01 r 10 ) t 01 t dd 0 ) and M cd = ( r 10 ρ t 01 t cc 0 ρ t 01 T 01 ( o ) / n 2 r 01 r 10 t 01 t cc 0 ) .
f r ( i , o ) = 1 π n 2 T 01 ( i ) T 01 ( o ) ρ ( 1 r 10 ρ ) .
M cc = ( 1 T cc 0 0 0 ) , M dd = ( 1 t 01 σ r 10 σ t 01 σ r 01 σ t 01 σ t 01 σ t 10 σ ( o ) r 01 σ r 10 σ t 01 σ ) and M cd = ( t 01 σ ( i ) T cc t 01 σ r 10 σ ( i ) t 01 σ t 01 σ r 01 σ ( i , o ) t 01 σ ( i ) r 01 σ T cc t 01 σ t 10 σ ( i , o ) t 01 σ r 10 σ ( i ) r 01 σ T cc t 01 σ ) ,
M cc = ( 1 T cc t cc 0 0 0 ) , M dd = ( 1 r 10 σ ρ t dd t 01 σ 0 r 01 σ + ρ ( t 01 σ t 10 σ ( o ) r 01 σ r 10 σ ) t dd t 01 σ 0 ) and M cd = ( t 01 σ ( i ) T cc t cc t 01 σ r 10 σ ρ t cc t 01 σ 0 t 01 σ r 01 σ ( i , o ) t 01 σ ( i ) r 01 σ T cc t cc t 01 σ + ρ t 01 σ t 10 σ ( o ) r 01 σ r 10 σ t cc t 01 σ 0 ) .
f r ( i , o ) = 1 π ( r 01 σ ( i , o ) + t 01 σ ( i ) t 10 σ ( o ) ρ 1 r 10 σ ρ ) .
f r ( i , o ) = r 01 σ ( i , o ) π + ρ π ( 1 r 10 ρ ) 1 n 2 ( i · n ) ( o · n ) 2 π T 01 ( i , m ) T 01 ( o , m ) D σ ( m ) G ( i , o , m ) ( i · m ) ( o · m ) d ω m ,
( t cc 0 0 0 t cd t d d 0 0 r cc 0 1 0 r cd r dd 0 1 ) ( I c k 1 I d k 1 J c k 1 J d k 1 ) = ( 1 0 r cc 0 0 1 r cd r dd 0 0 t cc 0 0 0 t cd t dd ) ( I c k I d k J c k J d k ) .
( T 0 2 , 2 R 1 2 , 2 ) ( I c k 1 I d k 1 J c k 1 J d k 1 ) = ( 1 2 , 2 R 0 2 , 2 T ) ( I c k I d k J c k J d k ) .
( I c k 1 I d k 1 J c k 1 J d k 1 ) = M ( I c k I d k J c k J d k ) , with M = ( T 1 0 2 , 2 RT 1 1 2 , 2 ) ( 1 2 , 2 R 0 2 , 2 T ) = ( T 1 T 1 R RT 1 T RT 1 R ) = ( M 11 M 12 M 21 M 22 ) .
{ R = M 21 M 11 1 T = M 11 1 R = M 11 1 M 12 T = M 12 M 21 M 11 1 M 12 .

Metrics