Abstract

We introduce a novel parametric bidirectional reflectance distribution function (BRDF) model that can accurately encode a wide variety of real-world isotropic BRDFs with a small number of parameters. The key observation we make is that a BRDF may be viewed as a statistical distribution on a unit hemisphere. We derive a novel directional statistics distribution, which we refer to as the hemispherical exponential power distribution, and model real-world isotropic BRDFs as mixtures of it. We derive a canonical probabilistic method for estimating the parameters, including the number of components, of this novel directional statistics BRDF model. We show that the model captures the full spectrum of real-world isotropic BRDFs with high accuracy, but a small footprint. We also demonstrate the advantages of the novel BRDF model by showing its use for reflection component separation and for exploring the space of isotropic BRDFs.

© 2011 Optical Society of America

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  1. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometric considerations and nomenclature for reflectance,” (National Bureau of Standards, 1977).
  2. J. H. Lambert, “Photometria sive de mensura de gratibus luminis colorum et umbrae,” (Eberhard Klett, 1760).
  3. B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
    [CrossRef]
  4. J. F. Blinn, “Models of light reflection for computer sythesized pictures,” in SIGGRAPH ’77 Proceedings of the 4th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1977), pp. 192–198.
    [CrossRef]
  5. C. Schlick, “An inexpensive BRDF model for physically-based rendering,” Computer Graphics Forum 13, 233–246 (1994).
    [CrossRef]
  6. R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” ACM Trans. Graph. 1, 7–24 (1982).
    [CrossRef]
  7. K. Torrance and E. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114(1967).
    [CrossRef]
  8. G. J. Ward, “Measuring and modeling anisotropic reflection,” in SIGGRAPH ’92 Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1992), pp. 265–272.
    [CrossRef]
  9. S. K. Nayar and M. Oren, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
    [CrossRef]
  10. J. J. Koenderink and A. J. van Doorn, “Phenomenological description of bidirectional surface reflection,” J. Opt. Soc. Am. A 15, 2903–2912 (1998).
    [CrossRef]
  11. R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for inverse rendering,” in SIGGRAPH ’01 Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2001), pp. 117–128.
    [CrossRef]
  12. E. P. F. Lafortune, S.-C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” in SIGGRAPH ’97 Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1997), pp. 117–126.
    [CrossRef]
  13. D. Edwards, S. Boulos, J. Johnson, P. Shirley, M. Ashikhmin, M. Stark, and C. Wyman, “The halfway vector disk for BRDF modeling,” ACM Trans. Graph. 25, 1–18 (2006).
    [CrossRef]
  14. P. Debevec, T. Hawkins, C. Tchou, H.-P.Duiker, and W. Sarokin, “Acquiring the reflectance field of a human face,” in SIGGRAPH ’00 Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.
    [CrossRef]
  15. W. Matusik, H. Pfister, M. Brand, and L. McMillan, “Efficient isotropic BRDF measurement,” in Proceedings of the 14th Eurographics Workshop on Rendering Techniques, P.H.Christensen, D.Cohen-Or, and S.N.Spencer, eds., Vol. 44 of ACM International Conference Proceeding Series (Eurographics Association, 2003), pp. 241–248.
  16. A. Ghosh, S. Achutha, W. Heidrich, and M. O’Toole., “BRDF acquisition with basis illumination,” in Proceedings of the IEEE 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.
  17. A. Ghosh, T. Chen, P. Peers, C. A. Wilson, and P. Debevec, “Estimating specular roughness and anisotropy from second order spherical gradient illumination,” Computer Graphics Forum 28, 1161–1170 (2009).
    [CrossRef]
  18. M. Ashikhmin and S. Premoze, “Distribution-based BRDFs,” Tech. Rep. (University of Utah, 2007).
  19. W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graph. 22, 759–769(2003).
    [CrossRef]
  20. F. Romeiro, Y. Vasilyev, and T. E. Zickler, “Passive reflectometry,” in Proceedings of the 10th European Conference on Computer Vision: Part IV (Springer, 2008), pp. 859–872.
  21. M. Stark, J. Arvo, and B. Smits, “Barycentric parameterizations for isotropic BRDFs,” IEEE Trans. Vis. Comput. Graph. 11, 126–138 (2005).
    [CrossRef]
  22. S. Rusinkiewicz, “A new change of variables for efficient BRDF representation,” presented at the 1998 Eurographics Workshop on Rendering, Vienna, Austria, 29 June–1July 1998.
  23. A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” in Proceedings of the Eurographics Symposium on Rendering 2005, (Eurographics Association, 2005), pages 117–226.
  24. R. A. Fisher, “Dispersion on a sphere,” Proc. R. Soc. Lond. A 217, 295–305 (1953).
    [CrossRef]
  25. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).
  26. S. K. Nayar, K. Ikeuchi, and T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
    [CrossRef]
  27. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2007).
  28. K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions for single-view relighting,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 25–35 (2008).
    [CrossRef]
  29. D. A. Williams, “A test for differences between treatment means when several dose levels are compared with a zero dose control,” Biometrics 27, 103–117 (1971).
    [CrossRef]
  30. S. Cang and D. Partridge, “Determining the number of components in mixture models using Williams’ statistical test,” presented at the 8th International Conference on Neural Information Processing, Shanghai, China, 14–18 Nov. 2001.
  31. P. Debevec, “Light probe image gallery,” http://www.debevec.org/Probes.
  32. M. Pharr and G. Humphreys, Physically Based Rendering: from Theory to Implementation (Morgan Kaufmann, 2004).
  33. J. O. Ramsay and B. W. Silverman, Functional Data Analysis, 2nd ed., Springer Series in Statistics (Springer, 2005).

2009

A. Ghosh, T. Chen, P. Peers, C. A. Wilson, and P. Debevec, “Estimating specular roughness and anisotropy from second order spherical gradient illumination,” Computer Graphics Forum 28, 1161–1170 (2009).
[CrossRef]

2008

K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions for single-view relighting,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 25–35 (2008).
[CrossRef]

2006

D. Edwards, S. Boulos, J. Johnson, P. Shirley, M. Ashikhmin, M. Stark, and C. Wyman, “The halfway vector disk for BRDF modeling,” ACM Trans. Graph. 25, 1–18 (2006).
[CrossRef]

2005

M. Stark, J. Arvo, and B. Smits, “Barycentric parameterizations for isotropic BRDFs,” IEEE Trans. Vis. Comput. Graph. 11, 126–138 (2005).
[CrossRef]

2003

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graph. 22, 759–769(2003).
[CrossRef]

1998

1995

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

1994

C. Schlick, “An inexpensive BRDF model for physically-based rendering,” Computer Graphics Forum 13, 233–246 (1994).
[CrossRef]

1991

S. K. Nayar, K. Ikeuchi, and T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

1982

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” ACM Trans. Graph. 1, 7–24 (1982).
[CrossRef]

1975

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
[CrossRef]

1971

D. A. Williams, “A test for differences between treatment means when several dose levels are compared with a zero dose control,” Biometrics 27, 103–117 (1971).
[CrossRef]

1967

1953

R. A. Fisher, “Dispersion on a sphere,” Proc. R. Soc. Lond. A 217, 295–305 (1953).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Achutha, S.

A. Ghosh, S. Achutha, W. Heidrich, and M. O’Toole., “BRDF acquisition with basis illumination,” in Proceedings of the IEEE 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Arvo, J.

M. Stark, J. Arvo, and B. Smits, “Barycentric parameterizations for isotropic BRDFs,” IEEE Trans. Vis. Comput. Graph. 11, 126–138 (2005).
[CrossRef]

Ashikhmin, M.

D. Edwards, S. Boulos, J. Johnson, P. Shirley, M. Ashikhmin, M. Stark, and C. Wyman, “The halfway vector disk for BRDF modeling,” ACM Trans. Graph. 25, 1–18 (2006).
[CrossRef]

M. Ashikhmin and S. Premoze, “Distribution-based BRDFs,” Tech. Rep. (University of Utah, 2007).

Bishop, C. M.

C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2007).

Blinn, J. F.

J. F. Blinn, “Models of light reflection for computer sythesized pictures,” in SIGGRAPH ’77 Proceedings of the 4th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1977), pp. 192–198.
[CrossRef]

Boulos, S.

D. Edwards, S. Boulos, J. Johnson, P. Shirley, M. Ashikhmin, M. Stark, and C. Wyman, “The halfway vector disk for BRDF modeling,” ACM Trans. Graph. 25, 1–18 (2006).
[CrossRef]

Brand, M.

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graph. 22, 759–769(2003).
[CrossRef]

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “Efficient isotropic BRDF measurement,” in Proceedings of the 14th Eurographics Workshop on Rendering Techniques, P.H.Christensen, D.Cohen-Or, and S.N.Spencer, eds., Vol. 44 of ACM International Conference Proceeding Series (Eurographics Association, 2003), pp. 241–248.

Cang, S.

S. Cang and D. Partridge, “Determining the number of components in mixture models using Williams’ statistical test,” presented at the 8th International Conference on Neural Information Processing, Shanghai, China, 14–18 Nov. 2001.

Chen, T.

A. Ghosh, T. Chen, P. Peers, C. A. Wilson, and P. Debevec, “Estimating specular roughness and anisotropy from second order spherical gradient illumination,” Computer Graphics Forum 28, 1161–1170 (2009).
[CrossRef]

Cook, R. L.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” ACM Trans. Graph. 1, 7–24 (1982).
[CrossRef]

Debevec, P.

A. Ghosh, T. Chen, P. Peers, C. A. Wilson, and P. Debevec, “Estimating specular roughness and anisotropy from second order spherical gradient illumination,” Computer Graphics Forum 28, 1161–1170 (2009).
[CrossRef]

P. Debevec, T. Hawkins, C. Tchou, H.-P.Duiker, and W. Sarokin, “Acquiring the reflectance field of a human face,” in SIGGRAPH ’00 Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.
[CrossRef]

P. Debevec, “Light probe image gallery,” http://www.debevec.org/Probes.

Duiker, H.-P.

P. Debevec, T. Hawkins, C. Tchou, H.-P.Duiker, and W. Sarokin, “Acquiring the reflectance field of a human face,” in SIGGRAPH ’00 Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.
[CrossRef]

Durand, F.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” in Proceedings of the Eurographics Symposium on Rendering 2005, (Eurographics Association, 2005), pages 117–226.

Edwards, D.

D. Edwards, S. Boulos, J. Johnson, P. Shirley, M. Ashikhmin, M. Stark, and C. Wyman, “The halfway vector disk for BRDF modeling,” ACM Trans. Graph. 25, 1–18 (2006).
[CrossRef]

Fisher, R. A.

R. A. Fisher, “Dispersion on a sphere,” Proc. R. Soc. Lond. A 217, 295–305 (1953).
[CrossRef]

Foo, S.-C.

E. P. F. Lafortune, S.-C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” in SIGGRAPH ’97 Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1997), pp. 117–126.
[CrossRef]

Ghosh, A.

A. Ghosh, T. Chen, P. Peers, C. A. Wilson, and P. Debevec, “Estimating specular roughness and anisotropy from second order spherical gradient illumination,” Computer Graphics Forum 28, 1161–1170 (2009).
[CrossRef]

A. Ghosh, S. Achutha, W. Heidrich, and M. O’Toole., “BRDF acquisition with basis illumination,” in Proceedings of the IEEE 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Ginsberg, I. W.

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

Greenberg, D. P.

E. P. F. Lafortune, S.-C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” in SIGGRAPH ’97 Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1997), pp. 117–126.
[CrossRef]

Hanrahan, P.

R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for inverse rendering,” in SIGGRAPH ’01 Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2001), pp. 117–128.
[CrossRef]

Hara, K.

K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions for single-view relighting,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 25–35 (2008).
[CrossRef]

Hawkins, T.

P. Debevec, T. Hawkins, C. Tchou, H.-P.Duiker, and W. Sarokin, “Acquiring the reflectance field of a human face,” in SIGGRAPH ’00 Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.
[CrossRef]

Heidrich, W.

A. Ghosh, S. Achutha, W. Heidrich, and M. O’Toole., “BRDF acquisition with basis illumination,” in Proceedings of the IEEE 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Hsia, J. J.

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

Humphreys, G.

M. Pharr and G. Humphreys, Physically Based Rendering: from Theory to Implementation (Morgan Kaufmann, 2004).

Ikeuchi, K.

K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions for single-view relighting,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 25–35 (2008).
[CrossRef]

S. K. Nayar, K. Ikeuchi, and T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

Johnson, J.

D. Edwards, S. Boulos, J. Johnson, P. Shirley, M. Ashikhmin, M. Stark, and C. Wyman, “The halfway vector disk for BRDF modeling,” ACM Trans. Graph. 25, 1–18 (2006).
[CrossRef]

Kanade, T.

S. K. Nayar, K. Ikeuchi, and T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

Koenderink, J. J.

Lafortune, E. P. F.

E. P. F. Lafortune, S.-C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” in SIGGRAPH ’97 Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1997), pp. 117–126.
[CrossRef]

Lambert, J. H.

J. H. Lambert, “Photometria sive de mensura de gratibus luminis colorum et umbrae,” (Eberhard Klett, 1760).

Limperis, T.

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

Matusik, W.

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graph. 22, 759–769(2003).
[CrossRef]

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “Efficient isotropic BRDF measurement,” in Proceedings of the 14th Eurographics Workshop on Rendering Techniques, P.H.Christensen, D.Cohen-Or, and S.N.Spencer, eds., Vol. 44 of ACM International Conference Proceeding Series (Eurographics Association, 2003), pp. 241–248.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” in Proceedings of the Eurographics Symposium on Rendering 2005, (Eurographics Association, 2005), pages 117–226.

McMillan, L.

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graph. 22, 759–769(2003).
[CrossRef]

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “Efficient isotropic BRDF measurement,” in Proceedings of the 14th Eurographics Workshop on Rendering Techniques, P.H.Christensen, D.Cohen-Or, and S.N.Spencer, eds., Vol. 44 of ACM International Conference Proceeding Series (Eurographics Association, 2003), pp. 241–248.

Nayar, S. K.

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

S. K. Nayar, K. Ikeuchi, and T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

Ngan, A.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” in Proceedings of the Eurographics Symposium on Rendering 2005, (Eurographics Association, 2005), pages 117–226.

Nicodemus, F. E.

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

Nishino, K.

K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions for single-view relighting,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 25–35 (2008).
[CrossRef]

O’Toole., M.

A. Ghosh, S. Achutha, W. Heidrich, and M. O’Toole., “BRDF acquisition with basis illumination,” in Proceedings of the IEEE 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Oren, M.

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

Partridge, D.

S. Cang and D. Partridge, “Determining the number of components in mixture models using Williams’ statistical test,” presented at the 8th International Conference on Neural Information Processing, Shanghai, China, 14–18 Nov. 2001.

Peers, P.

A. Ghosh, T. Chen, P. Peers, C. A. Wilson, and P. Debevec, “Estimating specular roughness and anisotropy from second order spherical gradient illumination,” Computer Graphics Forum 28, 1161–1170 (2009).
[CrossRef]

Pfister, H.

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graph. 22, 759–769(2003).
[CrossRef]

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “Efficient isotropic BRDF measurement,” in Proceedings of the 14th Eurographics Workshop on Rendering Techniques, P.H.Christensen, D.Cohen-Or, and S.N.Spencer, eds., Vol. 44 of ACM International Conference Proceeding Series (Eurographics Association, 2003), pp. 241–248.

Pharr, M.

M. Pharr and G. Humphreys, Physically Based Rendering: from Theory to Implementation (Morgan Kaufmann, 2004).

Phong, B. T.

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
[CrossRef]

Premoze, S.

M. Ashikhmin and S. Premoze, “Distribution-based BRDFs,” Tech. Rep. (University of Utah, 2007).

Ramamoorthi, R.

R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for inverse rendering,” in SIGGRAPH ’01 Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2001), pp. 117–128.
[CrossRef]

Ramsay, J. O.

J. O. Ramsay and B. W. Silverman, Functional Data Analysis, 2nd ed., Springer Series in Statistics (Springer, 2005).

Richmond, J. C.

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

Romeiro, F.

F. Romeiro, Y. Vasilyev, and T. E. Zickler, “Passive reflectometry,” in Proceedings of the 10th European Conference on Computer Vision: Part IV (Springer, 2008), pp. 859–872.

Rusinkiewicz, S.

S. Rusinkiewicz, “A new change of variables for efficient BRDF representation,” presented at the 1998 Eurographics Workshop on Rendering, Vienna, Austria, 29 June–1July 1998.

Sarokin, W.

P. Debevec, T. Hawkins, C. Tchou, H.-P.Duiker, and W. Sarokin, “Acquiring the reflectance field of a human face,” in SIGGRAPH ’00 Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.
[CrossRef]

Schlick, C.

C. Schlick, “An inexpensive BRDF model for physically-based rendering,” Computer Graphics Forum 13, 233–246 (1994).
[CrossRef]

Shirley, P.

D. Edwards, S. Boulos, J. Johnson, P. Shirley, M. Ashikhmin, M. Stark, and C. Wyman, “The halfway vector disk for BRDF modeling,” ACM Trans. Graph. 25, 1–18 (2006).
[CrossRef]

Silverman, B. W.

J. O. Ramsay and B. W. Silverman, Functional Data Analysis, 2nd ed., Springer Series in Statistics (Springer, 2005).

Smits, B.

M. Stark, J. Arvo, and B. Smits, “Barycentric parameterizations for isotropic BRDFs,” IEEE Trans. Vis. Comput. Graph. 11, 126–138 (2005).
[CrossRef]

Sparrow, E.

Stark, M.

D. Edwards, S. Boulos, J. Johnson, P. Shirley, M. Ashikhmin, M. Stark, and C. Wyman, “The halfway vector disk for BRDF modeling,” ACM Trans. Graph. 25, 1–18 (2006).
[CrossRef]

M. Stark, J. Arvo, and B. Smits, “Barycentric parameterizations for isotropic BRDFs,” IEEE Trans. Vis. Comput. Graph. 11, 126–138 (2005).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Tchou, C.

P. Debevec, T. Hawkins, C. Tchou, H.-P.Duiker, and W. Sarokin, “Acquiring the reflectance field of a human face,” in SIGGRAPH ’00 Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.
[CrossRef]

Torrance, K.

Torrance, K. E.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” ACM Trans. Graph. 1, 7–24 (1982).
[CrossRef]

E. P. F. Lafortune, S.-C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” in SIGGRAPH ’97 Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1997), pp. 117–126.
[CrossRef]

van Doorn, A. J.

Vasilyev, Y.

F. Romeiro, Y. Vasilyev, and T. E. Zickler, “Passive reflectometry,” in Proceedings of the 10th European Conference on Computer Vision: Part IV (Springer, 2008), pp. 859–872.

Ward, G. J.

G. J. Ward, “Measuring and modeling anisotropic reflection,” in SIGGRAPH ’92 Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1992), pp. 265–272.
[CrossRef]

Williams, D. A.

D. A. Williams, “A test for differences between treatment means when several dose levels are compared with a zero dose control,” Biometrics 27, 103–117 (1971).
[CrossRef]

Wilson, C. A.

A. Ghosh, T. Chen, P. Peers, C. A. Wilson, and P. Debevec, “Estimating specular roughness and anisotropy from second order spherical gradient illumination,” Computer Graphics Forum 28, 1161–1170 (2009).
[CrossRef]

Wyman, C.

D. Edwards, S. Boulos, J. Johnson, P. Shirley, M. Ashikhmin, M. Stark, and C. Wyman, “The halfway vector disk for BRDF modeling,” ACM Trans. Graph. 25, 1–18 (2006).
[CrossRef]

Zickler, T. E.

F. Romeiro, Y. Vasilyev, and T. E. Zickler, “Passive reflectometry,” in Proceedings of the 10th European Conference on Computer Vision: Part IV (Springer, 2008), pp. 859–872.

ACM Trans. Graph.

D. Edwards, S. Boulos, J. Johnson, P. Shirley, M. Ashikhmin, M. Stark, and C. Wyman, “The halfway vector disk for BRDF modeling,” ACM Trans. Graph. 25, 1–18 (2006).
[CrossRef]

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” ACM Trans. Graph. 1, 7–24 (1982).
[CrossRef]

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graph. 22, 759–769(2003).
[CrossRef]

Biometrics

D. A. Williams, “A test for differences between treatment means when several dose levels are compared with a zero dose control,” Biometrics 27, 103–117 (1971).
[CrossRef]

Commun. ACM

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
[CrossRef]

Computer Graphics Forum

A. Ghosh, T. Chen, P. Peers, C. A. Wilson, and P. Debevec, “Estimating specular roughness and anisotropy from second order spherical gradient illumination,” Computer Graphics Forum 28, 1161–1170 (2009).
[CrossRef]

C. Schlick, “An inexpensive BRDF model for physically-based rendering,” Computer Graphics Forum 13, 233–246 (1994).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

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Figures (10)

Fig. 1
Fig. 1

We derive a novel directional distribution model, (b) the hemi-EPD, which has a shape parameter γ in addition to the scale parameter κ that corresponds to the concentration of a (a) von Mises–Fisher distribution (shown as 1D profiles). The hemi-EPD model can represent a wide variety of hemispherical directional distributions (the corresponding colors represent different values of κ and the three distinct distributions in (b) are drawn with different values of γ for the same κ).

Fig. 2
Fig. 2

(a) DSBRDF model with three lobes (solid curve) fit to θ d = 0 slices of different measured BRDF data [19] (squares) shown as a 1D profile on the incident plane. The DSBRDF model accurately fits the measured data despite the dramatically different shapes of the measured distributions (RGB corresponds to RGB color channels). (b) Lobes (red, green, blue curves) of a three-lobe isotropic DSBRDF model (black solid curve) fit to measured data (black dots). Each lobe clearly captures a distinct characteristic reflectance component of the BRDF.

Fig. 3
Fig. 3

DSBRDF model (solid curve) fit to different slices ( θ d = { 10 ° , 40 ° , 70 ° } ) of two measured BRDF data (squares: (a) color-changing-paint3 and (b) teflon in [19]). The DSBRDF model successfully captures the variation of the BRDF distributions, including the overall intensity increase and shape changes of specular lobes/spikes as θ d is varied.

Fig. 4
Fig. 4

Synthetic spheres rendered using the DSBRDF model with parameter values estimated from measured BRDF data (nickel, specular-blue-phenolic, and orange-paint in [19]). Each row shows spheres rendered using the original measured BRDF data and those rendered using the DSBRDF model with one to five lobes from left to right, respectively. The estimated optimal number of lobes for the DSBRDF models (see Section 5) were 5, 4, and 2, respectively, which agree well with the visual quality of the rendered spheres.

Fig. 5
Fig. 5

Relative rms errors for all 100 BRDFs in [19] using the DSBRDF model with the optimal number of lobes and synthetic spheres rendered with the (top) DSBRDF model and (bottom) measured data. The large errors are mainly caused by subsurface scattering (see the left-most column); otherwise, the visual quality of the DSBRDF model is very high.

Fig. 6
Fig. 6

Relative rms errors of the DSBRDF model (solid curves) and a nonparametric representation with linear interpolation (dashed curves) for three different BRDFs (distinct colors) as the sampling of the measured data is reduced (we subsample θ h by the integer factors on the horizontal axis). The results show that the DSBRDF model achieves higher accuracy than nonparametric models even with moderate subsampling, demonstrating the fragility of nonparametric models and the robustness of the DSBRDF model.

Fig. 7
Fig. 7

Synthetic spheres rendered with the Lambertian model with two lobes of Torrance–Sparrow models, measured BRDF, three-lobe DSBRDF model, and first to third individual lobes of the DSBRDF model from left to right, respectively. The results clearly show that the conventional Lambertian diffuse plus Torrance–Sparrow specular reflection representation fails to capture the complex reflectance of these real-world materials that we encounter in our daily life, even with two Torrance–Sparrow specular lobes. On the other hand, the three-lobe DSBRDF model accurately reproduces the appearance under natural illumination, which validates its expressiveness and accuracy. The lobe decompositions clearly visualize the distinct reflectance characteristics of individual lobes, e.g., the color is solely encoded in the third lobe for the bottom two materials indicating body reflection.

Fig. 8
Fig. 8

Results of modeling the BRDF of polyoxymethylene plastic with a three-lobe DSBRDF model. (First column) Synthetic spheres rendered with (top) measured BRDF and the (bottom) three-lobe DSBRDF model. (Second column) DSBRDF model (solid curves) fit to θ d = 0 slices of the measured BRDF data [19] (squares), shown as the 1D profile on the incident plane for the red, green, and blue color channels. (Third column) Lobes (red, green, blue curves) of a three-lobe DSBRDF model (black solid curve) fit to measured data (black dots). (Fourth column) Synthetic sphere rendered with the DSBRDF model parameters for each lobe separately. Note that the second lobe (corresponding to the blue curve in the third column) encodes the body color while it exhibits specular reflection, which cannot be modeled with conventional dichromatic models.

Fig. 9
Fig. 9

By projecting each lobe onto the computed eigenfunctions, we can visualize how the eigenfunctions naturally characterize the space of isotropic BRDFs and offer physically intuitive interpretations. In this image, each sphere is a single synthetically rendered lobe of a BRDF whose x and y coordinates are the projections onto the first two eigenfunctions. The diffuse lobes are located at the top left with specular surfaces scattered at the bottom right, indicating that the first eigenfunction roughly encodes the variability of glossiness, while the second eigenfunction roughly encodes the variability of matteness in the BRDF space.

Fig. 10
Fig. 10

We project each BRDF (represented as 54 curves) onto the first two bivariate eigenfunctions obtained via FPCA on the ( κ , γ ) curves computed by representing each BRDF, as a whole, with a DSBRDF model with the optimal number of lobes. Note the natural clustering in this low-dimensional embedding of the BRDF space, with pure-diffuse surfaces located in a tight cluster in the upper right-hand corner. Metallic surfaces are scattered toward the left, and plastic surfaces are scattered toward the bottom. These results suggest that the DSBRDF model provides a sound foundation for extracting physically meaningful low-dimensional bases for encoding the otherwise massively high-dimensional space of real-world BRDFs.

Equations (7)

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f r ( ω i , ω o ) = d L o ( ω o ) d E i ( ω i ) = d L o ( ω o ) L i ( ω i ) ( ω i · n ) d ω i ,
ω h = ω i + ω o ω i + ω o ω d = R Y ( θ h ) R Z ( ϕ h ) ω i ,
p ( θ h | θ d , Θ ) = κ 4 π sinh κ exp [ κ cos θ h ] ,
p ( θ h | θ d , Θ ) = C ( Θ ) ( exp [ κ cos γ θ h ] 1 ) ,
p ( θ h | θ d , Θ ) = κ = 1 K α ( κ ) p ( θ h | θ d , Θ ( κ ) ) ,
p ( θ h | θ d , Θ ) = κ = 1 K 1 C ( Θ ( κ ) ) p ( θ h | θ d , Θ ( κ ) )
ξ 1 , ξ 2 = 1 μ κ 2 ξ 1 κ ( t ) ξ 2 κ ( t ) d t + 1 μ γ 2 ξ 1 γ ( t ) ξ 2 γ ( t ) d t ,

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