OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 24, Iss. 3 — Mar. 1, 2007
  • pp: 724–744

Statistical ray method for deriving reflection models of rough surfaces

Yinlong Sun  »View Author Affiliations

JOSA A, Vol. 24, Issue 3, pp. 724-744 (2007)

View Full Text Article

Enhanced HTML    Acrobat PDF (615 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A statistical ray method is developed for deriving reflection models of rough surfaces. Using this method, we derive generic equations of reflection due to single scattering. We have studied an isotropic surface with Gaussian statistics, and the derived bidirectional reflectance distribution function is found with four distinct regimes, namely, mirror reflection, grazing reflection, retroreflection, and normal reflection. An explicit form of self-shadowing for a surface with Gaussian statistics is also derived, and the result agrees well with computer simulation. Our solution is useful to describe the entire reflection of a highly or moderately smooth surface and offers a basis for studying multiple scattering. While focusing on optical reflection, the approach also applies to other waves if the assumptions are satisfied.

© 2007 Optical Society of America

OCIS Codes
(120.5700) Instrumentation, measurement, and metrology : Reflection
(290.5880) Scattering : Scattering, rough surfaces

ToC Category:

Original Manuscript: May 25, 2006
Revised Manuscript: September 22, 2006
Manuscript Accepted: September 26, 2006
Published: February 14, 2007

Yinlong Sun, "Statistical ray method for deriving reflection models of rough surfaces," J. Opt. Soc. Am. A 24, 724-744 (2007)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, 1990).
  2. M.Bass, E.W.van Stryland, D.R.Williams, and W.L.Wolfe, eds., Handbook of Optics, 2nd ed. (McGraw-Hall, 1995).
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  4. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, 1991).
  5. J. C. Leader, "Analysis and prediction of laser scattering from rough-surface materials," J. Opt. Soc. Am. 69, 610-628 (1979). [CrossRef]
  6. C. A. Depew and R. D. Weir, "Surface roughness determination by the measurement of reflectance," Appl. Opt. 10, 969-970 (1971). [CrossRef] [PubMed]
  7. P. J. Chandley, "Surface roughness measurements from coherent light scattering," Opt. Quantum Electron. 8, 323-327 (1976). [CrossRef]
  8. E. Marx and T. V. Vorburger, "Direct and inverse problems for light scattered by rough surfaces," Appl. Opt. 29, 3613-3626 (1990). [CrossRef] [PubMed]
  9. J. Caron, J. Lafait, and C. Andraud, 'Scalar Kirchhoff's model for light scattering from dielectric random rough surfaces," Opt. Commun. 207, 17-28 (2002). [CrossRef]
  10. K. A. O'Donnell and E. R. Mendez, "Experimental study of scattering from characterized random surfaces," J. Opt. Soc. Am. A 4, 1194-1205 (1987). [CrossRef]
  11. D. H. Berman and J. S. Perkins, "Exponential substitution for Kirchhoff scattering from Gaussian rough surfaces," J. Acoust. Soc. Am. 78, 1024-1028 (1985). [CrossRef]
  12. M. Nieto-Vesperinas, "Radiometry of rough surfaces," Opt. Acta 29, 961-971 (1982). [CrossRef]
  13. S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii., Principles of Statistical Radiophysics (Springer-Verlag, 1987). [CrossRef]
  14. H. Sizun, Radio Wave Propagation for Telecommunication Applications (Springer, 2005).
  15. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, Pergamon, 1963).
  16. T. M. Lillesand, R. W. Kiefer, and J. W. Chipman, Remote Sensing and Image Interpretation (Wiley, 2004).
  17. F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, 1982).
  18. B. F. Kavanagh, Geomatics (Prentice Hall, 1999).
  19. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
  20. T. Hagfors, "Backscattering from an undulating surface with applications to radar returns from the moon," J. Geophys. Res. 69, 3779-3784 (1964). [CrossRef]
  21. G.T.Ruck, D.Barrick, W.Stuart, and C.Krichbaum, eds., Radar Cross Section Handbook (Peninsula, 1970), p. 949.
  22. M. W. Long, Radar Reflectivity of Land and Sea, 3rd ed. (Artech House, 2001).
  23. A. W. Rihaczek and S. J. Hershkowitz, Theory and Practice of Radar Target Identification (Artech House, 2000).
  24. D. J. Whitehouse, Handbook of Surface Metrology (Institute of Physics, 1994).
  25. T. R. Thomas, Rough Surfaces, 2nd ed. (Imperial College Press, 1999).
  26. K. J. Stout and L. Blunt, Three-Dimensional Surface Topography (Penton, 2000).
  27. R. A. Hall, Illumination and Color in Computer Generated Imagery (Springer-Verlag, 1989). [CrossRef]
  28. A. S. Glassner, Principles of Digital Image Synthesis (Morgan Kaufmann, 1995).
  29. J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics: Principles and Practice, 2nd ed. (Addison-Wesley, 1996).
  30. A. Watt, 3D Computer Graphics, 3rd ed. (Addison-Wesley, 2000).
  31. 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]
  32. B. K. P. Horn and M.J. Brooks, eds., Shape from Shading (MIT, 1989).
  33. B. K. H. Horn, Robot Vision (MIT, 1986).
  34. Q.-T. Luong, "Color in computer vision," in Handbook of Pattern Recognition and Computer Vision, C.H.Chen, L.F.Pau, and P.W. P.Wand, eds. (World Scientific, 1993), pp. 311-368. [CrossRef]
  35. R. Bajcsy, S. W. Lee, and A. Leonardis, "Detection of diffuse and specular interface reflections and interreflections by color image segmentation," Int. J. Comput. Vis. 17, 241-271 (1996). [CrossRef]
  36. E. Trucco and A. Verri, Introductory Techniques for 3-D Computer Vision (Prentice Hall, 1998).
  37. L. B. Wolff, S. K. Nayar, and M. Oren, "Improved diffuse reflection models for computer vision," Int. J. Comput. Vis. 30, 55-71 (1998). [CrossRef]
  38. M. Suk and S. M. Bhandarkar, Computer Science Workbench: Three-Dimensional Object Recognition from Range Image, T.L.Kunii, ed. (Springer-Verlag, 1992). [CrossRef]
  39. J. D. Jackson, Classical Electrodynamics (Wiley, 1966).
  40. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Pergamon, 1975). [PubMed]
  41. E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).
  42. F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1979).
  43. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, 1991).
  44. A. G. Voronovich, Wave Scattering from Rough Surfaces (Spring-Verlag, Berlin, 1994).
  45. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  46. J. M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, 1989).
  47. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometric Considerations and Nomenclature for Reflectance (U.S. Department of Commerce, 1977).
  48. E. L. Church and P. Z. Takacs, "Surface scattering," in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), pp. 7.1-7.14.
  49. J. M. Palmer, "The measurement of transmission, absorption, emission, and reflection," in Handbook of Optics, M.Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), pp. 25.21-25.25.
  50. J. C. Stover, "Scatterometers," in Handbook of Optics, Vol. 2, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), pp. 26.1-26.16.
  51. Y. Sun, "Analytic framework for calculating BRDFs of randomly rough surfaces," Tech. Rep. CST TR 06-010 (Department of Computer Sciences, Purdue University, 2006), available at www.cs.purdue.edu/research/technical_reports.
  52. Y. Sun, "Statistically-based reflection model for rough surfaces," in Computational Imaging, C. A. Bouman and R. L. Stevenson, eds., Proc. SPIE 5016, 91-102 (2003).
  53. Y. Sun, "Physically-based reflection models for metallic surfaces," in Proceedings of the 3rd IASTED International Conference on Visualization, Imaging, and Image Processing (VIIP'03) (International Association of Science and Technology for Development, 2003), pp. 1028-1033. [PubMed]
  54. Y. Sun, "Self shadowing and local illumination of randomly rough surfaces," in Proeedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, CVPR, 2004), pp. 158-165.
  55. P. Beckmann, "Shadowing of random rough surfaces," IEEE Trans. Antennas Propag. AP-13, 384-388 (1965). [CrossRef]
  56. R. A. Brockelman and T. Hagfors, "Note on the effect of shadowing on the backscattering of waves from a random rough surface," IEEE Trans. Antennas Propag. AP-14, 621-626 (1965).
  57. R. J. Wagner, "Shadowing of randomly rough surfaces," J. Acoust. Soc. Am. 41, 138-147 (1967). [CrossRef]
  58. B. G. Smith, "Geometrical shadowing of a random rough surface," IEEE Trans. Antennas Propag. AP-15, 668-671 (1967). [CrossRef]
  59. M. I. Sancer, "Shadow-corrected electromagnetic scattering from a randomly rough surfaces," IEEE Trans. Antennas Propag. AP-17, 577-585 (1969). [CrossRef]
  60. A. Stogryn, "Electromagnetic scattering from rough, finitely conducting surfaces," Radio Sci. 2, 415-428 (1967).
  61. D. E. Barrick, "Rough surfaces scattering based on the specular point theory," IEEE Trans. Antennas Propag. 16, 449-454 (1968). [CrossRef]
  62. B.-T. Phong, "Illumination for computer generated images," Commun. ACM 18, 311-317 (1975). [CrossRef]
  63. E. L. Church and J. M. Zavada, "Residual surface roughness of diamond-turned optics," Appl. Opt. 14, 1788-1795 (1975). [CrossRef] [PubMed]
  64. E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Measurement of the finish of diamond-turned metal surfaces by differential light scattering," Opt. Eng. 16, 360-374 (1977).
  65. E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Relationship between surface scattering and microtopographic features," Opt. Eng. 18, 125-136 (1979).
  66. J. M. Elson, H. E. Bennett, and J. M. Bennett, "Scattering from optical surfaces," in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1979), pp. 191-244.
  67. K. Torrance and E. M. Sparrow, "Theory for off-specular reflection from roughened surfaces," J. Opt. Soc. Am. 57, 1105-1114 (1967). [CrossRef]
  68. R. L. Cook and K. E. Torrance, "A reflection model for computer graphics," ACM Trans. Graphics 1, 7-24 (1982). [CrossRef]
  69. T. A. Germer and M. E. Nadal, "Modeling the appearance of special effect pigment coatings," in Surface Scattering and Diffraction for Advanced Metrology, Z.-H. Gu and A. A. Maradudin, eds., Proc. SPIE 4447, pp. 77-86 (2001).
  70. T. A. Germer and E. Marx, "Ray model of light scattering by pigmented coatings and by coated rough surfaces," Appl. Opt. 43, 1266-1274 (2004). [CrossRef] [PubMed]
  71. T. A. Germer, "Angular dependence and polarization of out-of-plane optical scattering from particulate contamination, subsurface defects, and surface microroughness," Appl. Opt. 36, 8798-8805 (1997). [CrossRef]
  72. R. G. Priest and S. R. Meier, "Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces," Opt. Eng. 41, 988-993 (2002). [CrossRef]
  73. J. F. Blinn, "Models of light reflection for computer synthesized pictures," in Proceedings of ACM, SIGGRAPH 1977 (ACM, 1977), pp. 192-198.
  74. B. G. Smith, "Lunar surface roughness: shadowing and thermal emission," J. Geophys. Res. 72, 4059-4067 (1967). [CrossRef]
  75. P. Strauss, "A realistic lighting model for computer animators," IEEE Comput. Graphics Appl. 10, 56-64 (1990). [CrossRef]
  76. G. J. Ward, "Measuring and modeling anisotropic reflection," in Proceedings of ACM SIGGRAPH 1992 (ACM, 1992) pp. 265-272.
  77. E. P. F. Lafortune, S. Foo, K. E. Torrance, and D. P. Greenberg, "Non-linear approximation of reflectance functions," in Proceedings of SIGGRAPH (ACM, Addison-Wesley, 1997), pp. 117-126. [CrossRef]
  78. L. Neumann, A. Neumann, and L. Szirmay-Kalos, "Compact metetallic reflectance models," Comput. Graph. Forum 18, 161-172 (1999). [CrossRef]
  79. C. Schlick, "A survey of shading and reflectance models," Comput. Graph. Forum 13, 121-131 (1994). [CrossRef]
  80. C. Schlick, "A customizable reflectance model for everyday rendering," in Fourth Eurographics Workshop on Rendering (Eurographics, 1993) pp. 73-83.
  81. Y. Sun, F. D. Fracchia, M. S. Drew, and T. W. Calvert, "Rendering iridescent colors of optical disks," in Proceedings of the 11th EUROGRAPHICS Workshop on Rendering (Springer-Verlag, 2000), pp. 341-352.
  82. J. T. Kajiya, "Anisotropic reflection models," in Proceedings of ACM SIGGRAPH 1985 (ACM, 1985), pp. 15-21.
  83. X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," in Proceedings of ACM SIGGRAPH 1991 (ACM, 1991), pp. 175-186. [CrossRef]
  84. J. Stam, "Diffraction shaders," in Proceedings of ACM SIGGRAPH 1999 (ACM, Addison-Wesley, 1999), pp. 101-110.
  85. X. D. He, P. O. Heynen, R. L. Phillips, K. E. Torrance, D. H. Salesin, and D. P. Greenberg, "A fast and accurate light reflection model," in Proceedings of ACM SIGGRAPH 1992 (ACM, 1992), pp. 253-254.
  86. C. L. Vernold and J. E. Harvey, "A modified Beckmann-Kirchoff scattering theory for nonparaxial angles," in Scattering and Surface Roughness II, Z. H. Gu and A. A. Maradudin, eds., Proc. SPIE 3426, 51-56 (1998).
  87. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, "Diffracted radiance: a fundamental quantity in a nonparaxial scalar diffraction theory," Appl. Opt. 38, 6469-6481 (1999). [CrossRef]
  88. H. Ragheb and E. R. Hancock, "Estimating surface characteristics using physical reflectance models," in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR, 2003), pp. 177-184.
  89. M. Ashikhmin, S. Premoze, and P. Shirley, "A microfacet-based BRDF generator," in Proceedings of ACM SIGGRAPH 2000 (ACM, Addison-Wesley, 2000), pp. 65-74.
  90. B. van Ginneken, M. Stavridi, and J. J. Koenderink, "Diffuse and specular reflectance from rough surfaces," Appl. Opt. 37130-139 (1998). [CrossRef]
  91. C. Kelemen and L. Szirmay-Kalos, "A microfact based coupled specular-matte BRDF model with importance sampling," Proceedings of the Eurographics Conference (Eurographics, 2001), pp. 25-34.
  92. B. Cabral, N. Max, and R. Springmeyer, "Bidirectional reflectance functions from surface bump maps," in Proceedings of ACM SIGGRAPH (ACM, 1987), pp. 273-281. [CrossRef]
  93. M. Oren and S. K. Nayar, "Generalization of Lambert's reflectance," in Proceedings of ACM SIGGRAPH (ACM, 1994), pp. 239-246. [CrossRef]
  94. J. E. Freund, Mathematical Statistics (Prentice Hall, 1962).
  95. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, 1960).
  96. J. W. Goodman, Statistical Optics (Wiley, 1984).
  97. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).
  98. J. W. Harris and H. Stocker, Handbook of Mathematics and Computational Science (Springer, 1998). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited