Abstract

Precisely evaluating the geometrical attenuation factor is critical for constructing a more complete bidirectional reflectance distribution function (BRDF) model. Conventional theories for determining the geometrical attenuation factor neglect the correlation between height and slope and the self-shadowing or self-masking effects on microsurfaces, leading to results that are discrepant from reality, apparently. This paper presents a three-dimensional (3D) geometrical attenuation factor formulation on 3D Gaussian random rough surfaces. The proposed numerical analysis of 3D geometrical attenuation factor is much more precise for a practical application, especially near grazing angles. Our proposed numerical analysis of 3D geometrical attenuation factor can precisely evaluate the BRDF model.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antenn. Propag. 13(3), 384–388 (1965).
    [Crossref]
  2. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antenn. Propag. 15(5), 668–671 (1967).
    [Crossref]
  3. R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41(1), 138–147 (1967).
    [Crossref]
  4. K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57(9), 1105–1114 (1967).
    [Crossref]
  5. J. F. Blinn, “Models of light reflection for computer synthesized pictures,” in Proceedings of the 4th Annual Conference on Computer Graphics and Interactive Techniques, (1977), pp. 192–198.
    [Crossref]
  6. 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. Antenn. Propag. 50(3), 312–324 (2002).
    [Crossref]
  7. C. Bourlier, G. Berginc, and J. Saillard, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection on the infrared frequencies with shadowing effect,” IEEE Trans. Geosci. Remote Sens. 39(2), 379–392 (2001).
    [Crossref]
  8. C. Bourlier, “Unpolarized infrared emissivity with shadow from anisotropic rough sea surfaces with non-Gaussian statistics,” Appl. Opt. 44(20), 4335–4349 (2005).
    [Crossref] [PubMed]
  9. E. Heitz, “Understanding the masking-shadowing function in microfacet-based BRDFs,” J. Computer Graphics Techniques 3(2), 32–91 (2014).
  10. E. Heitz, J. Hanika, E. d’Eon, and C. D. Achsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. 35(4), 1–14 (2016).
  11. V. Schüssler, E. Heitz, J. Hanika, and C. Dachsbacher, “Microfacet-based normal mapping for robust Monte Carlo path tracing,” ACM Trans. Graph. 36(6), 1–12 (2017).
    [Crossref]
  12. E. Heitz, J. Dupuy, C. Crassin, and C. Dachsbacher, “The SGGX microflake distribution,” ACM Trans. Graph. 34(4), 48 (2015).
    [Crossref]
  13. E. Heitz, J. Dupuy, S. Hill, and D. Neubelt, “Real-time polygonal-light shading with linearly transformed cosines,” ACM Trans. Graph. 35(4), 1–8 (2016).
    [Crossref]
  14. B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Mircofacet models for refraction through rough surfaces,” in Eurographics Symposium on Rendering Techniques, (2007), pp.195–206.
  15. Y. Kuga and P. Phu, “Experimental studies of millimeter-wave scattering in discrete random media and from rough surfaces,” Prog. Electromagnetics Res. 14(3), 37–88 (1996).
    [Crossref]
  16. R. L. Wagner, J. Song, and W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antenn. Propag. 45(2), 235–245 (1997).
    [Crossref]
  17. K. Tang and R. O. Buckius, “The geometric optics approximation for reflection from two-dimensional random rough surfaces,” Int. J. Heat Mass Transf. 41(13), 2037–2047 (1998).
    [Crossref]
  18. M. Tembely, M. N. O. Sadiku, S. M. Musa, J. O. Attia, and P. Obiomon, “Electromagnetic scattering by random two-dimensional rough surface using the joint probability distribution function and Monte Carlo integration,” J. Multidiscip. Eng. Science Tech. 3(9), 5587–5592 (2016).
  19. E. I. Thorsos and D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86(1), 261–277 (1989).
    [Crossref]
  20. T. Chan, Y. Kuga, A. Ishimaru, and C. T. C. Le, “Experimental studies of bistatic scattering from two-dimensional conducting random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34(3), 674–680 (1996).
    [Crossref]
  21. B. E. Ewing and S. D. Butler, “Grazing angle experimental analysis of modification to microfacet BRDF Model for improved accuracy,” Proc. SPIE 10644, 1–14 (2018).
  22. S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Comparison of microfacet BRDF model to modified Beckmann-Kirchhoff BRDF model for rough and smooth surfaces,” Opt. Express 23(22), 29100–29112 (2015).
    [Crossref] [PubMed]
  23. A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of brdf models,” in Eurographic Symposium on Rendering (Academic, 2005), pp.117–126.
  24. L. Neumann, A. Neumann, and L. Szirmay-Kalos, “Compact metallic reflectance models,” Comput. Graph. Forum 18(3), 161–172 (1999).
    [Crossref]

2018 (1)

B. E. Ewing and S. D. Butler, “Grazing angle experimental analysis of modification to microfacet BRDF Model for improved accuracy,” Proc. SPIE 10644, 1–14 (2018).

2017 (1)

V. Schüssler, E. Heitz, J. Hanika, and C. Dachsbacher, “Microfacet-based normal mapping for robust Monte Carlo path tracing,” ACM Trans. Graph. 36(6), 1–12 (2017).
[Crossref]

2016 (3)

E. Heitz, J. Dupuy, S. Hill, and D. Neubelt, “Real-time polygonal-light shading with linearly transformed cosines,” ACM Trans. Graph. 35(4), 1–8 (2016).
[Crossref]

M. Tembely, M. N. O. Sadiku, S. M. Musa, J. O. Attia, and P. Obiomon, “Electromagnetic scattering by random two-dimensional rough surface using the joint probability distribution function and Monte Carlo integration,” J. Multidiscip. Eng. Science Tech. 3(9), 5587–5592 (2016).

E. Heitz, J. Hanika, E. d’Eon, and C. D. Achsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. 35(4), 1–14 (2016).

2015 (2)

2014 (1)

E. Heitz, “Understanding the masking-shadowing function in microfacet-based BRDFs,” J. Computer Graphics Techniques 3(2), 32–91 (2014).

2005 (1)

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. Antenn. Propag. 50(3), 312–324 (2002).
[Crossref]

2001 (1)

C. Bourlier, G. Berginc, and J. Saillard, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection on the infrared frequencies with shadowing effect,” IEEE Trans. Geosci. Remote Sens. 39(2), 379–392 (2001).
[Crossref]

1999 (1)

L. Neumann, A. Neumann, and L. Szirmay-Kalos, “Compact metallic reflectance models,” Comput. Graph. Forum 18(3), 161–172 (1999).
[Crossref]

1998 (1)

K. Tang and R. O. Buckius, “The geometric optics approximation for reflection from two-dimensional random rough surfaces,” Int. J. Heat Mass Transf. 41(13), 2037–2047 (1998).
[Crossref]

1997 (1)

R. L. Wagner, J. Song, and W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antenn. Propag. 45(2), 235–245 (1997).
[Crossref]

1996 (2)

Y. Kuga and P. Phu, “Experimental studies of millimeter-wave scattering in discrete random media and from rough surfaces,” Prog. Electromagnetics Res. 14(3), 37–88 (1996).
[Crossref]

T. Chan, Y. Kuga, A. Ishimaru, and C. T. C. Le, “Experimental studies of bistatic scattering from two-dimensional conducting random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34(3), 674–680 (1996).
[Crossref]

1989 (1)

E. I. Thorsos and D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86(1), 261–277 (1989).
[Crossref]

1967 (3)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antenn. Propag. 15(5), 668–671 (1967).
[Crossref]

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41(1), 138–147 (1967).
[Crossref]

K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57(9), 1105–1114 (1967).
[Crossref]

1965 (1)

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antenn. Propag. 13(3), 384–388 (1965).
[Crossref]

Achsbacher, C. D.

E. Heitz, J. Hanika, E. d’Eon, and C. D. Achsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. 35(4), 1–14 (2016).

Attia, J. O.

M. Tembely, M. N. O. Sadiku, S. M. Musa, J. O. Attia, and P. Obiomon, “Electromagnetic scattering by random two-dimensional rough surface using the joint probability distribution function and Monte Carlo integration,” J. Multidiscip. Eng. Science Tech. 3(9), 5587–5592 (2016).

Beckmann, P.

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antenn. Propag. 13(3), 384–388 (1965).
[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. Antenn. Propag. 50(3), 312–324 (2002).
[Crossref]

C. Bourlier, G. Berginc, and J. Saillard, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection on the infrared frequencies with shadowing effect,” IEEE Trans. Geosci. Remote Sens. 39(2), 379–392 (2001).
[Crossref]

Blinn, J. F.

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” in Proceedings of the 4th Annual Conference on Computer Graphics and Interactive Techniques, (1977), pp. 192–198.
[Crossref]

Bourlier, C.

C. Bourlier, “Unpolarized infrared emissivity with shadow from anisotropic rough sea surfaces with non-Gaussian statistics,” Appl. Opt. 44(20), 4335–4349 (2005).
[Crossref] [PubMed]

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. Antenn. Propag. 50(3), 312–324 (2002).
[Crossref]

C. Bourlier, G. Berginc, and J. Saillard, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection on the infrared frequencies with shadowing effect,” IEEE Trans. Geosci. Remote Sens. 39(2), 379–392 (2001).
[Crossref]

Buckius, R. O.

K. Tang and R. O. Buckius, “The geometric optics approximation for reflection from two-dimensional random rough surfaces,” Int. J. Heat Mass Transf. 41(13), 2037–2047 (1998).
[Crossref]

Butler, S. D.

B. E. Ewing and S. D. Butler, “Grazing angle experimental analysis of modification to microfacet BRDF Model for improved accuracy,” Proc. SPIE 10644, 1–14 (2018).

S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Comparison of microfacet BRDF model to modified Beckmann-Kirchhoff BRDF model for rough and smooth surfaces,” Opt. Express 23(22), 29100–29112 (2015).
[Crossref] [PubMed]

Chan, T.

T. Chan, Y. Kuga, A. Ishimaru, and C. T. C. Le, “Experimental studies of bistatic scattering from two-dimensional conducting random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34(3), 674–680 (1996).
[Crossref]

Chew, W. C.

R. L. Wagner, J. Song, and W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antenn. Propag. 45(2), 235–245 (1997).
[Crossref]

Crassin, C.

E. Heitz, J. Dupuy, C. Crassin, and C. Dachsbacher, “The SGGX microflake distribution,” ACM Trans. Graph. 34(4), 48 (2015).
[Crossref]

d’Eon, E.

E. Heitz, J. Hanika, E. d’Eon, and C. D. Achsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. 35(4), 1–14 (2016).

Dachsbacher, C.

V. Schüssler, E. Heitz, J. Hanika, and C. Dachsbacher, “Microfacet-based normal mapping for robust Monte Carlo path tracing,” ACM Trans. Graph. 36(6), 1–12 (2017).
[Crossref]

E. Heitz, J. Dupuy, C. Crassin, and C. Dachsbacher, “The SGGX microflake distribution,” ACM Trans. Graph. 34(4), 48 (2015).
[Crossref]

Dupuy, J.

E. Heitz, J. Dupuy, S. Hill, and D. Neubelt, “Real-time polygonal-light shading with linearly transformed cosines,” ACM Trans. Graph. 35(4), 1–8 (2016).
[Crossref]

E. Heitz, J. Dupuy, C. Crassin, and C. Dachsbacher, “The SGGX microflake distribution,” ACM Trans. Graph. 34(4), 48 (2015).
[Crossref]

Durand, F.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of brdf models,” in Eurographic Symposium on Rendering (Academic, 2005), pp.117–126.

Ewing, B. E.

B. E. Ewing and S. D. Butler, “Grazing angle experimental analysis of modification to microfacet BRDF Model for improved accuracy,” Proc. SPIE 10644, 1–14 (2018).

Hanika, J.

V. Schüssler, E. Heitz, J. Hanika, and C. Dachsbacher, “Microfacet-based normal mapping for robust Monte Carlo path tracing,” ACM Trans. Graph. 36(6), 1–12 (2017).
[Crossref]

E. Heitz, J. Hanika, E. d’Eon, and C. D. Achsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. 35(4), 1–14 (2016).

Heitz, E.

V. Schüssler, E. Heitz, J. Hanika, and C. Dachsbacher, “Microfacet-based normal mapping for robust Monte Carlo path tracing,” ACM Trans. Graph. 36(6), 1–12 (2017).
[Crossref]

E. Heitz, J. Hanika, E. d’Eon, and C. D. Achsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. 35(4), 1–14 (2016).

E. Heitz, J. Dupuy, S. Hill, and D. Neubelt, “Real-time polygonal-light shading with linearly transformed cosines,” ACM Trans. Graph. 35(4), 1–8 (2016).
[Crossref]

E. Heitz, J. Dupuy, C. Crassin, and C. Dachsbacher, “The SGGX microflake distribution,” ACM Trans. Graph. 34(4), 48 (2015).
[Crossref]

E. Heitz, “Understanding the masking-shadowing function in microfacet-based BRDFs,” J. Computer Graphics Techniques 3(2), 32–91 (2014).

Hill, S.

E. Heitz, J. Dupuy, S. Hill, and D. Neubelt, “Real-time polygonal-light shading with linearly transformed cosines,” ACM Trans. Graph. 35(4), 1–8 (2016).
[Crossref]

Ishimaru, A.

T. Chan, Y. Kuga, A. Ishimaru, and C. T. C. Le, “Experimental studies of bistatic scattering from two-dimensional conducting random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34(3), 674–680 (1996).
[Crossref]

Jackson, D. R.

E. I. Thorsos and D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86(1), 261–277 (1989).
[Crossref]

Kuga, Y.

Y. Kuga and P. Phu, “Experimental studies of millimeter-wave scattering in discrete random media and from rough surfaces,” Prog. Electromagnetics Res. 14(3), 37–88 (1996).
[Crossref]

T. Chan, Y. Kuga, A. Ishimaru, and C. T. C. Le, “Experimental studies of bistatic scattering from two-dimensional conducting random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34(3), 674–680 (1996).
[Crossref]

Le, C. T. C.

T. Chan, Y. Kuga, A. Ishimaru, and C. T. C. Le, “Experimental studies of bistatic scattering from two-dimensional conducting random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34(3), 674–680 (1996).
[Crossref]

Li, H.

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

Marciniak, M. A.

Marschner, S. R.

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

Matusik, W.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of brdf models,” in Eurographic Symposium on Rendering (Academic, 2005), pp.117–126.

Musa, S. M.

M. Tembely, M. N. O. Sadiku, S. M. Musa, J. O. Attia, and P. Obiomon, “Electromagnetic scattering by random two-dimensional rough surface using the joint probability distribution function and Monte Carlo integration,” J. Multidiscip. Eng. Science Tech. 3(9), 5587–5592 (2016).

Nauyoks, S. E.

Neubelt, D.

E. Heitz, J. Dupuy, S. Hill, and D. Neubelt, “Real-time polygonal-light shading with linearly transformed cosines,” ACM Trans. Graph. 35(4), 1–8 (2016).
[Crossref]

Neumann, A.

L. Neumann, A. Neumann, and L. Szirmay-Kalos, “Compact metallic reflectance models,” Comput. Graph. Forum 18(3), 161–172 (1999).
[Crossref]

Neumann, L.

L. Neumann, A. Neumann, and L. Szirmay-Kalos, “Compact metallic reflectance models,” Comput. Graph. Forum 18(3), 161–172 (1999).
[Crossref]

Ngan, A.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of brdf models,” in Eurographic Symposium on Rendering (Academic, 2005), pp.117–126.

Obiomon, P.

M. Tembely, M. N. O. Sadiku, S. M. Musa, J. O. Attia, and P. Obiomon, “Electromagnetic scattering by random two-dimensional rough surface using the joint probability distribution function and Monte Carlo integration,” J. Multidiscip. Eng. Science Tech. 3(9), 5587–5592 (2016).

Phu, P.

Y. Kuga and P. Phu, “Experimental studies of millimeter-wave scattering in discrete random media and from rough surfaces,” Prog. Electromagnetics Res. 14(3), 37–88 (1996).
[Crossref]

Sadiku, M. N. O.

M. Tembely, M. N. O. Sadiku, S. M. Musa, J. O. Attia, and P. Obiomon, “Electromagnetic scattering by random two-dimensional rough surface using the joint probability distribution function and Monte Carlo integration,” J. Multidiscip. Eng. Science Tech. 3(9), 5587–5592 (2016).

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. Antenn. Propag. 50(3), 312–324 (2002).
[Crossref]

C. Bourlier, G. Berginc, and J. Saillard, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection on the infrared frequencies with shadowing effect,” IEEE Trans. Geosci. Remote Sens. 39(2), 379–392 (2001).
[Crossref]

Schüssler, V.

V. Schüssler, E. Heitz, J. Hanika, and C. Dachsbacher, “Microfacet-based normal mapping for robust Monte Carlo path tracing,” ACM Trans. Graph. 36(6), 1–12 (2017).
[Crossref]

Smith, B. G.

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antenn. Propag. 15(5), 668–671 (1967).
[Crossref]

Song, J.

R. L. Wagner, J. Song, and W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antenn. Propag. 45(2), 235–245 (1997).
[Crossref]

Sparrow, E. M.

Szirmay-Kalos, L.

L. Neumann, A. Neumann, and L. Szirmay-Kalos, “Compact metallic reflectance models,” Comput. Graph. Forum 18(3), 161–172 (1999).
[Crossref]

Tang, K.

K. Tang and R. O. Buckius, “The geometric optics approximation for reflection from two-dimensional random rough surfaces,” Int. J. Heat Mass Transf. 41(13), 2037–2047 (1998).
[Crossref]

Tembely, M.

M. Tembely, M. N. O. Sadiku, S. M. Musa, J. O. Attia, and P. Obiomon, “Electromagnetic scattering by random two-dimensional rough surface using the joint probability distribution function and Monte Carlo integration,” J. Multidiscip. Eng. Science Tech. 3(9), 5587–5592 (2016).

Thorsos, E. I.

E. I. Thorsos and D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86(1), 261–277 (1989).
[Crossref]

Torrance, K. E.

K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57(9), 1105–1114 (1967).
[Crossref]

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

Wagner, R. J.

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41(1), 138–147 (1967).
[Crossref]

Wagner, R. L.

R. L. Wagner, J. Song, and W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antenn. Propag. 45(2), 235–245 (1997).
[Crossref]

Walter, B.

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

ACM Trans. Graph. (4)

E. Heitz, J. Hanika, E. d’Eon, and C. D. Achsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. 35(4), 1–14 (2016).

V. Schüssler, E. Heitz, J. Hanika, and C. Dachsbacher, “Microfacet-based normal mapping for robust Monte Carlo path tracing,” ACM Trans. Graph. 36(6), 1–12 (2017).
[Crossref]

E. Heitz, J. Dupuy, C. Crassin, and C. Dachsbacher, “The SGGX microflake distribution,” ACM Trans. Graph. 34(4), 48 (2015).
[Crossref]

E. Heitz, J. Dupuy, S. Hill, and D. Neubelt, “Real-time polygonal-light shading with linearly transformed cosines,” ACM Trans. Graph. 35(4), 1–8 (2016).
[Crossref]

Appl. Opt. (1)

Comput. Graph. Forum (1)

L. Neumann, A. Neumann, and L. Szirmay-Kalos, “Compact metallic reflectance models,” Comput. Graph. Forum 18(3), 161–172 (1999).
[Crossref]

IEEE Trans. Antenn. Propag. (4)

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. Antenn. Propag. 50(3), 312–324 (2002).
[Crossref]

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antenn. Propag. 13(3), 384–388 (1965).
[Crossref]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antenn. Propag. 15(5), 668–671 (1967).
[Crossref]

R. L. Wagner, J. Song, and W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antenn. Propag. 45(2), 235–245 (1997).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (2)

C. Bourlier, G. Berginc, and J. Saillard, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection on the infrared frequencies with shadowing effect,” IEEE Trans. Geosci. Remote Sens. 39(2), 379–392 (2001).
[Crossref]

T. Chan, Y. Kuga, A. Ishimaru, and C. T. C. Le, “Experimental studies of bistatic scattering from two-dimensional conducting random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34(3), 674–680 (1996).
[Crossref]

Int. J. Heat Mass Transf. (1)

K. Tang and R. O. Buckius, “The geometric optics approximation for reflection from two-dimensional random rough surfaces,” Int. J. Heat Mass Transf. 41(13), 2037–2047 (1998).
[Crossref]

J. Acoust. Soc. Am. (2)

E. I. Thorsos and D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86(1), 261–277 (1989).
[Crossref]

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41(1), 138–147 (1967).
[Crossref]

J. Computer Graphics Techniques (1)

E. Heitz, “Understanding the masking-shadowing function in microfacet-based BRDFs,” J. Computer Graphics Techniques 3(2), 32–91 (2014).

J. Multidiscip. Eng. Science Tech. (1)

M. Tembely, M. N. O. Sadiku, S. M. Musa, J. O. Attia, and P. Obiomon, “Electromagnetic scattering by random two-dimensional rough surface using the joint probability distribution function and Monte Carlo integration,” J. Multidiscip. Eng. Science Tech. 3(9), 5587–5592 (2016).

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Figures (10)

Fig. 1
Fig. 1 Gaussian random rough surface. We generated the surface by using Algorithm 1, and by defining truncation length Lx = Ly = 12.8, correlation length lx = ly = 0.5, discrete interval τ x = τ y = 0.05 andσ = 0.1.
Fig. 2
Fig. 2 (a) Illustration of the shadowing effect: the incident ray can reach the surface, and (b) Illustration of the masking effect: the outgoing ray is blocked by adjacent microsurfaces.
Fig. 3
Fig. 3 Illustration on masking function. The outgoing ray o can be seen depending on two conditions: i). the incident ray can illuminate the microfacet, ii). the outgoing ray o is not blocked by adjacent microfacets.
Fig. 4
Fig. 4 Top surface M0 and the bottom surface M1. The top surface simulates a coaxial light source, and so the parallel incidence light emitting out from it can be moved along y axis. The M1 is the orthographic projection of surface Z on xoy plane.
Fig. 5
Fig. 5 Illustration on self-shadowing. (a) L intersects Z. L represents an incident ray coming from the parallel light source M0, assuming it intersects surface Z and reached at the bottom surface M1 at the same index of matrix of M0 and M1. (b) Reflection occurs at points P0 and P1 in the first bounce. Assuming an incident ray intersects the surface Z with several intersections, apparently the reflectance occurs at P1.
Fig. 6
Fig. 6 Illustration on self-masking. At point P1, the incidence ray L and the reflectance ray R go without shadowing and masking. At point P2,the reflectance ray is blocked by the surface Z at point P3, the self-shadowing occurs.
Fig. 7
Fig. 7 Our proposed algorithm is applied to ten rough surfaces with σ = 0.1 to 1.0. The numerical solutions for geometrical attenuation factor curves are continuous, coinciding with the law of physics. As the incidence angle is 0, the geometrical attenuation factor is 1. The surface is illuminated, the incidence angle is 90, while the surface is invisible, and the factor is 0. From the result, demarcation point, between the lighted and not lighted surfaces, is given.
Fig. 8
Fig. 8 Our 3D geometrical attenuation factor and the Smith’s. σ = 0.1, 0.3, 0.5 respectively.
Fig. 9
Fig. 9 The comparison between the Cook-Torrance BRDF model (black) and the Neumann- Cook-Torrance BRDF model (blue). The geometrical attenuation factor is Smith (dotted markers) and ours (solid curves).
Fig. 10
Fig. 10 Smith geometrical attenuation factor (dotted markers) and Ours(solid line) are shown for θ i = 30(black), θ i = 45 (bottle green), θ i = 60 (blue), θ i = 75 (rose red) and θ i = 85 (grass green) . σis from 0.1 to1.0 respectively.

Tables (8)

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Notation used throughout this paper

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Algorithm 1. Generation of a Gaussian random surface z(x,y)

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Algorithm 2. Local shadowing function G 1 local ( ω i , ω m )

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Algorithm 3. Local masking function G 1 local ( ω m , ω o )

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Algorithm 4. Self-shadowing function G 1 dist ( ω i )

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Algorithm 5. Self-masking function G 1 dist ( ω o )

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Algorithm 6. Geometrical attenuation factor G 2 ( ω i , ω o )

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Table 1 Illumination factor

Equations (8)

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G( θ i )= 1- 1 2 erfc(μ/ 2 σ) Λ(μ)+1
[z(x,y)]=[ z(0,0) z(0,1) z(0,Y-1) z(1,0) z(1,1) z(1,Y-1) z(X-1,0) z(X-1,1) z(X-1,Y-1) ]
G 1 local ( ω i , ω m , ω o )= G 1 local ( ω i , ω m ) G 1 local ( ω m , ω o )
[( G 2 ( ω i , ω o ))(x,y)]=[ G 2 (0,0) G 2 (0,1) G 2 (0,Y-1) G 2 (1,0) G 2 (1,1) G 2 (1,Y-1) G 2 (X-1,0) G 2 (X-1,1) G 2 (X-1,Y-1) ]
G 2 ( ω i , ω o )=1- N s XY
G 1 local ( ω i , ω m )= χ + (i·m) χ + (m·n)
G 1 local ( ω m , ω o )= χ + (n·o) χ + (m·n) χ + (m·o)
f( ω i , ω m )= F( ω o ) G 2 ( ω i , ω o )D( ω m ) 4| ω i · ω g || ω o · ω g |

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