Abstract

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.

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References

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  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).

2011 (1)

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]

2007 (1)

S. Liang, “Recent developments in estimating land surface biogeophysical variables from optical remote sensing,” Prog. Phys. Geogr. 31, 501516 (2007).
[CrossRef]

2006 (1)

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]

2000 (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]

1994 (1)

R. R. Lewis, “Making shaders more physically plausible,” Comput. Graph. Forum 13, 109–120 (1994).
[CrossRef]

1975 (1)

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

Ballard, J. R.

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.

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.

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.

Berry, T.

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.

Bruegge, C. J.

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]

Carrrillo, A.

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.

Cummins, C. L.

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]

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.

Dangel, S.

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]

Durst, P. J.

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]

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.

Eslinger, O. J.

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.

Fairley, J. R.

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.

Farthing, M. W.

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.

Gates, B. Q.

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]

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.

George, T. R.

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]

Ginsberg, I. W.

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).

Goodin, C.

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]

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.

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.

Hensley, J. L.

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.

Hines, A. M.

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.

Howington, S. E.

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.

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.

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.

Hsia, J. J.

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).

Kala, R.

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.

Kees, C.

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.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover, 2000).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover, 2000).

Lafortune, E. P.

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).

Lewis, R. R.

R. R. Lewis, “Making shaders more physically plausible,” Comput. Graph. Forum 13, 109–120 (1994).
[CrossRef]

Li, H.

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.

Liang, S.

S. Liang, “Recent developments in estimating land surface biogeophysical variables from optical remote sensing,” Prog. Phys. Geogr. 31, 501516 (2007).
[CrossRef]

Limperis, T.

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).

Liu, L. Y.

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.

Lynch, L.

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.

Lynch, L. N.

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.

Martonchik, J. V.

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]

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]

Nicodemus, F. E.

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).

Painter, T. H.

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]

Peters, J. F.

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.

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.

Phong, B. T.

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

Priddy, J. D.

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]

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.

Richmond, J. C.

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).

Schaepman, M. E.

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]

Schaepman-Strub, G.

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]

Strahler, A. H.

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]

Torrance, K. E.

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.

Westin, S. H.

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.

Willens, Y. D.

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).

Commun. ACM (1)

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

Comput. Graph. Forum (1)

R. R. Lewis, “Making shaders more physically plausible,” Comput. Graph. Forum 13, 109–120 (1994).
[CrossRef]

Proc. SPIE (1)

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]

Prog. Phys. Geogr. (1)

S. Liang, “Recent developments in estimating land surface biogeophysical variables from optical remote sensing,” Prog. Phys. Geogr. 31, 501516 (2007).
[CrossRef]

Remote Sens. Environ. (1)

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]

Remote Sens. Rev. (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]

Other (9)

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.

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.

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.

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.

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.

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.

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).

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).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover, 2000).

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Figures (9)

Fig. 1.
Fig. 1.

Example of the result of a physics-based sensor simulation. The surface is composed of triangles with reflectance parameterized by the cosine lobe BRDF. Sparse vegetation and the resulting shadows on the surface are accurately rendered using ray tracing.

Fig. 2.
Fig. 2.

Definition of the geometry convention used in this work. The incident direction is defined to be ϕ=180°.

Fig. 3.
Fig. 3.

Comparison of the numerical solution for the black-sky albedo over the hemisphere to the analytic solution over the quadrisphere for n=4.

Fig. 4.
Fig. 4.

Plot of the δ function defined in Eq. (19) for several values of n.

Fig. 5.
Fig. 5.

Comparison of black-sky albedo found from numerical integration and analytical solution for n=2.

Fig. 6.
Fig. 6.

Comparison of black-sky albedo found from numerical integration and analytical solution for n=4.

Fig. 7.
Fig. 7.

Comparison of black-sky albedo found from numerical integration and analytical solution for n=11.

Fig. 8.
Fig. 8.

Comparison of black-sky albedo found from numerical integration and analytical solution for n=100.

Fig. 9.
Fig. 9.

Comparison of white-sky albedo found from numerical integration and analytic solution in Eq. (25) as a function of n.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

fr(θi,ϕi;θr,ϕr)=dLr(θi,ϕi;θr,ϕr)dEi(θi,ϕi)[sr1],
fr(θi,θr,ϕ)=dLr(θi,θr,ϕ)dEi(θi)[sr1],
For allθi,ππ0π2fr(θi,θr,ϕ)sinθrcosθrdθrdϕ1.
v⃗i=[sinθi,0,cosθi],
v⃗r=[sinθrcosϕ,sinθrsinϕ,cosθr].
ρ(θi)=ππ0π2fr(θi,θr,ϕ)sinθrcosθrdθrdϕ.
α=1πππ0π2ρ(θi)cosθisinθidθidϕ.
v⃗s=[sinθi,0,cosθi].
Cs=v⃗s·v⃗r=cosθicosθr+sinθisinθrcosϕ,
fr(θi,θr,ϕ)=kdπ+frspec,
frspec={ksns+22πCsns,ForCs>00,otherwise.
ρd(θi)=kd,αd=kd.
ρ(θi)=ρd(θi)+ρn(θi),α=αd+αn,
ρnq(θi)=n+22ππ2π20π2(cosθicosθr+sinθisinθrcosϕ)nsinθrcosθrdθrdϕ.
ρnq(θi)=n+2π0π20π2k=0n(nk)(cosnkθicosnk+1θrsinkθisink+1θrcoskϕ)dθrdϕ,
η(p,q)=0π2sinpxcosqxdx=0π2sinqxcospxdx,η(1,1)=12,η(2,1)=13,η(2,2)=π16,η(m+2,n)=m+1m+n+2η(m,n),
β(n)=0π2sinnxdx=0π2cosnxdx,β(0)=π2,β(1)=1,β(n)=1·3·5(n1)2·4·6(n)π2,Forneven,β(n)=2·4·6(n1)1·3·5(n),Fornodd.
ρnq(θi)=n+2πk=0n(nk)cosnkθisinkθi0π2cosnk+1θrsink+1θrdθr0π2coskϕdϕ,=n+2πk=0n(nk)cosnkθisinkθiη(nk+1,k+1)β(nk),=n+2πk=0nAnskcosnkθisinkθi,
δn(θi)=ρn(θi)ρnq(θi)1.
[1ξln(θi+1)]γn,
γn=2β(n)+β(n).
ρn(θi)=ρnq(θi)[(1ξln(θi+1))γn+1].
ρn(θi)=n+2π[[1ξln(θi+1)]γn+1]k=0nAnkcosnkθisinkθi.
αn=n+2π2ππ0π2[(1ξln(θi+1))γn+1]k=0nAnkcosnkθisinkθisinθicosθidθidϕ.
αn=1π(1+12(n1)).

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