Abstract

A polarimetric bidirectional reflectance distribution function (pBRDF), based on geometrical optics, is presented. The pBRDF incorporates a visibility (shadowing/masking) function and a Lambertian (diffuse) component which distinguishes it from other geometrical optics pBRDFs in literature. It is shown that these additions keep the pBRDF bounded (and thus a more realistic physical model) as the angle of incidence or observation approaches grazing and better able to model the behavior of light scattered from rough, reflective surfaces. In this paper, the theoretical development of the pBRDF is shown and discussed. Simulation results of a rough, perfect reflecting surface obtained using an exact, electromagnetic solution and experimental Mueller matrix results of two, rough metallic samples are presented to validate the pBRDF.

© 2009 Optical Society of America

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References

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    [Crossref]
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2008 (1)

G. Zonios, I. Bassukas, and A. Dimou, “Comparative evaluation of two simple diffuse reflectance models for biological tissue applications,” Appl. Opt. 47(27), 4965–4973 ( 2008).
[Crossref]

2007 (4)

J. Xia and G. Yao, “Angular distribution of diffuse reflectance in biological tissue,” Appl. Opt. 46(26), 6552–6560 ( 2007).
[Crossref]

M. A. Greiner, B. D. Duncan, and M. P. Dierking, “Bidirectional scattering distribution functions of maple and cottonwood leaves,” Appl. Opt. 46(25), 6485–6494 ( 2007).
[Crossref]

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 ( 2007).
[Crossref]

Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A 24(3), 724–744 ( 2007).
[Crossref]

2006 (1)

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8(10), 914–925 ( 2006).
[Crossref]

2002 (1)

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]

1999 (1)

E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 ( 1999).
[Crossref]

1996 (1)

K. K. Ellis, “Polarimetric bidirectional reflectance distribution function of glossy coatings,” J. Opt. Soc. Am. A 13(8), 1758–1762 ( 1996).
[Crossref]

1995 (1)

D. S. Flynn and C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function,” Opt. Eng. 34(6), 1646–1650 ( 1995).
[Crossref]

1991 (1)

R. Anderson, “Matrix description of radiometric quantities,” Appl. Opt. 30(7), 858–867 ( 1991).
[Crossref]

1990 (1)

M. F. Chen and S. Y. Bai, “Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case,” J. Electromagn. Waves Appl. 4(10), 963–982 ( 1990).
[Crossref]

1988 (1)

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 ( 1988).
[Crossref]

1985 (2)

A. K. Fung and M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2(12), 2274–2284 ( 1985).
[Crossref]

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53(5), 468–478 ( 1985).
[Crossref]

1984 (1)

D. S. Kimes, “Modeling the directional reflectance from complete homogeneous vegetation canopies with various leaf-orientation distributions,” J. Opt. Soc. Am. A 1(7), 725–737 ( 1984).
[Crossref]

1978 (1)

R. M. Axline and A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. AP-26(3), 482–488 ( 1978).
[Crossref]

1975 (1)

E. Y. Harper and F. M. Labianca, “Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean,” J. Acoust. Soc. Am. 58(2), 349–364 ( 1975).
[Crossref]

1971 (2)

K. Krishen, “Correlation of radar backscattering cross sections with ocean wave height and wind velocity,” J. Geophys. Res. 76, 6528–6539 ( 1971).
[Crossref]

D. E. Barrick, “Theory of HF and VHF propagation across the rough sea—parts I and II,” Radio Sci. 6, 517–533 ( 1971).
[Crossref]

1967 (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]

1965 (1)

F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4(7), 368–377 ( 1965).

1963 (2)

F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368–377 ( 1963).
[Crossref]

B. W. Hapke, “A theoretical photometric function for the lunar surface,” J. Geophys. Res. 68(15), 4571–4586 ( 1963).

1953 (1)

C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. 25, 566–570 ( 1953).
[Crossref]

Alexander, C.

D. S. Flynn and C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function,” Opt. Eng. 34(6), 1646–1650 ( 1995).
[Crossref]

An, C.-H.

C.-H. An and K. J. Zeringue, “Polarization scattering from rough surfaces based on the vector Kirchoff diffraction model,” in Proc. SPIE, vol. 5158, pp. 205–216 (The International Society for Optical Engineering (SPIE), 2003).

Anderson, R.

R. Anderson, “Matrix description of radiometric quantities,” Appl. Opt. 30(7), 858–867 ( 1991).
[Crossref]

Arko, S. A.

D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).

Axline, R. M.

R. M. Axline and A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. AP-26(3), 482–488 ( 1978).
[Crossref]

Bai, S. Y.

M. F. Chen and S. Y. Bai, “Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case,” J. Electromagn. Waves Appl. 4(10), 963–982 ( 1990).
[Crossref]

Bailey, W. M.

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53(5), 468–478 ( 1985).
[Crossref]

Barrick, D. E.

D. E. Barrick, “Theory of HF and VHF propagation across the rough sea—parts I and II,” Radio Sci. 6, 517–533 ( 1971).
[Crossref]

Barta, A.

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 ( 2007).
[Crossref]

Bassukas, I.

G. Zonios, I. Bassukas, and A. Dimou, “Comparative evaluation of two simple diffuse reflectance models for biological tissue applications,” Appl. Opt. 47(27), 4965–4973 ( 2008).
[Crossref]

Beard, J.

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional Reflectance Model Validation and Utilization,” Tech. Rep. AFAL-TR-73-303, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, OH ( 1973).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Inc., Norwood, MA, 1963).

Bernáth, B.

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 ( 2007).
[Crossref]

Bickel, W. S.

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53(5), 468–478 ( 1985).
[Crossref]

Bishop, K. P.

D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).

M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, “Results of a new polarization simulation,” in Proc. SPIE, vol. 4481, pp. 149–162 (The International Society for Optical Engineering (SPIE), 2002).

Blinn, J. F.

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” in SIGGRAPH 1977 Proceedings, vol. 11, pp. 192–198, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1977).

Boger, J.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8(10), 914–925 ( 2006).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, New York, NY, 1999).

Bowers, D.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8(10), 914–925 ( 2006).
[Crossref]

Brown, S. D.

M. G. Gartley, S. D. Brown, and J. R. Schott, “Micro-scale surface and contaminate modeling for polarimetric signature prediction,” in Proc. SPIE, vol. 6972 (The International Society for Optical Engineering (SPIE), 2008).
[Crossref]

Caudill, T. R.

D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).

M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, “Results of a new polarization simulation,” in Proc. SPIE, vol. 4481, pp. 149–162 (The International Society for Optical Engineering (SPIE), 2002).

Chen, M. F.

M. F. Chen and S. Y. Bai, “Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case,” J. Electromagn. Waves Appl. 4(10), 963–982 ( 1990).
[Crossref]

A. K. Fung and M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2(12), 2274–2284 ( 1985).
[Crossref]

Compain, E.

E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 ( 1999).
[Crossref]

Cook, R. L.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in SIGGRAPH 1981 Proceedings, vol. 15, pp. 307–316, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1981).

Davis, M. L.

M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, “Results of a new polarization simulation,” in Proc. SPIE, vol. 4481, pp. 149–162 (The International Society for Optical Engineering (SPIE), 2002).

Dierking, M. P.

M. A. Greiner, B. D. Duncan, and M. P. Dierking, “Bidirectional scattering distribution functions of maple and cottonwood leaves,” Appl. Opt. 46(25), 6485–6494 ( 2007).
[Crossref]

Dimou, A.

G. Zonios, I. Bassukas, and A. Dimou, “Comparative evaluation of two simple diffuse reflectance models for biological tissue applications,” Appl. Opt. 47(27), 4965–4973 ( 2008).
[Crossref]

Drevillon, B.

E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 ( 1999).
[Crossref]

Duncan, B. D.

M. A. Greiner, B. D. Duncan, and M. P. Dierking, “Bidirectional scattering distribution functions of maple and cottonwood leaves,” Appl. Opt. 46(25), 6485–6494 ( 2007).
[Crossref]

Eckart, C.

C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. 25, 566–570 ( 1953).
[Crossref]

Ellis, K. K.

K. K. Ellis, “Polarimetric bidirectional reflectance distribution function of glossy coatings,” J. Opt. Soc. Am. A 13(8), 1758–1762 ( 1996).
[Crossref]

Fetrow, M.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8(10), 914–925 ( 2006).
[Crossref]

Fetrow, M. P.

D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).

M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, “Results of a new polarization simulation,” in Proc. SPIE, vol. 4481, pp. 149–162 (The International Society for Optical Engineering (SPIE), 2002).

Flynn, D. S.

D. S. Flynn and C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function,” Opt. Eng. 34(6), 1646–1650 ( 1995).
[Crossref]

Fung, A. K.

A. K. Fung and M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2(12), 2274–2284 ( 1985).
[Crossref]

R. M. Axline and A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. AP-26(3), 482–488 ( 1978).
[Crossref]

Gartley, M. G.

M. G. Gartley, “Polarimetric Modeling of Remotely Sensed Scenes in the Thermal Infrared,” Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY ( 2007).

M. G. Gartley, S. D. Brown, and J. R. Schott, “Micro-scale surface and contaminate modeling for polarimetric signature prediction,” in Proc. SPIE, vol. 6972 (The International Society for Optical Engineering (SPIE), 2008).
[Crossref]

Germer, T. A.

R. G. Priest and T. A. Germer, “Polarimetric BRDF in the microfacet model: theory and measurements,” in Proceedings of the 2000 Meeting of the Military Sensing Symposia Specialty Group on Passive Sensors, pp. 169–181 (Infrared Information Analysis Center, 2000).

Greenberg, D. P.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH 1991 Proceedings, vol. 25, pp. 175–186, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1991).

Greiner, M. A.

M. A. Greiner, B. D. Duncan, and M. P. Dierking, “Bidirectional scattering distribution functions of maple and cottonwood leaves,” Appl. Opt. 46(25), 6485–6494 ( 2007).
[Crossref]

Hapke, B. W.

B. W. Hapke, “A theoretical photometric function for the lunar surface,” J. Geophys. Res. 68(15), 4571–4586 ( 1963).

Harper, E. Y.

E. Y. Harper and F. M. Labianca, “Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean,” J. Acoust. Soc. Am. 58(2), 349–364 ( 1975).
[Crossref]

He, X. D.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH 1991 Proceedings, vol. 25, pp. 175–186, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1991).

Hegedüs, R.

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 ( 2007).
[Crossref]

Horváth, G.

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 ( 2007).
[Crossref]

Irene, E. A.

H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, Inc., Norwich, NY, 2005).
[Crossref]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, NY, 1997).

Kimes, D. S.

D. S. Kimes, “Modeling the directional reflectance from complete homogeneous vegetation canopies with various leaf-orientation distributions,” J. Opt. Soc. Am. A 1(7), 725–737 ( 1984).
[Crossref]

Krishen, K.

K. Krishen, “Correlation of radar backscattering cross sections with ocean wave height and wind velocity,” J. Geophys. Res. 76, 6528–6539 ( 1971).
[Crossref]

Labianca, F. M.

E. Y. Harper and F. M. Labianca, “Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean,” J. Acoust. Soc. Am. 58(2), 349–364 ( 1975).
[Crossref]

Ladd, D.

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional Reflectance Model Validation and Utilization,” Tech. Rep. AFAL-TR-73-303, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, OH ( 1973).

Ladd, S.

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional Reflectance Model Validation and Utilization,” Tech. Rep. AFAL-TR-73-303, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, OH ( 1973).

Malherbe, J. A. G.

D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, Inc., Norwood, MA, 1990).

Maxwell, J. R.

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional Reflectance Model Validation and Utilization,” Tech. Rep. AFAL-TR-73-303, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, OH ( 1973).

McNamara, D. A.

D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, Inc., Norwood, MA, 1990).

Meier, S. R.

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]

Meyer-Rochow, V. B.

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 ( 2007).
[Crossref]

Mittra, R.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE Press, New York, NY, 1998).

Nicodemus, F. E.

F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4(7), 368–377 ( 1965).

F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368–377 ( 1963).
[Crossref]

Ortega, S.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8(10), 914–925 ( 2006).
[Crossref]

Peterson, A. F.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE Press, New York, NY, 1998).

Pistorius, C. W. I.

D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, Inc., Norwood, MA, 1990).

Poirier, S.

E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 ( 1999).
[Crossref]

Priest, R. G.

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]

R. G. Priest and T. A. Germer, “Polarimetric BRDF in the microfacet model: theory and measurements,” in Proceedings of the 2000 Meeting of the Military Sensing Symposia Specialty Group on Passive Sensors, pp. 169–181 (Infrared Information Analysis Center, 2000).

Ray, S. L.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE Press, New York, NY, 1998).

Robertson, D. C.

B. P. Sandford and D. C. Robertson, “Infrared reflectance properties of aircraft paints,” in Proceedings of IRIS Targets, Backgrounds and Discrimination ( 1985).

Sandford, B. P.

B. P. Sandford and D. C. Robertson, “Infrared reflectance properties of aircraft paints,” in Proceedings of IRIS Targets, Backgrounds and Discrimination ( 1985).

Schott, J. R.

J. R. Schott, Fundamentals of Polarimetric Remote Sensing (SPIE Press, Bellingham, WA, 2009).
[Crossref]

M. G. Gartley, S. D. Brown, and J. R. Schott, “Micro-scale surface and contaminate modeling for polarimetric signature prediction,” in Proc. SPIE, vol. 6972 (The International Society for Optical Engineering (SPIE), 2008).
[Crossref]

Serna, M.

D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).

Shell, J. R.

J. R. Shell, “Polarimetric Remote Sensing in the Visible to Near Infrared,” Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY ( 2005).

Sillion, F. X.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH 1991 Proceedings, vol. 25, pp. 175–186, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1991).

Simrell, E. R.

M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, “Results of a new polarization simulation,” in Proc. SPIE, vol. 4481, pp. 149–162 (The International Society for Optical Engineering (SPIE), 2002).

Sparrow, E. M.

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]

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Inc., Norwood, MA, 1963).

Sposato, S. H.

M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, “Results of a new polarization simulation,” in Proc. SPIE, vol. 4481, pp. 149–162 (The International Society for Optical Engineering (SPIE), 2002).

D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).

Sun, Y.

Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A 24(3), 724–744 ( 2007).
[Crossref]

Thorsos, E. I.

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 ( 1988).
[Crossref]

Tompkins, H. G.

H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, Inc., Norwich, NY, 2005).
[Crossref]

Torrance, K. E.

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]

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH 1991 Proceedings, vol. 25, pp. 175–186, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1991).

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in SIGGRAPH 1981 Proceedings, vol. 15, pp. 307–316, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1981).

Ufimtsev, P. Y.

P. Y. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (John Wiley & Sons, Inc., Hoboken, NJ, 2007).
[Crossref]

Weiner, S.

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional Reflectance Model Validation and Utilization,” Tech. Rep. AFAL-TR-73-303, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, OH ( 1973).

Wellems, D.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8(10), 914–925 ( 2006).
[Crossref]

D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).

M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, “Results of a new polarization simulation,” in Proc. SPIE, vol. 4481, pp. 149–162 (The International Society for Optical Engineering (SPIE), 2002).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, New York, NY, 1999).

Xia, J.

J. Xia and G. Yao, “Angular distribution of diffuse reflectance in biological tissue,” Appl. Opt. 46(26), 6552–6560 ( 2007).
[Crossref]

Yao, G.

J. Xia and G. Yao, “Angular distribution of diffuse reflectance in biological tissue,” Appl. Opt. 46(26), 6552–6560 ( 2007).
[Crossref]

Zeringue, K. J.

C.-H. An and K. J. Zeringue, “Polarization scattering from rough surfaces based on the vector Kirchoff diffraction model,” in Proc. SPIE, vol. 5158, pp. 205–216 (The International Society for Optical Engineering (SPIE), 2003).

Zonios, G.

G. Zonios, I. Bassukas, and A. Dimou, “Comparative evaluation of two simple diffuse reflectance models for biological tissue applications,” Appl. Opt. 47(27), 4965–4973 ( 2008).
[Crossref]

Am. J. Phys. (2)

F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368–377 ( 1963).
[Crossref]

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53(5), 468–478 ( 1985).
[Crossref]

Appl. Opt. (7)

F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4(7), 368–377 ( 1965).

R. Anderson, “Matrix description of radiometric quantities,” Appl. Opt. 30(7), 858–867 ( 1991).
[Crossref]

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 ( 2007).
[Crossref]

G. Zonios, I. Bassukas, and A. Dimou, “Comparative evaluation of two simple diffuse reflectance models for biological tissue applications,” Appl. Opt. 47(27), 4965–4973 ( 2008).
[Crossref]

J. Xia and G. Yao, “Angular distribution of diffuse reflectance in biological tissue,” Appl. Opt. 46(26), 6552–6560 ( 2007).
[Crossref]

M. A. Greiner, B. D. Duncan, and M. P. Dierking, “Bidirectional scattering distribution functions of maple and cottonwood leaves,” Appl. Opt. 46(25), 6485–6494 ( 2007).
[Crossref]

E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 ( 1999).
[Crossref]

IEEE Trans. Antennas Propag. (1)

R. M. Axline and A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. AP-26(3), 482–488 ( 1978).
[Crossref]

J. Acoust. Soc. Am. (3)

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 ( 1988).
[Crossref]

C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. 25, 566–570 ( 1953).
[Crossref]

E. Y. Harper and F. M. Labianca, “Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean,” J. Acoust. Soc. Am. 58(2), 349–364 ( 1975).
[Crossref]

J. Electromagn. Waves Appl. (1)

M. F. Chen and S. Y. Bai, “Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case,” J. Electromagn. Waves Appl. 4(10), 963–982 ( 1990).
[Crossref]

J. Geophys. Res. (2)

K. Krishen, “Correlation of radar backscattering cross sections with ocean wave height and wind velocity,” J. Geophys. Res. 76, 6528–6539 ( 1971).
[Crossref]

B. W. Hapke, “A theoretical photometric function for the lunar surface,” J. Geophys. Res. 68(15), 4571–4586 ( 1963).

J. Opt. A: Pure Appl. Opt. (1)

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8(10), 914–925 ( 2006).
[Crossref]

J. Opt. Soc. Am. (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]

J. Opt. Soc. Am. A (4)

D. S. Kimes, “Modeling the directional reflectance from complete homogeneous vegetation canopies with various leaf-orientation distributions,” J. Opt. Soc. Am. A 1(7), 725–737 ( 1984).
[Crossref]

K. K. Ellis, “Polarimetric bidirectional reflectance distribution function of glossy coatings,” J. Opt. Soc. Am. A 13(8), 1758–1762 ( 1996).
[Crossref]

Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A 24(3), 724–744 ( 2007).
[Crossref]

A. K. Fung and M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2(12), 2274–2284 ( 1985).
[Crossref]

Opt. Eng. (2)

D. S. Flynn and C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function,” Opt. Eng. 34(6), 1646–1650 ( 1995).
[Crossref]

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]

Radio Sci. (1)

D. E. Barrick, “Theory of HF and VHF propagation across the rough sea—parts I and II,” Radio Sci. 6, 517–533 ( 1971).
[Crossref]

Other (22)

D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).

H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, Inc., Norwich, NY, 2005).
[Crossref]

J. R. Schott, Fundamentals of Polarimetric Remote Sensing (SPIE Press, Bellingham, WA, 2009).
[Crossref]

J. R. Shell, “Polarimetric Remote Sensing in the Visible to Near Infrared,” Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY ( 2005).

M. G. Gartley, S. D. Brown, and J. R. Schott, “Micro-scale surface and contaminate modeling for polarimetric signature prediction,” in Proc. SPIE, vol. 6972 (The International Society for Optical Engineering (SPIE), 2008).
[Crossref]

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional Reflectance Model Validation and Utilization,” Tech. Rep. AFAL-TR-73-303, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, OH ( 1973).

M. G. Gartley, “Polarimetric Modeling of Remotely Sensed Scenes in the Thermal Infrared,” Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY ( 2007).

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE Press, New York, NY, 1998).

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” in SIGGRAPH 1977 Proceedings, vol. 11, pp. 192–198, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1977).

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in SIGGRAPH 1981 Proceedings, vol. 15, pp. 307–316, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1981).

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH 1991 Proceedings, vol. 25, pp. 175–186, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1991).

B. P. Sandford and D. C. Robertson, “Infrared reflectance properties of aircraft paints,” in Proceedings of IRIS Targets, Backgrounds and Discrimination ( 1985).

M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, “Results of a new polarization simulation,” in Proc. SPIE, vol. 4481, pp. 149–162 (The International Society for Optical Engineering (SPIE), 2002).

R. G. Priest and T. A. Germer, “Polarimetric BRDF in the microfacet model: theory and measurements,” in Proceedings of the 2000 Meeting of the Military Sensing Symposia Specialty Group on Passive Sensors, pp. 169–181 (Infrared Information Analysis Center, 2000).

P. Y. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (John Wiley & Sons, Inc., Hoboken, NJ, 2007).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, New York, NY, 1999).

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Inc., Norwood, MA, 1963).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, NY, 1997).

C.-H. An and K. J. Zeringue, “Polarization scattering from rough surfaces based on the vector Kirchoff diffraction model,” in Proc. SPIE, vol. 5158, pp. 205–216 (The International Society for Optical Engineering (SPIE), 2003).

D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, Inc., Norwood, MA, 1990).

LabSphere, Inc., “A guide to reflectance coatings and materials,” http://www.labsphere.com/tecdocs.aspx.

Luxpop, Inc. http://www.luxpop.com/.

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Figures (9)

Fig. 1.
Fig. 1.

Macroscopic surface scattering geometry. Light subtending solid angle dωi is incident from the (θi ,ϕi ) direction on a small area dA of a much larger rough surface with complex index of refraction η=n-jκ. Light is scattered and observed within solid angle dωr in the (θr ,ϕr ) direction.

Fig. 2.
Fig. 2.

Scattering geometry of a single microfacet. The angle α is the polar angle from the mean surface normal to the microfacet normal n. The angle β is the incident angle onto and reflected angle from a microfacet as measured from the microfacet normal. The angle γi is the angle between the macroscopic plane of incidence and the scattering plane of the microfacet (depicted in the figure as the plane containing the vectors n and t). Likewise, the angle γr is the angle between the macroscopic plane of reflection and the scattering plane of the microfacet.

Fig. 3.
Fig. 3.

Scattering geometry of a v-shaped groove. The top subfigure depicts shadowing while the bottom subfigure depicts masking. Shadowing occurs when the angle of incidence approaches grazing. Similarly, masking occurs when the angle of observation nears grazing.

Fig. 4.
Fig. 4.

Comparisons of the F00 elements of the Priest and Germer pBRDF [23, 24] and the pBRDF in Eq. (13) for θi =45°, 60°, 75°, and 85° with 21/2σ h /=0.3. The pBRDFs are evaluated in the specular plane (ϕ=π) and using a perfect reflecting surface.

Fig. 5.
Fig. 5.

Scattering geometry of the MoM solutions. The surface is a 15,000λ long, random (surface height is Gaussian distributed) PEC surface. The surface is assumed to be invariant in the z direction.

Fig. 6.
Fig. 6.

Comparisons of the reflectance distributions predicted by MoM solutions of a 15,000λ long, random (surface height is Gaussian distributed) PEC surface with those of the pBRDF in Eq. (17) for θi =10°, 30°, 45°, 60°, and 75° and 21/2σ h /=0.3. Note that the reflectance distributions in the figure are normalized with respect to their values at the specular angles (θi =θr ). Observation for both the MoM and the pBRDF predictions is in the specular plane (ϕ=π).

Fig. 7.
Fig. 7.

Photograph of the Mueller matrix ellipsometer used in this experiment. The ellipsometer is located at the Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio.

Fig. 8.
Fig. 8.

Mueller matrix measurement results for LabSphere Infragold [46] compared to predictions made using the pBRDF. The measurement results are plotted as symbols; the pBRDF predictions are plotted as solid lines. Note that the measurements are made in the specular plane (ϕ=π). The complex index of refraction used for gold is η=0.285-j7.3523 [47] and 21/2σ h /=0.44. The plotted values for the measured Mueller matrix elements of LabSphere Infragold are the means of 256 irradiance measurements. The bars on the figure represent ±1σ, i.e., one standard deviation of those 256 measurements.

Fig. 9.
Fig. 9.

Mueller matrix measurement results for flame sprayed aluminum (FSA) compared to predictions made using the pBRDF. The measurement results are plotted as symbols; the pBRDF predictions are plotted as solid lines. Note that the measurements are made in the specular plane (ϕ=π). The complex index of refraction used for aluminum is η=1.226-j10.413 [47] and 21/2σ h /=0.43. The plotted values for the measured Mueller matrix elements of FSA are the means of 256 irradiance measurements. The bars on the figure represent ±1σ, i.e., one standard deviation of those 256 measurements.

Equations (34)

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

f ( θ i , θ r , ϕ ) = d L r ( θ r , ϕ ) d E i ( θ i ) = d L r ( θ r , ϕ ) L i ( θ i ) cos θ i d ω i
F ( θ i , θ r , ϕ ) = d L r ( θ r , ϕ ) L i ( θ i ) cos θ i d ω i .
L r ( θ r , ϕ ) = L r sin gle ( θ r , ϕ ) + L r multiple ( θ r , ϕ )
F = F sin gle + F multiple .
F s = F sin gle , F d = F multiple
F = F s + F d .
F s ( θ i , θ r , ϕ ; σ h , ; η ) = P ( α ; σ h , ) M ( β ; η ) G ( θ i , θ r , ϕ ) 4 cos θ i cos θ r cos α
cos α = ( cos θ i + cos θ r ) ( 2 cos β )
cos 2 β = cos θ i cos θ r + sin θ i sin θ r cos ϕ .
P ( α ; σ h , ) = 2 exp ( 2 tan 2 α 4 σ h 2 ) 4 π σ h 2 cos 3 α .
[ E r s E r p ] = [ cos γ r sin γ r sin γ r cos γ r ] [ r s 0 0 r p ] [ cos γ i sin γ i sin γ i cos γ i ] [ E i s E i p ]
[ E r s E r p ] = [ T ss T ps T sp T pp ] [ E i s E i p ]
cos γ i = ( cos α cos θ i cos β ) ( sin θ i sin β )
cos γ r = ( cos α cos θ r cos β ) ( sin θ r sin β ) .
M = 1 2 [ M 00 M 01 0 M 01 M 00 0 0 0 M 22 j M 23 0 0 j M 23 M 22 ]
M 00 = T ss 2 + T sp 2 + T ps 2 + T pp 2
M 01 = T ss 2 + T sp 2 T ps 2 T pp 2
M 22 = T ss T pp * + T ss * T pp + T ps T sp * + T ps * T sp
M 23 = T ps T sp * T ps * T sp T ss T pp * + T ss * T pp
G ( θ i , θ r , ϕ ) = min ( 1 ; 2 cos α cos θ r cos β ; 2 cos α cos θ i cos β ) .
F jk s ( θ i , θ r , ϕ ; σ h , ; η ) = 2 exp ( 2 tan 2 α 4 σ h 2 ) 16 π σ h 2 cos θ i cos θ r cos 4 α G ( θ i , θ r , ϕ ) M jk ( β ; η ) .
ρ DHR ( θ i ; σ h , ) = 0 2 π 0 π 2 F 00 ( θ i , θ r , ϕ ; σ h , ; η ) cos θ r sin θ r d θ r d ϕ .
1 = 0 2 π 0 π 2 F 00 s , PEC cos θ r sin θ r d θ r d ϕ + 0 2 π 0 π 2 F 00 d , PEC cos θ r sin θ r d θ r d ϕ .
F 00 d , PEC ( θ i ; σ h , ) = 1 π ( 1 0 2 π 0 π 2 F 00 s , PEC cos θ r sin θ r d θ r d ϕ ) .
F 00 d , PEC ( θ i ; σ h , ) = 1 π [ 1 ρ DHR s , PEC ( θ i ; σ h , ) ]
F 00 ( θ i , θ r , ϕ ; σ h , ; η ) = F 00 s ( θ i , θ r , ϕ ; σ h , ; η ) + 1 π [ 1 ρ DHR s , PEC ( θ i ; σ h , ) ] M 00 ( β ; η ) .
F jk ( θ i , θ r , ϕ ; σ h , ; η ) = F jk s ( θ i , θ r , ϕ ; σ h , ; η ) j , k 0
π Z 0 2 λ c J z ( ρ ) H 0 ( 2 ) ( 2 π λ ρ ρ ) d C ' = exp [ j 2 π λ ( k i · ρ ) ] ρ C
J z ( ρ ) = Σ n = 1 N α n p n ( ρ ) .
[ α 11 α 12 α 1 N α 21 α 22 α 2 N α N 1 α N 2 α NN ] [ α 1 α 2 α N ] = [ E i , 1 z E i , 2 z E i , N N ] .
E r z ( x , y ) = π Z 0 2 λ Σ n = 1 N α n C n H 0 ( 2 ) ( 2 π λ ρ ρ ) d C n lim ρ E r z ( ρ , θ r ) = Z 0 2 ρλ exp [ j ( 2 π λ ρ π 4 ) ] Σ n = 1 N α n C n exp [ j 2 π λ ( x sin θ r + y cos θ r ) ] d C n
W m ( x ) = exp [ ( x x m w ) 2 ]
σ ( θ r ) = 1 w π / 2 ( 1 M lim ρ 2 π ρ Σ m = 1 M E r , m z 2 )
M = A 1 SW 1

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