Abstract

In a scene observed from a fixed viewpoint, the set of shadow boundaries in an image changes as a point light source (nearby or at infinity) assumes different locations. We show that for any finite set of point light sources illuminating an object viewed under either orthographic or perspective projection, there is an equivalence class of object shapes having the same set of shadows. Members of this equivalence class differ by a four-parameter family of projective transformations, and the shadows of a transformed object are identical when the same transformation is applied to the light source locations. Under orthographic projection, this family is the generalized bas-relief (GBR) transformation, and we show that the GBR transformation is the only family of transformations of an object’s shape for which the complete set of imaged shadows is identical. Finally, we show that given multiple images under differing and unknown light source directions, it is possible to reconstruct both an object’s surface and the light source locations up to this family of transformations from the shadows alone.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D.L. Waltz, “Understanding line drawings of scenes with shadows,” in The Psychology of Computer Vision,P. H. Winston, ed. (McGraw-Hill, New York, 1975), pp. 19–91.
  2. S. Shafer, T. Kanade, “Using shadows in finding surface orientation,” Comput. Vision Graph. Image Process. 22, 145–176 (1983).
    [CrossRef]
  3. L. N. Hambrick, M. H. Loew, R. L. Carroll, “The entry–exit method of shadow boundary segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 597–607 (1987).
    [CrossRef]
  4. M. Hatzitheodorou, “The derivation of 3-D surface shape from shadows,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 1012–1020.
  5. J. R. Kender, E. M. Smith, “Shape from darkness,” in International Conference on Computer Vision (Morgan Kaufmann, Los Altos, Calif., 1987), pp. 539–546.
  6. D. Yang, J. R. Kender, “Shape from shadows under error,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1993), pp. 1083–1090.
  7. M. Daum, G. Dudek, “On 3-D surface reconstruction using shape from shadows,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 461–468.
  8. F. Cheng, K. H. Thiel, “Delimiting the building heights in a city from the shadow in a panchromatic spot-image. 1 test of 42 buildings,” Int. J. Remote Sens. 16, 409–415 (1995).
    [CrossRef]
  9. A. Huertas, R. Nevatia, “Detection of buildings in aerial images using shape and shadows,” in Proceedings of the International Joint Conference on Artificial Intelligence (Morgan Kaufmann, San Francisco, Calif., 1983), pp. 1099–1103.
  10. R. B. Irvin, D. M. McKeown, “Methods for exploiting the relationship between buildings and their shadows in aerial imagery,” IEEE Trans. Syst. Man Cybern. 19, 1564–1575 (1989).
    [CrossRef]
  11. G. G. Medioni, “Obtaining 3-D from shadows in aerial images,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1983), pp. 73–76.
  12. P. Belhumeur, D. Kriegman, A. Yuille, “The bas-relief ambiguity,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 1040–1046.
  13. D. Kriegman, P. Belhumeur, “What shadows reveal about object structure,” in Proceedings of the European Conference on Computer Vision (Springer, Berlin, 1998), pp. 399–414.
  14. M. S. Langer, S. W. Zucker, “What is a light source?” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 172–178.
  15. J. Mundy, A. Zisserman, Geometric Invariance in Computer Vision (MIT Press, Cambridge, Mass., 1992).
  16. M. Baxandall, Shadows and Enlightenment (Yale U. Press, New Haven, Conn., 1995).
  17. A. Shashua, “Geometry and photometry in 3D visual recognition,” Ph.D. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1992).
  18. O. Faugeras, Three Dimensional Computer Vision (MIT Press, Cambridge, Mass., 1993).
  19. A. Criminisi, I. Reid, A. Zisserman, “Duality, rigidity and planar parallax,” in Proceedings of the European Conference on Computer Vision (Springer, Berlin, 1998), Vol. II, pp. 846–861.
  20. J. J. Koenderink, A. J. Van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–385 (1991).
    [CrossRef] [PubMed]
  21. R. Rosenholtz, J. J. Koenderink, “Affine structure and photometry,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 18–20.
  22. L. S. Shapiro, A. Zisserman, M. Brady, “3D motion recovery via affine epipolar geometry,” Int. J. Comput. Vis. 16, 147–182 (1995).
    [CrossRef]
  23. S. Ullman, R. Basri, “Recognition by a linear combination of models,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 992–1006 (1991).
    [CrossRef]
  24. O. Faugeras, “Stratification of 3-D vision: projective, affine, and metric representations,” J. Opt. Soc. Am. A 12, 465–484 (1995).
    [CrossRef]
  25. C. Fermuller, Y. Aloimonos, “Ordinal representations of visual space,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Francisco, Calif., 1996), pp. 897–904.
  26. P. N. Belhumeur, D. J. Kriegman, A. L. Yuille, “The bas-relief ambiguity,” Int. J. Comput. Vis. 35, 33–44 (1999).
    [CrossRef]
  27. M. P. DoCarmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, N.J., 1976).
  28. L. Donati, N. Stolfi, “Singularities of illuminated surfaces,” Int. J. Comput. Vision 23, 207–216 (1997).
    [CrossRef]
  29. E. Artin, Geometric Algebra (Interscience, New York, 1957).
  30. P. N. Belhumeur, D. J. Kriegman, “What is the set of images of an object under all possible lighting conditions,” in Proceedings of the IEEE Conference on Computer Visionand Pattern Recognition (IEEE Computer Science Press, Los Alamitos, Calif., 1996), pp. 270–277.
  31. J. Fan, L. B. Wolff, “Surface curvature and shape reconstruction from unknown multiple illumination and integrability,” Comput. Vision Image Underst. 65, 347–359 (1997).
    [CrossRef]
  32. P. Breton, S. W. Zucker, “Shadows and shading flow fields,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 782–789.
  33. C. Jiang, M. O. Ward, “Shadow segmentation and clas-sification in a constrained environment,” CVGIP: Graph. Models Image Process. 59, 213–225 (1994).
    [CrossRef]
  34. A. P. Witkin, “Intensity-based edge classification,” in Proceedings of the American Association for Artificial Intelligence (AIII Press, Menlo Park, Calif., 1982), pp. 36–41.
  35. H. von Helmholtz, Treatise on Physiological Optics (Dover, New York, 1925).
  36. J. J. Koenderink, A. J. van Doorn, “Geometry of binocular vision and a model for stereopsis,” Biol. Cybern. 21, 29–35 (1976).
    [CrossRef] [PubMed]

1999

P. N. Belhumeur, D. J. Kriegman, A. L. Yuille, “The bas-relief ambiguity,” Int. J. Comput. Vis. 35, 33–44 (1999).
[CrossRef]

1997

L. Donati, N. Stolfi, “Singularities of illuminated surfaces,” Int. J. Comput. Vision 23, 207–216 (1997).
[CrossRef]

J. Fan, L. B. Wolff, “Surface curvature and shape reconstruction from unknown multiple illumination and integrability,” Comput. Vision Image Underst. 65, 347–359 (1997).
[CrossRef]

1995

F. Cheng, K. H. Thiel, “Delimiting the building heights in a city from the shadow in a panchromatic spot-image. 1 test of 42 buildings,” Int. J. Remote Sens. 16, 409–415 (1995).
[CrossRef]

L. S. Shapiro, A. Zisserman, M. Brady, “3D motion recovery via affine epipolar geometry,” Int. J. Comput. Vis. 16, 147–182 (1995).
[CrossRef]

O. Faugeras, “Stratification of 3-D vision: projective, affine, and metric representations,” J. Opt. Soc. Am. A 12, 465–484 (1995).
[CrossRef]

1994

C. Jiang, M. O. Ward, “Shadow segmentation and clas-sification in a constrained environment,” CVGIP: Graph. Models Image Process. 59, 213–225 (1994).
[CrossRef]

1991

S. Ullman, R. Basri, “Recognition by a linear combination of models,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 992–1006 (1991).
[CrossRef]

J. J. Koenderink, A. J. Van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–385 (1991).
[CrossRef] [PubMed]

1989

R. B. Irvin, D. M. McKeown, “Methods for exploiting the relationship between buildings and their shadows in aerial imagery,” IEEE Trans. Syst. Man Cybern. 19, 1564–1575 (1989).
[CrossRef]

1987

L. N. Hambrick, M. H. Loew, R. L. Carroll, “The entry–exit method of shadow boundary segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 597–607 (1987).
[CrossRef]

1983

S. Shafer, T. Kanade, “Using shadows in finding surface orientation,” Comput. Vision Graph. Image Process. 22, 145–176 (1983).
[CrossRef]

1976

J. J. Koenderink, A. J. van Doorn, “Geometry of binocular vision and a model for stereopsis,” Biol. Cybern. 21, 29–35 (1976).
[CrossRef] [PubMed]

Aloimonos, Y.

C. Fermuller, Y. Aloimonos, “Ordinal representations of visual space,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Francisco, Calif., 1996), pp. 897–904.

Artin, E.

E. Artin, Geometric Algebra (Interscience, New York, 1957).

Basri, R.

S. Ullman, R. Basri, “Recognition by a linear combination of models,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 992–1006 (1991).
[CrossRef]

Baxandall, M.

M. Baxandall, Shadows and Enlightenment (Yale U. Press, New Haven, Conn., 1995).

Belhumeur, P.

P. Belhumeur, D. Kriegman, A. Yuille, “The bas-relief ambiguity,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 1040–1046.

D. Kriegman, P. Belhumeur, “What shadows reveal about object structure,” in Proceedings of the European Conference on Computer Vision (Springer, Berlin, 1998), pp. 399–414.

Belhumeur, P. N.

P. N. Belhumeur, D. J. Kriegman, A. L. Yuille, “The bas-relief ambiguity,” Int. J. Comput. Vis. 35, 33–44 (1999).
[CrossRef]

P. N. Belhumeur, D. J. Kriegman, “What is the set of images of an object under all possible lighting conditions,” in Proceedings of the IEEE Conference on Computer Visionand Pattern Recognition (IEEE Computer Science Press, Los Alamitos, Calif., 1996), pp. 270–277.

Brady, M.

L. S. Shapiro, A. Zisserman, M. Brady, “3D motion recovery via affine epipolar geometry,” Int. J. Comput. Vis. 16, 147–182 (1995).
[CrossRef]

Breton, P.

P. Breton, S. W. Zucker, “Shadows and shading flow fields,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 782–789.

Carroll, R. L.

L. N. Hambrick, M. H. Loew, R. L. Carroll, “The entry–exit method of shadow boundary segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 597–607 (1987).
[CrossRef]

Cheng, F.

F. Cheng, K. H. Thiel, “Delimiting the building heights in a city from the shadow in a panchromatic spot-image. 1 test of 42 buildings,” Int. J. Remote Sens. 16, 409–415 (1995).
[CrossRef]

Criminisi, A.

A. Criminisi, I. Reid, A. Zisserman, “Duality, rigidity and planar parallax,” in Proceedings of the European Conference on Computer Vision (Springer, Berlin, 1998), Vol. II, pp. 846–861.

Daum, M.

M. Daum, G. Dudek, “On 3-D surface reconstruction using shape from shadows,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 461–468.

DoCarmo, M. P.

M. P. DoCarmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, N.J., 1976).

Donati, L.

L. Donati, N. Stolfi, “Singularities of illuminated surfaces,” Int. J. Comput. Vision 23, 207–216 (1997).
[CrossRef]

Dudek, G.

M. Daum, G. Dudek, “On 3-D surface reconstruction using shape from shadows,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 461–468.

Fan, J.

J. Fan, L. B. Wolff, “Surface curvature and shape reconstruction from unknown multiple illumination and integrability,” Comput. Vision Image Underst. 65, 347–359 (1997).
[CrossRef]

Faugeras, O.

Fermuller, C.

C. Fermuller, Y. Aloimonos, “Ordinal representations of visual space,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Francisco, Calif., 1996), pp. 897–904.

Hambrick, L. N.

L. N. Hambrick, M. H. Loew, R. L. Carroll, “The entry–exit method of shadow boundary segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 597–607 (1987).
[CrossRef]

Hatzitheodorou, M.

M. Hatzitheodorou, “The derivation of 3-D surface shape from shadows,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 1012–1020.

Huertas, A.

A. Huertas, R. Nevatia, “Detection of buildings in aerial images using shape and shadows,” in Proceedings of the International Joint Conference on Artificial Intelligence (Morgan Kaufmann, San Francisco, Calif., 1983), pp. 1099–1103.

Irvin, R. B.

R. B. Irvin, D. M. McKeown, “Methods for exploiting the relationship between buildings and their shadows in aerial imagery,” IEEE Trans. Syst. Man Cybern. 19, 1564–1575 (1989).
[CrossRef]

Jiang, C.

C. Jiang, M. O. Ward, “Shadow segmentation and clas-sification in a constrained environment,” CVGIP: Graph. Models Image Process. 59, 213–225 (1994).
[CrossRef]

Kanade, T.

S. Shafer, T. Kanade, “Using shadows in finding surface orientation,” Comput. Vision Graph. Image Process. 22, 145–176 (1983).
[CrossRef]

Kender, J. R.

J. R. Kender, E. M. Smith, “Shape from darkness,” in International Conference on Computer Vision (Morgan Kaufmann, Los Altos, Calif., 1987), pp. 539–546.

D. Yang, J. R. Kender, “Shape from shadows under error,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1993), pp. 1083–1090.

Koenderink, J. J.

J. J. Koenderink, A. J. Van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–385 (1991).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “Geometry of binocular vision and a model for stereopsis,” Biol. Cybern. 21, 29–35 (1976).
[CrossRef] [PubMed]

R. Rosenholtz, J. J. Koenderink, “Affine structure and photometry,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 18–20.

Kriegman, D.

P. Belhumeur, D. Kriegman, A. Yuille, “The bas-relief ambiguity,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 1040–1046.

D. Kriegman, P. Belhumeur, “What shadows reveal about object structure,” in Proceedings of the European Conference on Computer Vision (Springer, Berlin, 1998), pp. 399–414.

Kriegman, D. J.

P. N. Belhumeur, D. J. Kriegman, A. L. Yuille, “The bas-relief ambiguity,” Int. J. Comput. Vis. 35, 33–44 (1999).
[CrossRef]

P. N. Belhumeur, D. J. Kriegman, “What is the set of images of an object under all possible lighting conditions,” in Proceedings of the IEEE Conference on Computer Visionand Pattern Recognition (IEEE Computer Science Press, Los Alamitos, Calif., 1996), pp. 270–277.

Langer, M. S.

M. S. Langer, S. W. Zucker, “What is a light source?” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 172–178.

Loew, M. H.

L. N. Hambrick, M. H. Loew, R. L. Carroll, “The entry–exit method of shadow boundary segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 597–607 (1987).
[CrossRef]

McKeown, D. M.

R. B. Irvin, D. M. McKeown, “Methods for exploiting the relationship between buildings and their shadows in aerial imagery,” IEEE Trans. Syst. Man Cybern. 19, 1564–1575 (1989).
[CrossRef]

Medioni, G. G.

G. G. Medioni, “Obtaining 3-D from shadows in aerial images,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1983), pp. 73–76.

Mundy, J.

J. Mundy, A. Zisserman, Geometric Invariance in Computer Vision (MIT Press, Cambridge, Mass., 1992).

Nevatia, R.

A. Huertas, R. Nevatia, “Detection of buildings in aerial images using shape and shadows,” in Proceedings of the International Joint Conference on Artificial Intelligence (Morgan Kaufmann, San Francisco, Calif., 1983), pp. 1099–1103.

Reid, I.

A. Criminisi, I. Reid, A. Zisserman, “Duality, rigidity and planar parallax,” in Proceedings of the European Conference on Computer Vision (Springer, Berlin, 1998), Vol. II, pp. 846–861.

Rosenholtz, R.

R. Rosenholtz, J. J. Koenderink, “Affine structure and photometry,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 18–20.

Shafer, S.

S. Shafer, T. Kanade, “Using shadows in finding surface orientation,” Comput. Vision Graph. Image Process. 22, 145–176 (1983).
[CrossRef]

Shapiro, L. S.

L. S. Shapiro, A. Zisserman, M. Brady, “3D motion recovery via affine epipolar geometry,” Int. J. Comput. Vis. 16, 147–182 (1995).
[CrossRef]

Shashua, A.

A. Shashua, “Geometry and photometry in 3D visual recognition,” Ph.D. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1992).

Smith, E. M.

J. R. Kender, E. M. Smith, “Shape from darkness,” in International Conference on Computer Vision (Morgan Kaufmann, Los Altos, Calif., 1987), pp. 539–546.

Stolfi, N.

L. Donati, N. Stolfi, “Singularities of illuminated surfaces,” Int. J. Comput. Vision 23, 207–216 (1997).
[CrossRef]

Thiel, K. H.

F. Cheng, K. H. Thiel, “Delimiting the building heights in a city from the shadow in a panchromatic spot-image. 1 test of 42 buildings,” Int. J. Remote Sens. 16, 409–415 (1995).
[CrossRef]

Ullman, S.

S. Ullman, R. Basri, “Recognition by a linear combination of models,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 992–1006 (1991).
[CrossRef]

Van Doorn, A. J.

J. J. Koenderink, A. J. Van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–385 (1991).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “Geometry of binocular vision and a model for stereopsis,” Biol. Cybern. 21, 29–35 (1976).
[CrossRef] [PubMed]

von Helmholtz, H.

H. von Helmholtz, Treatise on Physiological Optics (Dover, New York, 1925).

Waltz, D.L.

D.L. Waltz, “Understanding line drawings of scenes with shadows,” in The Psychology of Computer Vision,P. H. Winston, ed. (McGraw-Hill, New York, 1975), pp. 19–91.

Ward, M. O.

C. Jiang, M. O. Ward, “Shadow segmentation and clas-sification in a constrained environment,” CVGIP: Graph. Models Image Process. 59, 213–225 (1994).
[CrossRef]

Witkin, A. P.

A. P. Witkin, “Intensity-based edge classification,” in Proceedings of the American Association for Artificial Intelligence (AIII Press, Menlo Park, Calif., 1982), pp. 36–41.

Wolff, L. B.

J. Fan, L. B. Wolff, “Surface curvature and shape reconstruction from unknown multiple illumination and integrability,” Comput. Vision Image Underst. 65, 347–359 (1997).
[CrossRef]

Yang, D.

D. Yang, J. R. Kender, “Shape from shadows under error,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1993), pp. 1083–1090.

Yuille, A.

P. Belhumeur, D. Kriegman, A. Yuille, “The bas-relief ambiguity,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 1040–1046.

Yuille, A. L.

P. N. Belhumeur, D. J. Kriegman, A. L. Yuille, “The bas-relief ambiguity,” Int. J. Comput. Vis. 35, 33–44 (1999).
[CrossRef]

Zisserman, A.

L. S. Shapiro, A. Zisserman, M. Brady, “3D motion recovery via affine epipolar geometry,” Int. J. Comput. Vis. 16, 147–182 (1995).
[CrossRef]

A. Criminisi, I. Reid, A. Zisserman, “Duality, rigidity and planar parallax,” in Proceedings of the European Conference on Computer Vision (Springer, Berlin, 1998), Vol. II, pp. 846–861.

J. Mundy, A. Zisserman, Geometric Invariance in Computer Vision (MIT Press, Cambridge, Mass., 1992).

Zucker, S. W.

M. S. Langer, S. W. Zucker, “What is a light source?” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 172–178.

P. Breton, S. W. Zucker, “Shadows and shading flow fields,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 782–789.

Biol. Cybern.

J. J. Koenderink, A. J. van Doorn, “Geometry of binocular vision and a model for stereopsis,” Biol. Cybern. 21, 29–35 (1976).
[CrossRef] [PubMed]

Comput. Vision Graph. Image Process.

S. Shafer, T. Kanade, “Using shadows in finding surface orientation,” Comput. Vision Graph. Image Process. 22, 145–176 (1983).
[CrossRef]

Comput. Vision Image Underst.

J. Fan, L. B. Wolff, “Surface curvature and shape reconstruction from unknown multiple illumination and integrability,” Comput. Vision Image Underst. 65, 347–359 (1997).
[CrossRef]

CVGIP: Graph. Models Image Process.

C. Jiang, M. O. Ward, “Shadow segmentation and clas-sification in a constrained environment,” CVGIP: Graph. Models Image Process. 59, 213–225 (1994).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

S. Ullman, R. Basri, “Recognition by a linear combination of models,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 992–1006 (1991).
[CrossRef]

L. N. Hambrick, M. H. Loew, R. L. Carroll, “The entry–exit method of shadow boundary segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 597–607 (1987).
[CrossRef]

IEEE Trans. Syst. Man Cybern.

R. B. Irvin, D. M. McKeown, “Methods for exploiting the relationship between buildings and their shadows in aerial imagery,” IEEE Trans. Syst. Man Cybern. 19, 1564–1575 (1989).
[CrossRef]

Int. J. Comput. Vis.

P. N. Belhumeur, D. J. Kriegman, A. L. Yuille, “The bas-relief ambiguity,” Int. J. Comput. Vis. 35, 33–44 (1999).
[CrossRef]

L. S. Shapiro, A. Zisserman, M. Brady, “3D motion recovery via affine epipolar geometry,” Int. J. Comput. Vis. 16, 147–182 (1995).
[CrossRef]

Int. J. Comput. Vision

L. Donati, N. Stolfi, “Singularities of illuminated surfaces,” Int. J. Comput. Vision 23, 207–216 (1997).
[CrossRef]

Int. J. Remote Sens.

F. Cheng, K. H. Thiel, “Delimiting the building heights in a city from the shadow in a panchromatic spot-image. 1 test of 42 buildings,” Int. J. Remote Sens. 16, 409–415 (1995).
[CrossRef]

J. Opt. Soc. Am. A

Other

D.L. Waltz, “Understanding line drawings of scenes with shadows,” in The Psychology of Computer Vision,P. H. Winston, ed. (McGraw-Hill, New York, 1975), pp. 19–91.

E. Artin, Geometric Algebra (Interscience, New York, 1957).

P. N. Belhumeur, D. J. Kriegman, “What is the set of images of an object under all possible lighting conditions,” in Proceedings of the IEEE Conference on Computer Visionand Pattern Recognition (IEEE Computer Science Press, Los Alamitos, Calif., 1996), pp. 270–277.

P. Breton, S. W. Zucker, “Shadows and shading flow fields,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 782–789.

A. P. Witkin, “Intensity-based edge classification,” in Proceedings of the American Association for Artificial Intelligence (AIII Press, Menlo Park, Calif., 1982), pp. 36–41.

H. von Helmholtz, Treatise on Physiological Optics (Dover, New York, 1925).

R. Rosenholtz, J. J. Koenderink, “Affine structure and photometry,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 18–20.

A. Huertas, R. Nevatia, “Detection of buildings in aerial images using shape and shadows,” in Proceedings of the International Joint Conference on Artificial Intelligence (Morgan Kaufmann, San Francisco, Calif., 1983), pp. 1099–1103.

M. P. DoCarmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, N.J., 1976).

M. Hatzitheodorou, “The derivation of 3-D surface shape from shadows,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 1012–1020.

J. R. Kender, E. M. Smith, “Shape from darkness,” in International Conference on Computer Vision (Morgan Kaufmann, Los Altos, Calif., 1987), pp. 539–546.

D. Yang, J. R. Kender, “Shape from shadows under error,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1993), pp. 1083–1090.

M. Daum, G. Dudek, “On 3-D surface reconstruction using shape from shadows,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 461–468.

G. G. Medioni, “Obtaining 3-D from shadows in aerial images,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1983), pp. 73–76.

P. Belhumeur, D. Kriegman, A. Yuille, “The bas-relief ambiguity,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 1040–1046.

D. Kriegman, P. Belhumeur, “What shadows reveal about object structure,” in Proceedings of the European Conference on Computer Vision (Springer, Berlin, 1998), pp. 399–414.

M. S. Langer, S. W. Zucker, “What is a light source?” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 172–178.

J. Mundy, A. Zisserman, Geometric Invariance in Computer Vision (MIT Press, Cambridge, Mass., 1992).

M. Baxandall, Shadows and Enlightenment (Yale U. Press, New Haven, Conn., 1995).

A. Shashua, “Geometry and photometry in 3D visual recognition,” Ph.D. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1992).

O. Faugeras, Three Dimensional Computer Vision (MIT Press, Cambridge, Mass., 1993).

A. Criminisi, I. Reid, A. Zisserman, “Duality, rigidity and planar parallax,” in Proceedings of the European Conference on Computer Vision (Springer, Berlin, 1998), Vol. II, pp. 846–861.

C. Fermuller, Y. Aloimonos, “Ordinal representations of visual space,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Francisco, Calif., 1996), pp. 897–904.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Illustration of the effect of applying a generalized perspective bas-relief (GPBR) transformation to a scene composed of a teapot resting on a supporting plane. The first image shows the original teapot. The second image shows the teapot after having undergone a GPBR transformation (a1, a2, a3, a4)=(0.05,0.05,0.05,1) with respect to the viewpoint used to generate the first image. Note that the attached and cast shadows as well as the occluding contour are identical in the first two images. The third image shows the original teapot from a second viewpoint. The fourth image reveals the nature of the GPBR transformation, showing the transformed teapot from the second viewpoint.

Fig. 2
Fig. 2

In this 2-D illustration of the GPBR transformation, the lower shadow is an attached shadow, while the upper one is composed of both attached and cast components. A GPBR transformation has been applied to the left surface, yielding the right one. Note that under a GPBR, all surface points and the light source are transformed along the optical rays through the center of projection. Transforming the light source from s to s preserves the shadows.

Fig. 3
Fig. 3

Under orthographic projection, the image points that lie in shadow for a surface under light source s are identical to those in shadow for a transformed surface under light source s. In this 2-D illustration, the lower shadow is an attached shadow, while the upper one is composed of both attached and cast components. A generalized bas-relief (GBR) transformation with both flattening and an additive plane has been applied to the left surface, yielding the right one.

Fig. 4
Fig. 4

Relation of different spaces in the proof of Proposition 3.1.    

Fig. 5
Fig. 5

Illustration of some of the ideas in the proof of Proposition 3.1. In two dimensions, consider the shadows of a nonconvex object observed under orthographic projection with viewing direction v. Over all light source directions, only points on the thickened arcs can fall on a visible global attached shadow boundary. The points between the thickened arcs and the dotted horizontal lines are visible to the observer, but there is no light source direction for which these points satisfy the global attached shadow conditions. Nonetheless, there exist light source directions for which each of these points can lie on a cast shadow boundary (e.g., p lies on a cast shadow boundary for light source direction s). Since p is a cast shadow, it lies in the tangent plane to an attached shadow point q; when the portion of the surface about q is transformed by a GBR transformation, p must also be transformed by the same GBR transformation. Similarly, points such as p and q are, respectively, cast and attached shadow points for light source s . Note that p can also be an attached shadow boundary point. Because p lies in the tangent plane to q , the same GBR that is applied to q and its tangent plane must be applied to p . Hence the two thickened arcs containing p and q must be transformed by the same GBR if they are to be strongly shadow equivalent.

Fig. 6
Fig. 6

Reconstruction up to a GBR from attached shadows. For a single object in fixed pose, these figures show superimposed attached shadow contours Ci for light source direction si. The surface normal where Ci intersects the occluding contour is denoted by ni. The normal at the intersection of Ci and Cj is denoted by ni,j. (a) The three contours intersect at three points in the image. (b) The three contours meet at a common point, implying that s1, s2, and s3 lie on a great circle of the illumination sphere. (c) Eight attached shadow boundaries, of which four intersect at p1,2 and four intersect at p1,3; the direction of the light sources s1,s8 and the surface normals at the intersection points can be determined up to a GBR. (d) Structure of the illumination sphere S2 for the light source directions generating the attached shadow boundaries in (c).

Equations (11)

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

Γp=100001000010.
A=100001000010a1a2a3a4.
p=1ap+a4p,s=1as+a4s,
Γo=100001000001.
G=10000100g1g2g3g40001,
xyz=xyg1x+g2y+g3f(x, y)+g4.
P=10p101p200p3,
s1(θ1)=cos θ1n1+sin θ1zˆ,
n1,2=s1(θ1)×s2(θ2).
s4(θ1,θ2)=n4×[s1(θ1)×s2(θ2)].
si(θ1,θ3)=ni×[s1(θ1)×s3(θ3)].

Metrics