OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 5 — May. 1, 2013
  • pp: 854–858

Analytic expressions for the black-sky and white-sky albedos of the cosine lobe model

Christopher Goodin  »View Author Affiliations

JOSA A, Vol. 30, Issue 5, pp. 854-858 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (406 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The cosine lobe model is a bidirectional reflectance distribution function (BRDF) that is commonly used in computer graphics to model specular reflections. The model is both simple and physically plausible, but physical quantities such as albedo have not been related to the parameterization of the model. In this paper, analytic expressions for calculating the black-sky and white-sky albedos from the cosine lobe BRDF model with integer exponents will be derived, to the author’s knowledge for the first time. These expressions for albedo can be used to place constraints on physics-based simulations of radiative transfer such as high-fidelity ray-tracing simulations.

OCIS Codes
(160.4760) Materials : Optical properties
(290.1483) Scattering : BSDF, BRDF, and BTDF
(280.1350) Remote sensing and sensors : Backscattering

ToC Category:

Original Manuscript: September 7, 2012
Revised Manuscript: March 12, 2013
Manuscript Accepted: March 15, 2013
Published: April 10, 2013

Christopher Goodin, "Analytic expressions for the black-sky and white-sky albedos of the cosine lobe model," J. Opt. Soc. Am. A 30, 854-858 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. V. Martonchik, C. J. Bruegge, and A. H. Strahler, “A review of reflectance nomenclature used in remote sensing,” Remote Sens. Rev. 19, 9–20 (2000). [CrossRef]
  2. S. H. Westin, H. Li, and K. E. Torrance, “A comparison of four BRDF models,” in Proceedings of the Eurographics Symposium on Rendering (European Association of Computer Graphics, 2004), pp. 1–10.
  3. S. Liang, “Recent developments in estimating land surface biogeophysical variables from optical remote sensing,” Prog. Phys. Geogr. 31, 501516 (2007). [CrossRef]
  4. C. Goodin, R. Kala, A. Carrrillo, and L. Y. Liu, “Sensor modeling for the virtual autonomous navigation environment,” in Sensors, 2009 (IEEE, 2009), pp. 1588–1592.
  5. C. Goodin, P. J. Durst, B. Q. Gates, C. L. Cummins, and J. D. Priddy, “High fidelity sensor simulations for the virtual autonomous navigation environment,” in Simulation, Modeling, and Programming for Autonomous Robots (Springer-Verlag, 2010), pp. 75–86.
  6. C. Goodin, B. Q. Gates, C. L. Cummins, T. R. George, P. J. Durst, and J. D. Priddy, “High-fidelity physics-based simulation of a UGV reconnaissance mission in a complex urban environment,” Proc. SPIE 8045, 80450X (2011). [CrossRef]
  7. S. E. Howington, J. F. Peters, J. R. Ballard, T. Berry, L. Lynch, and C. Kees, “A suite of models for producing synthetic, small-scale thermal imagery of vegetated soil surfaces,” in Proceedings of CMWR XVI, International Conference on Computational Methods in Water Resources [Technical University of Denmark (DTU), 2006], 19–22.
  8. J. F. Peters, J. R. Ballard, S. E. Howington, and L. N. Lynch, “Signature evaluation for thermal infrared countermine and IED detection systems,” in Proceedings of the 2009 High-Performance Computing Users Group Conference (IEEE, 2007), pp. 238–246.
  9. S. E. Howington, O. J. Eslinger, J. L. Hensley, A. M. Hines, J. R. Ballard, M. W. Farthing, and J. R. Fairley, “The role of hydrogeology in remote sensing for threat detection,” in Proceedings of the 27th Army Science Conference Vol. 29 (U.S. Army Office of the Assistant Secretary for Acquisition, Logistics, and Technology, 2010), pp. 1–6.
  10. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Technical report 160 (National Bureau of Standards, 1977).
  11. R. R. Lewis, “Making shaders more physically plausible,” Comput. Graph. Forum 13, 109–120 (1994). [CrossRef]
  12. G. Schaepman-Strub, M. E. Schaepman, T. H. Painter, S. Dangel, and J. V. Martonchik, “Reflectance quantities in optical remote sensingdefinitions and case studies,” Remote Sens. Environ. 103, 27–42 (2006). [CrossRef]
  13. B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975). [CrossRef]
  14. E. P. Lafortune and Y. D. Willens, “Using the modified Phong reflectance model for physically based rendering,” Technical report CW197 (Department of Computing Science, K. U. Leuven, 1994).
  15. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover, 2000).

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