## Four-parameter model for polarization-resolved rough-surface BRDF |

Optics Express, Vol. 19, Issue 2, pp. 1027-1036 (2011)

http://dx.doi.org/10.1364/OE.19.001027

Acrobat PDF (1027 KB)

### Abstract

A modeling procedure is demonstrated, which allows representation of polarization-resolved BRDF data using only four parameters: the real and imaginary parts of an effective refractive index with an added parameter taking grazing incidence absorption into account and an angular-scattering parameter determined from the BRDF measurement of a chosen angle of incidence, preferably close to normal incidence. These parameters allow accurate predictions of *s*- and *p*-polarized BRDF for a painted rough surface, over three decades of variation in BRDF magnitude. To characterize any particular surface of interest, the measurements required to determine these four parameters are the directional hemispherical reflectance (DHR) for *s*- and *p*-polarized input radiation and the BRDF at a selected angle of incidence. The DHR data describes the angular and polarization dependence, as well as providing the overall normalization constraint. The resulting model conserves energy and fulfills the reciprocity criteria.

© 2011 OSA

## 1. Introduction

1. 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]

3. J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE **3699**, 2–15 (1999). [CrossRef]

5. X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Comput. Graph. **25**(4), 175–186 (1991). [CrossRef]

6. D. B. Goldman, B. Curless, A. Hertzmann, and S. M. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Pattern Anal. Mach. Intell. **32**(6), 1060–1071 (2010). [CrossRef] [PubMed]

## 2. Theory

*q*stands for s- or p-polarization. Conservation of energy is secured by scaling the pBRDF with DHR results.

14. R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. **41**(5), 988–993 (2002). [CrossRef]

*S*component of the Stokes vector. From this it is obvious that

_{0}*β*= 0. For

*β*= 1,

*g*= 0 resulting in

## 3. Experimental results

*β*= 0), over a range from −80 to 80 degrees. The surface was chosen as an example for the fitting technique because it showed a significant amount of forward scatter for high angles of incidence, which is governed by the variation of reflectance as a function of both angles of incidence and scattering. This is representative of the challenge in modelling rough-surface BRDF and is why this particular surface data was chosen for illustration. Smoother surfaces can generally be handled more easily, and would also be amenable to description by our analytical model.

17. M. J. Persky and M. Szczesniak, “Infrared, spectral, directional-hemispherical reflectance of fused silica, Teflon polytetrafluoroethylene polymer, chrome oxide ceramic particle surface, Pyromark 2500 paint, Krylon 1602 paint, and Duraflect coating,” Appl. Opt. **47**(10), 1389–1396 (2008). [CrossRef] [PubMed]

*s*and

*p*polarizations. The DHR data quoted in this article were evaluated at 3.39 μm, to match the wavelength at which the BRDF data were measured. In this type of measurement, the depolarization of the scattered radiation is not determined. The effective complex index of refraction obtained from the fitting of the observed measurements to Fresnel equations therefore will include also the depolarization part. The result will therefore strictly speaking be relevant only for non-polarimetric imaging. If depolarization is small, results will also be relevant for polarimetric imaging. In order to take depolarization into account in more detail, the model has to be further developed. This will also require more elaborate measurement techniques in order to determine the degree of depolarization.

18. H. Holl, “Specular reflection and characteristics of reflected light,” J. Opt. Soc. Am. **57**(5), 683–690 (1967). [CrossRef]

*n*-

*ik*, which yields a significant simplification in the model, while preserving the main features of the behavior. As seen in Fig. 2 , the DHR data sets for both polarizations were fitted (including a small additive offset) to the Fresnel equations using a least-squares fit, which allowed determination of an effective value for

*n*-

*ik*. The good fit to the Fresnel equation can possibly be understood from the fact that the diffuse scattering dominates at small angles where the reflectance is changing only slowly while the scattering at large angles is more specular.

*ρ*was not constant as illustrated in Fig. 3 . It was noticed that the angular behavior seen in Fig. 3 has an approximately complementary dependence to that seen in Fig. 2. This behavior was included in Eq. (5). This equation can certainly be refined by including further shielding at large angles not accounted for in the model. Deviations from predicted values are therefore expected at large angles.

_{q}*ρ*= 0.47 together with

*n*= 1.526 and

*k*= 0.193 has been used in the predictions.

## 4. Discussion and conclusions

## References and links

1. | K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. |

2. | J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional reflectance model validation and utilization,” Technical Report AFAL-TR-73–303, Environmental Research Institute of Michigan (ERIM), October 1973. |

3. | J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE |

4. | P. Beckmann and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces,” (Pergamon Press, 1963). |

5. | X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Comput. Graph. |

6. | D. B. Goldman, B. Curless, A. Hertzmann, and S. M. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Pattern Anal. Mach. Intell. |

7. | R. Priest, and T. Germer, “Polarimetric BRDF in the microfacet model: theory and measurements,” Proc. Military Sensing Symposia Specialty Group on Passive Sensors, Vol. 1, pp. 169–181 (Infrared Information Analysis Center, Ann Arbor, MI, August 2000), available at http://physics.nist.gov/Divisions/Div844/publications/germer/ GermerPriestMicroFacet.pdf |

8. | T. A. Germer and E. Marx, “Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings,” Appl. Opt. |

9. | I. G. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express |

10. | J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s Law,” J. Opt. Soc. Am. A |

11. | J. Stover, |

12. | B. G. Hoover and V. L. Gamiz, “Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics,” J. Opt. Soc. Am. A |

13. | C. F. Bohren, and D. R. Huffman, |

14. | R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. |

15. | F. Nicodemus, J. Richmond, J. Hsia, I. Ginsburg, and T. Lamparis, “Geometrical considerations and nomenclature for reflectance,” Nat. Bur. Stand. (U.S.) Monograph 160 (1977). |

16. | D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. |

17. | M. J. Persky and M. Szczesniak, “Infrared, spectral, directional-hemispherical reflectance of fused silica, Teflon polytetrafluoroethylene polymer, chrome oxide ceramic particle surface, Pyromark 2500 paint, Krylon 1602 paint, and Duraflect coating,” Appl. Opt. |

18. | H. Holl, “Specular reflection and characteristics of reflected light,” J. Opt. Soc. Am. |

**OCIS Codes**

(290.5820) Scattering : Scattering measurements

(290.5880) Scattering : Scattering, rough surfaces

(290.1483) Scattering : BSDF, BRDF, and BTDF

**ToC Category:**

Scattering

**History**

Original Manuscript: July 15, 2010

Revised Manuscript: October 27, 2010

Manuscript Accepted: December 20, 2010

Published: January 10, 2011

**Virtual Issues**

Vol. 6, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Ingmar G. E. Renhorn, Tomas Hallberg, David Bergström, and Glenn D. Boreman, "Four-parameter model for polarization-resolved rough-surface BRDF," Opt. Express **19**, 1027-1036 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-1027

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### References

- 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]
- J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional reflectance model validation and utilization,” Technical Report AFAL-TR-73–303, Environmental Research Institute of Michigan (ERIM), October 1973.
- J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE 3699, 2–15 (1999). [CrossRef]
- P. Beckmann and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces,” (Pergamon Press, 1963).
- X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Comput. Graph. 25(4), 175–186 (1991). [CrossRef]
- D. B. Goldman, B. Curless, A. Hertzmann, and S. M. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 32(6), 1060–1071 (2010). [CrossRef] [PubMed]
- R. Priest, and T. Germer, “Polarimetric BRDF in the microfacet model: theory and measurements,” Proc. Military Sensing Symposia Specialty Group on Passive Sensors, Vol. 1, pp. 169–181 (Infrared Information Analysis Center, Ann Arbor, MI, August 2000), available at http://physics.nist.gov/Divisions/Div844/publications/germer/ GermerPriestMicroFacet.pdf
- T. A. Germer and E. Marx, “Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings,” Appl. Opt. 43(6), 1266–1274 (2004). [CrossRef] [PubMed]
- I. G. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express 16(17), 12892–12898 (2008). [CrossRef] [PubMed]
- J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s Law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998). [CrossRef]
- J. Stover, Optical Scattering, Measurement and Analysis (SPIE Press, 1995).
- B. G. Hoover and V. L. Gamiz, “Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics,” J. Opt. Soc. Am. A 23(2), 314–328 (2006). [CrossRef]
- C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley-Interscience, New York, 1983).
- R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002). [CrossRef]
- F. Nicodemus, J. Richmond, J. Hsia, I. Ginsburg, and T. Lamparis, “Geometrical considerations and nomenclature for reflectance,” Nat. Bur. Stand. (U.S.) Monograph 160 (1977).
- D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998). [CrossRef]
- M. J. Persky and M. Szczesniak, “Infrared, spectral, directional-hemispherical reflectance of fused silica, Teflon polytetrafluoroethylene polymer, chrome oxide ceramic particle surface, Pyromark 2500 paint, Krylon 1602 paint, and Duraflect coating,” Appl. Opt. 47(10), 1389–1396 (2008). [CrossRef] [PubMed]
- H. Holl, “Specular reflection and characteristics of reflected light,” J. Opt. Soc. Am. 57(5), 683–690 (1967). [CrossRef]

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