Abstract

We present an approach to recover scenes deteriorated by reflections off a semireflecting medium (e.g., a glass window). The method, based on imaging through a polarizer at two or more orientations, separates the reflected and transmitted scenes and determines which is which. We analyze the polarization effects, taking into account internal reflections within the medium. The scene reconstruction requires the estimation of the orientation (inclination and tilt angles) of the transparent (invisible) surface. The inclination angle is estimated by seeking the value that leads to the minimal mutual information of the estimated scenes. The limitations and the consequences of noise and angle error are discussed, including a fundamental ambiguity in the determination of the plane of incidence. Experimental results demonstrate the success of angle estimation and consequent scene separation and labeling.

© 2000 Optical Society of America

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References

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  1. W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).
  2. T. Darrell, E. Simoncelli, “Separation of transparent motion into layers using velocity-tuned mechanisms,” (Massachusetts Institute of Technology, Cambridge, Mass., 1993).
  3. H. Fujikake, K. Takizawa, T. Aida, H. Kikuchi, T. Fujii, M. Kawakita, “Electrically-controllable liquid crystal polarizing filter for eliminating reflected light,” Opt. Rev. 5, 93–98 (1998).
    [CrossRef]
  4. Y. Y. Schechner, N. Kiryati, R. Basri, “Separation of transparent layers using focus,” in Proceedings of the International Conference on Computer Vision (Narosa, Bombay, India, 1998), pp. 1061–1066.
  5. N. Ohnishi, K. Kumaki, T. Yamamura, T. Tanaka, “Separating real and virtual objects from their overlapping images,” in Proceedings of the European Conference on Computer Vision, Vol. 1065 of Lecture Notes in Computer Science, B. Buxton, R. Cipolla, eds. (Springer, New York, 1996), Vol. II, pp. 636–646.
  6. M. Oren, S. K. Nayar, “A theory of specular surface geometry,” Int. J. Comput. Vision 24, 105–124 (1997).
    [CrossRef]
  7. J. R. Bergen, P. J. Burt, R. Hingorani, S. Peleg, “A three-frame algorithm for estimating two component image motion,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 886–895 (1990).
    [CrossRef]
  8. J. Y. A. Wang, E. H. Adelson, “Representing moving images with layers,” IEEE Trans. Image Process. 3, 625–638 (1994).
    [CrossRef] [PubMed]
  9. M. Shizawa, “Direct estimation of multiple disparities for transparent multiple surfaces in binocular stereo,” in Proceedings of the International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1993), pp. 447–454.
  10. D. Weinshall, “Perception of multiple transparent planes in stereo vision,” Nature (London) 341, 737–739 (1989).
    [CrossRef]
  11. M. Shizawa, “On visual ambiguities due to transparency in motion and stereo,” in Proceedings of the European Conference on Computer Vision, Vol. 588 of Lecture Notes in Computer Science, G. Sandini, ed. (Springer-Verlag, New York, 1992), pp. 411–419.
  12. J. E. Solomon, “Polarization imaging,” Appl. Opt. 20, 1537–1544 (1981).
    [CrossRef] [PubMed]
  13. R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).
    [CrossRef]
  14. A. M. Shutov, “Videopolarimeters,” Sov. J. Opt. Technol. 60, 295–301 (1993).
  15. L. B. Wolff, “Using polarization to separate reflection components,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1989), pp. 363–369.
  16. L. B. Wolff, “Polarization camera for computer vision with a beam splitter,” J. Opt. Soc. Am. A 11, 2935–2945 (1994).
    [CrossRef]
  17. C. S. L. Chun, D. L. Fleming, W. A. Harvey, E. J. Torok, “Polarization-sensitive infrared sensor for target discrimination,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 55–62 (1997).
    [CrossRef]
  18. S. Lin, S. W. Lee, “Detection of specularity using stereo in color and polarization space,” Comput. Vision Image Understand. 65, 336–346 (1997).
    [CrossRef]
  19. S. K. Nayar, X. S. Fang, T. Boult, “Separation of reflection components using color and polarization,” Int. J. Comput. Vision 21, 163–186 (1997).
    [CrossRef]
  20. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), pp. 38–45; J. Shamir, Optical Systems and Processes (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1999), pp. 13–17.
  21. H. Farid, E. H. Adelson, “Separating reflections from images using independent components analysis,” J. Opt. Soc. Am. A 16, 2136–2145 (1999).
    [CrossRef]
  22. Y. Y. Schechner, N. Kiryati, J. Shamir, “Separation of transparent layers by polarization analysis,” in Scandinavian Conference on Image Analysis, B. K. Ersboll, P. Johansen, ed. (Pattern Recognition Society of Denmark, Lyngby, Denmark, 1999), Vol. I, pp. 235–242.
  23. Y. Y. Schechner, J. Shamir, N. Kiryati, “Vision through semi-reflecting media: polarization analysis,” Opt. Lett. 24, 1088–1090 (1999).
    [CrossRef]
  24. This is an example of the principle described in C. F. Bohren, “Maximum degree of polarization of the resultant of two partially polarized incoherent beams,” Appl. Opt. 26, 606–607 (1987): The degree of polarization of the superposition of partially polarized incoherent sources is less than (or equal to) that of the beam with the highest degree and may even be zero.
    [CrossRef] [PubMed]
  25. Mount Shuksan photograph, courtesy of Bonnie Lorimer. http://members.aol.com/tetonbon/bonzweb.htm .
  26. T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), pp. 12–21.

1999

1998

H. Fujikake, K. Takizawa, T. Aida, H. Kikuchi, T. Fujii, M. Kawakita, “Electrically-controllable liquid crystal polarizing filter for eliminating reflected light,” Opt. Rev. 5, 93–98 (1998).
[CrossRef]

1997

M. Oren, S. K. Nayar, “A theory of specular surface geometry,” Int. J. Comput. Vision 24, 105–124 (1997).
[CrossRef]

S. Lin, S. W. Lee, “Detection of specularity using stereo in color and polarization space,” Comput. Vision Image Understand. 65, 336–346 (1997).
[CrossRef]

S. K. Nayar, X. S. Fang, T. Boult, “Separation of reflection components using color and polarization,” Int. J. Comput. Vision 21, 163–186 (1997).
[CrossRef]

1994

L. B. Wolff, “Polarization camera for computer vision with a beam splitter,” J. Opt. Soc. Am. A 11, 2935–2945 (1994).
[CrossRef]

J. Y. A. Wang, E. H. Adelson, “Representing moving images with layers,” IEEE Trans. Image Process. 3, 625–638 (1994).
[CrossRef] [PubMed]

1993

A. M. Shutov, “Videopolarimeters,” Sov. J. Opt. Technol. 60, 295–301 (1993).

1990

J. R. Bergen, P. J. Burt, R. Hingorani, S. Peleg, “A three-frame algorithm for estimating two component image motion,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 886–895 (1990).
[CrossRef]

1989

D. Weinshall, “Perception of multiple transparent planes in stereo vision,” Nature (London) 341, 737–739 (1989).
[CrossRef]

1987

1981

J. E. Solomon, “Polarization imaging,” Appl. Opt. 20, 1537–1544 (1981).
[CrossRef] [PubMed]

R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).
[CrossRef]

Adelson, E. H.

H. Farid, E. H. Adelson, “Separating reflections from images using independent components analysis,” J. Opt. Soc. Am. A 16, 2136–2145 (1999).
[CrossRef]

J. Y. A. Wang, E. H. Adelson, “Representing moving images with layers,” IEEE Trans. Image Process. 3, 625–638 (1994).
[CrossRef] [PubMed]

Aida, T.

H. Fujikake, K. Takizawa, T. Aida, H. Kikuchi, T. Fujii, M. Kawakita, “Electrically-controllable liquid crystal polarizing filter for eliminating reflected light,” Opt. Rev. 5, 93–98 (1998).
[CrossRef]

Ballard, S. S.

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).

Basri, R.

Y. Y. Schechner, N. Kiryati, R. Basri, “Separation of transparent layers using focus,” in Proceedings of the International Conference on Computer Vision (Narosa, Bombay, India, 1998), pp. 1061–1066.

Bergen, J. R.

J. R. Bergen, P. J. Burt, R. Hingorani, S. Peleg, “A three-frame algorithm for estimating two component image motion,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 886–895 (1990).
[CrossRef]

Bohren, C. F.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), pp. 38–45; J. Shamir, Optical Systems and Processes (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1999), pp. 13–17.

Boult, T.

S. K. Nayar, X. S. Fang, T. Boult, “Separation of reflection components using color and polarization,” Int. J. Comput. Vision 21, 163–186 (1997).
[CrossRef]

Burt, P. J.

J. R. Bergen, P. J. Burt, R. Hingorani, S. Peleg, “A three-frame algorithm for estimating two component image motion,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 886–895 (1990).
[CrossRef]

Chun, C. S. L.

C. S. L. Chun, D. L. Fleming, W. A. Harvey, E. J. Torok, “Polarization-sensitive infrared sensor for target discrimination,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 55–62 (1997).
[CrossRef]

Cover, T. M.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), pp. 12–21.

Darrell, T.

T. Darrell, E. Simoncelli, “Separation of transparent motion into layers using velocity-tuned mechanisms,” (Massachusetts Institute of Technology, Cambridge, Mass., 1993).

Fang, X. S.

S. K. Nayar, X. S. Fang, T. Boult, “Separation of reflection components using color and polarization,” Int. J. Comput. Vision 21, 163–186 (1997).
[CrossRef]

Farid, H.

Fleming, D. L.

C. S. L. Chun, D. L. Fleming, W. A. Harvey, E. J. Torok, “Polarization-sensitive infrared sensor for target discrimination,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 55–62 (1997).
[CrossRef]

Fujii, T.

H. Fujikake, K. Takizawa, T. Aida, H. Kikuchi, T. Fujii, M. Kawakita, “Electrically-controllable liquid crystal polarizing filter for eliminating reflected light,” Opt. Rev. 5, 93–98 (1998).
[CrossRef]

Fujikake, H.

H. Fujikake, K. Takizawa, T. Aida, H. Kikuchi, T. Fujii, M. Kawakita, “Electrically-controllable liquid crystal polarizing filter for eliminating reflected light,” Opt. Rev. 5, 93–98 (1998).
[CrossRef]

Harvey, W. A.

C. S. L. Chun, D. L. Fleming, W. A. Harvey, E. J. Torok, “Polarization-sensitive infrared sensor for target discrimination,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 55–62 (1997).
[CrossRef]

Hingorani, R.

J. R. Bergen, P. J. Burt, R. Hingorani, S. Peleg, “A three-frame algorithm for estimating two component image motion,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 886–895 (1990).
[CrossRef]

Kawakita, M.

H. Fujikake, K. Takizawa, T. Aida, H. Kikuchi, T. Fujii, M. Kawakita, “Electrically-controllable liquid crystal polarizing filter for eliminating reflected light,” Opt. Rev. 5, 93–98 (1998).
[CrossRef]

Kikuchi, H.

H. Fujikake, K. Takizawa, T. Aida, H. Kikuchi, T. Fujii, M. Kawakita, “Electrically-controllable liquid crystal polarizing filter for eliminating reflected light,” Opt. Rev. 5, 93–98 (1998).
[CrossRef]

Kiryati, N.

Y. Y. Schechner, J. Shamir, N. Kiryati, “Vision through semi-reflecting media: polarization analysis,” Opt. Lett. 24, 1088–1090 (1999).
[CrossRef]

Y. Y. Schechner, N. Kiryati, R. Basri, “Separation of transparent layers using focus,” in Proceedings of the International Conference on Computer Vision (Narosa, Bombay, India, 1998), pp. 1061–1066.

Y. Y. Schechner, N. Kiryati, J. Shamir, “Separation of transparent layers by polarization analysis,” in Scandinavian Conference on Image Analysis, B. K. Ersboll, P. Johansen, ed. (Pattern Recognition Society of Denmark, Lyngby, Denmark, 1999), Vol. I, pp. 235–242.

Kumaki, K.

N. Ohnishi, K. Kumaki, T. Yamamura, T. Tanaka, “Separating real and virtual objects from their overlapping images,” in Proceedings of the European Conference on Computer Vision, Vol. 1065 of Lecture Notes in Computer Science, B. Buxton, R. Cipolla, eds. (Springer, New York, 1996), Vol. II, pp. 636–646.

Lee, S. W.

S. Lin, S. W. Lee, “Detection of specularity using stereo in color and polarization space,” Comput. Vision Image Understand. 65, 336–346 (1997).
[CrossRef]

Lin, S.

S. Lin, S. W. Lee, “Detection of specularity using stereo in color and polarization space,” Comput. Vision Image Understand. 65, 336–346 (1997).
[CrossRef]

Nayar, S. K.

M. Oren, S. K. Nayar, “A theory of specular surface geometry,” Int. J. Comput. Vision 24, 105–124 (1997).
[CrossRef]

S. K. Nayar, X. S. Fang, T. Boult, “Separation of reflection components using color and polarization,” Int. J. Comput. Vision 21, 163–186 (1997).
[CrossRef]

Ohnishi, N.

N. Ohnishi, K. Kumaki, T. Yamamura, T. Tanaka, “Separating real and virtual objects from their overlapping images,” in Proceedings of the European Conference on Computer Vision, Vol. 1065 of Lecture Notes in Computer Science, B. Buxton, R. Cipolla, eds. (Springer, New York, 1996), Vol. II, pp. 636–646.

Oren, M.

M. Oren, S. K. Nayar, “A theory of specular surface geometry,” Int. J. Comput. Vision 24, 105–124 (1997).
[CrossRef]

Peleg, S.

J. R. Bergen, P. J. Burt, R. Hingorani, S. Peleg, “A three-frame algorithm for estimating two component image motion,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 886–895 (1990).
[CrossRef]

Schechner, Y. Y.

Y. Y. Schechner, J. Shamir, N. Kiryati, “Vision through semi-reflecting media: polarization analysis,” Opt. Lett. 24, 1088–1090 (1999).
[CrossRef]

Y. Y. Schechner, N. Kiryati, R. Basri, “Separation of transparent layers using focus,” in Proceedings of the International Conference on Computer Vision (Narosa, Bombay, India, 1998), pp. 1061–1066.

Y. Y. Schechner, N. Kiryati, J. Shamir, “Separation of transparent layers by polarization analysis,” in Scandinavian Conference on Image Analysis, B. K. Ersboll, P. Johansen, ed. (Pattern Recognition Society of Denmark, Lyngby, Denmark, 1999), Vol. I, pp. 235–242.

Shamir, J.

Y. Y. Schechner, J. Shamir, N. Kiryati, “Vision through semi-reflecting media: polarization analysis,” Opt. Lett. 24, 1088–1090 (1999).
[CrossRef]

Y. Y. Schechner, N. Kiryati, J. Shamir, “Separation of transparent layers by polarization analysis,” in Scandinavian Conference on Image Analysis, B. K. Ersboll, P. Johansen, ed. (Pattern Recognition Society of Denmark, Lyngby, Denmark, 1999), Vol. I, pp. 235–242.

Shizawa, M.

M. Shizawa, “On visual ambiguities due to transparency in motion and stereo,” in Proceedings of the European Conference on Computer Vision, Vol. 588 of Lecture Notes in Computer Science, G. Sandini, ed. (Springer-Verlag, New York, 1992), pp. 411–419.

M. Shizawa, “Direct estimation of multiple disparities for transparent multiple surfaces in binocular stereo,” in Proceedings of the International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1993), pp. 447–454.

Shurcliff, W. A.

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).

Shutov, A. M.

A. M. Shutov, “Videopolarimeters,” Sov. J. Opt. Technol. 60, 295–301 (1993).

Simoncelli, E.

T. Darrell, E. Simoncelli, “Separation of transparent motion into layers using velocity-tuned mechanisms,” (Massachusetts Institute of Technology, Cambridge, Mass., 1993).

Solomon, J. E.

Takizawa, K.

H. Fujikake, K. Takizawa, T. Aida, H. Kikuchi, T. Fujii, M. Kawakita, “Electrically-controllable liquid crystal polarizing filter for eliminating reflected light,” Opt. Rev. 5, 93–98 (1998).
[CrossRef]

Tanaka, T.

N. Ohnishi, K. Kumaki, T. Yamamura, T. Tanaka, “Separating real and virtual objects from their overlapping images,” in Proceedings of the European Conference on Computer Vision, Vol. 1065 of Lecture Notes in Computer Science, B. Buxton, R. Cipolla, eds. (Springer, New York, 1996), Vol. II, pp. 636–646.

Thomas, J. A.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), pp. 12–21.

Torok, E. J.

C. S. L. Chun, D. L. Fleming, W. A. Harvey, E. J. Torok, “Polarization-sensitive infrared sensor for target discrimination,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 55–62 (1997).
[CrossRef]

Walraven, R.

R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).
[CrossRef]

Wang, J. Y. A.

J. Y. A. Wang, E. H. Adelson, “Representing moving images with layers,” IEEE Trans. Image Process. 3, 625–638 (1994).
[CrossRef] [PubMed]

Weinshall, D.

D. Weinshall, “Perception of multiple transparent planes in stereo vision,” Nature (London) 341, 737–739 (1989).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), pp. 38–45; J. Shamir, Optical Systems and Processes (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1999), pp. 13–17.

Wolff, L. B.

L. B. Wolff, “Polarization camera for computer vision with a beam splitter,” J. Opt. Soc. Am. A 11, 2935–2945 (1994).
[CrossRef]

L. B. Wolff, “Using polarization to separate reflection components,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1989), pp. 363–369.

Yamamura, T.

N. Ohnishi, K. Kumaki, T. Yamamura, T. Tanaka, “Separating real and virtual objects from their overlapping images,” in Proceedings of the European Conference on Computer Vision, Vol. 1065 of Lecture Notes in Computer Science, B. Buxton, R. Cipolla, eds. (Springer, New York, 1996), Vol. II, pp. 636–646.

Appl. Opt.

Comput. Vision Image Understand.

S. Lin, S. W. Lee, “Detection of specularity using stereo in color and polarization space,” Comput. Vision Image Understand. 65, 336–346 (1997).
[CrossRef]

IEEE Trans. Image Process.

J. Y. A. Wang, E. H. Adelson, “Representing moving images with layers,” IEEE Trans. Image Process. 3, 625–638 (1994).
[CrossRef] [PubMed]

IEEE Trans. Pattern. Anal. Mach. Intell.

J. R. Bergen, P. J. Burt, R. Hingorani, S. Peleg, “A three-frame algorithm for estimating two component image motion,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 886–895 (1990).
[CrossRef]

Int. J. Comput. Vision

M. Oren, S. K. Nayar, “A theory of specular surface geometry,” Int. J. Comput. Vision 24, 105–124 (1997).
[CrossRef]

S. K. Nayar, X. S. Fang, T. Boult, “Separation of reflection components using color and polarization,” Int. J. Comput. Vision 21, 163–186 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Nature (London)

D. Weinshall, “Perception of multiple transparent planes in stereo vision,” Nature (London) 341, 737–739 (1989).
[CrossRef]

Opt. Eng.

R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).
[CrossRef]

Opt. Lett.

Opt. Rev.

H. Fujikake, K. Takizawa, T. Aida, H. Kikuchi, T. Fujii, M. Kawakita, “Electrically-controllable liquid crystal polarizing filter for eliminating reflected light,” Opt. Rev. 5, 93–98 (1998).
[CrossRef]

Sov. J. Opt. Technol.

A. M. Shutov, “Videopolarimeters,” Sov. J. Opt. Technol. 60, 295–301 (1993).

Other

L. B. Wolff, “Using polarization to separate reflection components,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1989), pp. 363–369.

C. S. L. Chun, D. L. Fleming, W. A. Harvey, E. J. Torok, “Polarization-sensitive infrared sensor for target discrimination,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 55–62 (1997).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), pp. 38–45; J. Shamir, Optical Systems and Processes (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1999), pp. 13–17.

Y. Y. Schechner, N. Kiryati, J. Shamir, “Separation of transparent layers by polarization analysis,” in Scandinavian Conference on Image Analysis, B. K. Ersboll, P. Johansen, ed. (Pattern Recognition Society of Denmark, Lyngby, Denmark, 1999), Vol. I, pp. 235–242.

Mount Shuksan photograph, courtesy of Bonnie Lorimer. http://members.aol.com/tetonbon/bonzweb.htm .

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), pp. 12–21.

Y. Y. Schechner, N. Kiryati, R. Basri, “Separation of transparent layers using focus,” in Proceedings of the International Conference on Computer Vision (Narosa, Bombay, India, 1998), pp. 1061–1066.

N. Ohnishi, K. Kumaki, T. Yamamura, T. Tanaka, “Separating real and virtual objects from their overlapping images,” in Proceedings of the European Conference on Computer Vision, Vol. 1065 of Lecture Notes in Computer Science, B. Buxton, R. Cipolla, eds. (Springer, New York, 1996), Vol. II, pp. 636–646.

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).

T. Darrell, E. Simoncelli, “Separation of transparent motion into layers using velocity-tuned mechanisms,” (Massachusetts Institute of Technology, Cambridge, Mass., 1993).

M. Shizawa, “Direct estimation of multiple disparities for transparent multiple surfaces in binocular stereo,” in Proceedings of the International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1993), pp. 447–454.

M. Shizawa, “On visual ambiguities due to transparency in motion and stereo,” in Proceedings of the European Conference on Computer Vision, Vol. 588 of Lecture Notes in Computer Science, G. Sandini, ed. (Springer-Verlag, New York, 1992), pp. 411–419.

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Figures (10)

Fig. 1
Fig. 1

The image of a real object is partially transmitted through a transparent window inclined at an angle φ. The window also creates a virtual image by partially reflecting the image of another object. The combined scene can be viewed through a polarization analyzer (filter) at angle α. The polarization component perpendicular to the plane of incidence is best transmitted for α=θ.

Fig. 2
Fig. 2

Polarizing effects (PE’s) of reflection and transmission through a single air–glass interface as a function of the angle of incidence (AOI) (solid curves) and PE’s of reflection and transmission through a glass window (dashed curves).

Fig. 3
Fig. 3

Real layer IT (top left) and virtual layer IR (top right), and simulated images at an AOI of 80° (bottom). Although the reflected component in f (left) is weak (especially when compared with f on the right), significant cross talk of the layers exists.

Fig. 4
Fig. 4

Relative contamination of each layer, per 1° of error in the AOI, if the reflected contribution is as bright as the transmitted one (after the incidence on a glass window).

Fig. 5
Fig. 5

Mutual information (normalized) of the estimated layers as a function of the assumed AOI φ for φtrue=18°, 43°, 56°, 68°, 80°. The minimum mutual information was achieved when φ=φtrue.

Fig. 6
Fig. 6

For each φtrue there are domains (white) of assumed angles φ for which a zero of the correlation exists at a wrong angle (besides the correct one).

Fig. 7
Fig. 7

Decorrelation occurred at φ=19°, while the minimum mutual information correctly appeared at 20°. Additive noise increased the estimated angle of the first crossing (as theoretically predicted), while consecutive low-pass filtering (LPF) of the raw images recovered the noiseless results. Similar effects are observed in the minimum mutual information.

Fig. 8
Fig. 8

Combined scene, seen without an analyzer (left); although the reflected component is smaller in f, the image is still unclear (right).

Fig. 9
Fig. 9

At the estimated angle 27°, the estimated layers are decorrelated (solid curve); the mutual information of the estimated layers has a local minimum at 25.5° (dotted–dashed curve).

Fig. 10
Fig. 10

Reconstructions (top), real object photographed without the interfering glass window (bottom left), and virtual object photographed by removing the objects behind the glass window (bottom right).

Equations (30)

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

R˜=R+T2Rl=0(R2)l=21+RR.
PER(window)=R˜-R˜R˜+R˜,
PET(window)=T˜-T˜T˜+T˜,
PET(window)PER(window)=R˜avT˜av,
f(α)=f+f2+f-f2cos[2(α-θ)],
f-f=PERR˜avIR-PETT˜avIT=PERR˜av(IR-IT).
IˆT(φ)=2R˜(φ)R˜(φ)-R˜(φ)fˆ-2R˜(φ)R˜(φ)-R˜(φ)fˆ,
IˆR(φ)=2-2R˜(φ)R˜(φ)-R˜(φ)fˆ-2-2R˜(φ)R˜(φ)-R˜(φ)fˆ.
IˆT=IT+2R˜(φ)n-R˜(φ)nR˜(φ)-R˜(φ),
IˆR=IR+2T˜(φ)n-T˜(φ)nT˜(φ)-T˜(φ).
σT2=2σ21PER2(φ)+1,σR2=2σ21PET2(φ)+1.
IˆT(φ)=(1-ρ)IT+ρIR,
IˆR(φ)=(1-τ)IR+τIT,
ρ(φtrue, φ)=R˜(φ)R˜(φtrue)-R˜(φtrue)R˜(φ)R˜(φ)-R˜(φ),
τ(φtrue, φ)=T˜(φ)T˜(φtrue)-T˜(φtrue)T˜(φ)T˜(φ)-T˜(φ).
IˆT(without analyzer)=T˜avIT(1+contaminationT).
IˆR(withoutanalyzer)=R˜avIR(1+contaminationR).
IˆT(φ)=(1-ρ)IT(1+cTcontaminationT),
IˆR(φ)=(1-τ)IR(1+cRcontaminationR),
cT(φ)=ρ(φ)1-ρ(φ)T˜avR˜av,cR(φ)=τ(φ)1-τ(φ)R˜avT˜av.
dcTdφ=T˜avR˜av1R˜-R˜dR˜dφR˜-dR˜dφR˜,
dcRdφ=R˜avT˜av1T˜-T˜dT˜dφT˜-dT˜dφT˜.
I(IˆT, IˆR)=I˜T,I˜RP(I˜T, I˜R)logP(I˜T, I˜R)P(I˜T)P(I˜R).
H(IˆT)=-I˜TP(I˜T)log P(I˜T),
In(IˆT, IˆR)=I(IˆT, IˆR)[H(IˆT)+H(IˆR)]/2,
φˆ=arg minφ In[IˆT(φ), IˆR(φ)].
Cov(IˆI, IˆR)=τ(1-ρ)Var(IR)Var(IT)Var(IR)-η,
-4σ2R˜(φ)T˜(φ)+R˜(φ)T˜(φ)[R˜(φ)-R˜(φ)]2<0.
f=C+A cos[2(α-θ)].
fˆ=C+A,fˆ=C-A.

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