Abstract

Both surface contours and texture patterns can provide strong cues to the three-dimensional shape of a surface in space. Many of the most perceptually salient texture patterns have a strong flowlike structure, resulting from the directional nature of the surface textures from which they project. Under the minimal assumption that an oriented surface texture is homogeneous, the texture flow on a developable surface can be shown to follow parallel geodesics of the surface. The geometry of texture flow is therefore equivalent to that of an important class of surface contours: those that project from parallel geodesics of a developable surface. I derive a set of differential equations that support the estimation of surface shape from geodesic surface contours under spherical perspective, for both parallel and nonparallel contours. For perfectly oriented textures, the equations apply directly to the integrated flow lines in a texture image. For weakly oriented textures, perspective projection distorts the projected orientation of flow lines away from the idealized case of pure contours; however, simulations show that for a large class of textures, these distortions will be small and limited largely to extreme surface poses. The geometrical analysis, along with a number of phenomenal demonstrations and psychophysical results, suggests that the human visual system co-opts shape from contour mechanisms to estimate surface shape from texture flow.

© 2001 Optical Society of America

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References

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  1. J. J. Koenderink, A. J. van Doorn, “The shape of smooth objects and the way contours end,” Perception 11, 129–137 (1982).
    [CrossRef] [PubMed]
  2. J. J. Koenderink, “What does occluding contour tell us about solid shape,” Perception 13, 321–330 (1984).
    [CrossRef]
  3. D. A. Huffman, “Realizable configurations of lines in pictures of polyhedra,” Mach. Intell. 8, 493–509 (1971).
  4. M. B. Clowes, “On seeing things,” Artif. Intel. 2, 79–116 (1971).
    [CrossRef]
  5. D. Waltz, “Understanding line drawings of scenes with shadows,” in P. H. Winston, ed., The Psychology of Computer Vision (McGraw-Hill, New York, 1975), pp. 19–91.
  6. V. S. Nalwa, “Line-drawing interpretation: a mathematical framework,” Int. J. Comput. Vision 2(2), 103–124 (1988).
    [CrossRef]
  7. F. Ulupinar, R. Nevatia, “Perception of 3-D surfaces from 2-D contours,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 3–18 (1993).
    [CrossRef]
  8. S. A. Shafer, T. Kanade, “Using shadows in finding surface orientations,” Comput. Vision Graph. Image Process. 22, 145–176 (1983).
    [CrossRef]
  9. D. C. Knill, P. Mamassian, D. Kersten, “The geometry of shadows,” J. Opt. Soc. Am. A 14, 3216–3232 (1997).
    [CrossRef]
  10. A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intel. 17, 17–47 (1981).
    [CrossRef]
  11. M. Brady, A. L. Yuille, “An extremum principle for shape from contour,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 288–301 (1984).
    [CrossRef]
  12. I. Weiss, “3D shape representation by contours,” Comput. Vis. Graph. Image Process. 41, 80–100 (1988).
    [CrossRef]
  13. T. Kanade, “Recovery of the three-dimensional shape of an object from a single view,” Artif. Intel. 17, 409–460 (1981).
    [CrossRef]
  14. K. A. Stevens, “The visual interpretation of surface contours,” Artif. Intel. 17, 47–73 (1981).
    [CrossRef]
  15. D. C. Knill, “Perception of surface contours and surface shape: from computation to psychophysics,” J. Opt. Soc. Am. A 9, 1449–1464 (1992).
    [CrossRef] [PubMed]
  16. R. Horaud, M. Brady, “On the geometric interpretation of image contours,” Artif. Intel. 37, 333–353 (1988).
    [CrossRef]
  17. F. Ulupinar, R. Nevatia, “Shape form contour—straight homogeneous generalized cylinders and constant cross-section generalized cylinders,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 120–135 (1995).
    [CrossRef]
  18. P. Mamassian, M. S. Landy, “Observer biases in the 3D interpretation of line drawings,” Vision Res. 38, 2817–2832 (1998).
    [CrossRef] [PubMed]
  19. J. Wagemans, “Perceptual use of nonaccidental properties,” Can. J. Psychol. 46, 236–279 (1992).
    [CrossRef] [PubMed]
  20. M. P. Do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, N.J., 1976).
  21. A. Jepson, W. Richards, “What makes a good feature?” in Spatial Vision in Humans and Robots, L. Harris, M. Jenkin, eds. (Cambridge U. Press, Cambridge, UK, 1992).
  22. Garding23took a different tack to show that Rosenholtz and Malik’s definition of homogeneity implies that surface texture orientations at neighboring points on a surface are equivalent under parallel transport. This result also implies that homogeneous, oriented textures flow along geodesics of developable surfaces.
  23. J. Garding, “Surface orientation and curvature from differential texture distortion,” presented at the 5th International Conference on Computer Vision, Cambridge, Mass., June 1995.
  24. A. Blake, C. Marinos, “Shape from texture: estimation, isotropy and moments,” (Oxford U. Press, Oxford, UK, 1989).
  25. J. Garding, “Shape from texture and contour by weak isotropy,” Artif. Intel. 64, 243–297 (1993).
    [CrossRef]
  26. B. Super, A. Bovik, “Shape from texture using local spectral moments,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 333–343 (1995).
    [CrossRef]
  27. J. T. Todd, F. D. Reichel, “The visual perception of smoothly curved surfaces from double projected contour patterns,” J. Exp. Psychol. 16, 665–674 (1990).
  28. D. C. Knill, “From contour to texture: static texture flow is a strong cue to surface shape,” presented at the European Conference on Visual Perception, Helsinki, Finland, August 24–29, 1997.
  29. A. Li, Q. Zaidi, “Shape from natural textures,” Invest. Ophthalmol. Visual Sci. 40, 2097 (1999).
  30. J. Malik, R. Rosenholtz, “Recovering surface curvature and orientation from texture distortion: a least squares algorithm and sensitivity analysis,” in Proceedings of the 3rd European Conference on Computer Vision, Vol. 800 of Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1995), pp. 353–364.
  31. A. R. Rao, R. C. Jain, “Computerized flow field analysis—oriented texture fields,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 693–709 (1992).
    [CrossRef]
  32. Y. Bonneh, D. Sagi, “Effects of spatial configuration on contrast detection,” Vision Res. 38, 3541–3553 (1998).
    [CrossRef]
  33. M. Usher, Y. Bonneh, D. Sagi, M. Herrmann, “Mechanisms for spatial integration in visual detection: a model based on lateral interactions,” Spatial Vision 12, 187–209 (1999).
    [CrossRef] [PubMed]

1999 (2)

A. Li, Q. Zaidi, “Shape from natural textures,” Invest. Ophthalmol. Visual Sci. 40, 2097 (1999).

M. Usher, Y. Bonneh, D. Sagi, M. Herrmann, “Mechanisms for spatial integration in visual detection: a model based on lateral interactions,” Spatial Vision 12, 187–209 (1999).
[CrossRef] [PubMed]

1998 (2)

Y. Bonneh, D. Sagi, “Effects of spatial configuration on contrast detection,” Vision Res. 38, 3541–3553 (1998).
[CrossRef]

P. Mamassian, M. S. Landy, “Observer biases in the 3D interpretation of line drawings,” Vision Res. 38, 2817–2832 (1998).
[CrossRef] [PubMed]

1997 (1)

1995 (2)

F. Ulupinar, R. Nevatia, “Shape form contour—straight homogeneous generalized cylinders and constant cross-section generalized cylinders,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 120–135 (1995).
[CrossRef]

B. Super, A. Bovik, “Shape from texture using local spectral moments,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 333–343 (1995).
[CrossRef]

1993 (2)

J. Garding, “Shape from texture and contour by weak isotropy,” Artif. Intel. 64, 243–297 (1993).
[CrossRef]

F. Ulupinar, R. Nevatia, “Perception of 3-D surfaces from 2-D contours,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 3–18 (1993).
[CrossRef]

1992 (3)

J. Wagemans, “Perceptual use of nonaccidental properties,” Can. J. Psychol. 46, 236–279 (1992).
[CrossRef] [PubMed]

D. C. Knill, “Perception of surface contours and surface shape: from computation to psychophysics,” J. Opt. Soc. Am. A 9, 1449–1464 (1992).
[CrossRef] [PubMed]

A. R. Rao, R. C. Jain, “Computerized flow field analysis—oriented texture fields,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 693–709 (1992).
[CrossRef]

1990 (1)

J. T. Todd, F. D. Reichel, “The visual perception of smoothly curved surfaces from double projected contour patterns,” J. Exp. Psychol. 16, 665–674 (1990).

1988 (3)

R. Horaud, M. Brady, “On the geometric interpretation of image contours,” Artif. Intel. 37, 333–353 (1988).
[CrossRef]

I. Weiss, “3D shape representation by contours,” Comput. Vis. Graph. Image Process. 41, 80–100 (1988).
[CrossRef]

V. S. Nalwa, “Line-drawing interpretation: a mathematical framework,” Int. J. Comput. Vision 2(2), 103–124 (1988).
[CrossRef]

1984 (2)

J. J. Koenderink, “What does occluding contour tell us about solid shape,” Perception 13, 321–330 (1984).
[CrossRef]

M. Brady, A. L. Yuille, “An extremum principle for shape from contour,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 288–301 (1984).
[CrossRef]

1983 (1)

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

1982 (1)

J. J. Koenderink, A. J. van Doorn, “The shape of smooth objects and the way contours end,” Perception 11, 129–137 (1982).
[CrossRef] [PubMed]

1981 (3)

A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intel. 17, 17–47 (1981).
[CrossRef]

T. Kanade, “Recovery of the three-dimensional shape of an object from a single view,” Artif. Intel. 17, 409–460 (1981).
[CrossRef]

K. A. Stevens, “The visual interpretation of surface contours,” Artif. Intel. 17, 47–73 (1981).
[CrossRef]

1971 (2)

D. A. Huffman, “Realizable configurations of lines in pictures of polyhedra,” Mach. Intell. 8, 493–509 (1971).

M. B. Clowes, “On seeing things,” Artif. Intel. 2, 79–116 (1971).
[CrossRef]

Blake, A.

A. Blake, C. Marinos, “Shape from texture: estimation, isotropy and moments,” (Oxford U. Press, Oxford, UK, 1989).

Bonneh, Y.

M. Usher, Y. Bonneh, D. Sagi, M. Herrmann, “Mechanisms for spatial integration in visual detection: a model based on lateral interactions,” Spatial Vision 12, 187–209 (1999).
[CrossRef] [PubMed]

Y. Bonneh, D. Sagi, “Effects of spatial configuration on contrast detection,” Vision Res. 38, 3541–3553 (1998).
[CrossRef]

Bovik, A.

B. Super, A. Bovik, “Shape from texture using local spectral moments,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 333–343 (1995).
[CrossRef]

Brady, M.

R. Horaud, M. Brady, “On the geometric interpretation of image contours,” Artif. Intel. 37, 333–353 (1988).
[CrossRef]

M. Brady, A. L. Yuille, “An extremum principle for shape from contour,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 288–301 (1984).
[CrossRef]

Clowes, M. B.

M. B. Clowes, “On seeing things,” Artif. Intel. 2, 79–116 (1971).
[CrossRef]

Do Carmo, M. P.

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

Garding, J.

J. Garding, “Shape from texture and contour by weak isotropy,” Artif. Intel. 64, 243–297 (1993).
[CrossRef]

J. Garding, “Surface orientation and curvature from differential texture distortion,” presented at the 5th International Conference on Computer Vision, Cambridge, Mass., June 1995.

Herrmann, M.

M. Usher, Y. Bonneh, D. Sagi, M. Herrmann, “Mechanisms for spatial integration in visual detection: a model based on lateral interactions,” Spatial Vision 12, 187–209 (1999).
[CrossRef] [PubMed]

Horaud, R.

R. Horaud, M. Brady, “On the geometric interpretation of image contours,” Artif. Intel. 37, 333–353 (1988).
[CrossRef]

Huffman, D. A.

D. A. Huffman, “Realizable configurations of lines in pictures of polyhedra,” Mach. Intell. 8, 493–509 (1971).

Jain, R. C.

A. R. Rao, R. C. Jain, “Computerized flow field analysis—oriented texture fields,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 693–709 (1992).
[CrossRef]

Jepson, A.

A. Jepson, W. Richards, “What makes a good feature?” in Spatial Vision in Humans and Robots, L. Harris, M. Jenkin, eds. (Cambridge U. Press, Cambridge, UK, 1992).

Kanade, T.

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

T. Kanade, “Recovery of the three-dimensional shape of an object from a single view,” Artif. Intel. 17, 409–460 (1981).
[CrossRef]

Kersten, D.

Knill, D. C.

D. C. Knill, P. Mamassian, D. Kersten, “The geometry of shadows,” J. Opt. Soc. Am. A 14, 3216–3232 (1997).
[CrossRef]

D. C. Knill, “Perception of surface contours and surface shape: from computation to psychophysics,” J. Opt. Soc. Am. A 9, 1449–1464 (1992).
[CrossRef] [PubMed]

D. C. Knill, “From contour to texture: static texture flow is a strong cue to surface shape,” presented at the European Conference on Visual Perception, Helsinki, Finland, August 24–29, 1997.

Koenderink, J. J.

J. J. Koenderink, “What does occluding contour tell us about solid shape,” Perception 13, 321–330 (1984).
[CrossRef]

J. J. Koenderink, A. J. van Doorn, “The shape of smooth objects and the way contours end,” Perception 11, 129–137 (1982).
[CrossRef] [PubMed]

Landy, M. S.

P. Mamassian, M. S. Landy, “Observer biases in the 3D interpretation of line drawings,” Vision Res. 38, 2817–2832 (1998).
[CrossRef] [PubMed]

Li, A.

A. Li, Q. Zaidi, “Shape from natural textures,” Invest. Ophthalmol. Visual Sci. 40, 2097 (1999).

Malik, J.

J. Malik, R. Rosenholtz, “Recovering surface curvature and orientation from texture distortion: a least squares algorithm and sensitivity analysis,” in Proceedings of the 3rd European Conference on Computer Vision, Vol. 800 of Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1995), pp. 353–364.

Mamassian, P.

P. Mamassian, M. S. Landy, “Observer biases in the 3D interpretation of line drawings,” Vision Res. 38, 2817–2832 (1998).
[CrossRef] [PubMed]

D. C. Knill, P. Mamassian, D. Kersten, “The geometry of shadows,” J. Opt. Soc. Am. A 14, 3216–3232 (1997).
[CrossRef]

Marinos, C.

A. Blake, C. Marinos, “Shape from texture: estimation, isotropy and moments,” (Oxford U. Press, Oxford, UK, 1989).

Nalwa, V. S.

V. S. Nalwa, “Line-drawing interpretation: a mathematical framework,” Int. J. Comput. Vision 2(2), 103–124 (1988).
[CrossRef]

Nevatia, R.

F. Ulupinar, R. Nevatia, “Shape form contour—straight homogeneous generalized cylinders and constant cross-section generalized cylinders,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 120–135 (1995).
[CrossRef]

F. Ulupinar, R. Nevatia, “Perception of 3-D surfaces from 2-D contours,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 3–18 (1993).
[CrossRef]

Rao, A. R.

A. R. Rao, R. C. Jain, “Computerized flow field analysis—oriented texture fields,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 693–709 (1992).
[CrossRef]

Reichel, F. D.

J. T. Todd, F. D. Reichel, “The visual perception of smoothly curved surfaces from double projected contour patterns,” J. Exp. Psychol. 16, 665–674 (1990).

Richards, W.

A. Jepson, W. Richards, “What makes a good feature?” in Spatial Vision in Humans and Robots, L. Harris, M. Jenkin, eds. (Cambridge U. Press, Cambridge, UK, 1992).

Rosenholtz, R.

J. Malik, R. Rosenholtz, “Recovering surface curvature and orientation from texture distortion: a least squares algorithm and sensitivity analysis,” in Proceedings of the 3rd European Conference on Computer Vision, Vol. 800 of Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1995), pp. 353–364.

Sagi, D.

M. Usher, Y. Bonneh, D. Sagi, M. Herrmann, “Mechanisms for spatial integration in visual detection: a model based on lateral interactions,” Spatial Vision 12, 187–209 (1999).
[CrossRef] [PubMed]

Y. Bonneh, D. Sagi, “Effects of spatial configuration on contrast detection,” Vision Res. 38, 3541–3553 (1998).
[CrossRef]

Shafer, S. A.

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

Stevens, K. A.

K. A. Stevens, “The visual interpretation of surface contours,” Artif. Intel. 17, 47–73 (1981).
[CrossRef]

Super, B.

B. Super, A. Bovik, “Shape from texture using local spectral moments,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 333–343 (1995).
[CrossRef]

Todd, J. T.

J. T. Todd, F. D. Reichel, “The visual perception of smoothly curved surfaces from double projected contour patterns,” J. Exp. Psychol. 16, 665–674 (1990).

Ulupinar, F.

F. Ulupinar, R. Nevatia, “Shape form contour—straight homogeneous generalized cylinders and constant cross-section generalized cylinders,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 120–135 (1995).
[CrossRef]

F. Ulupinar, R. Nevatia, “Perception of 3-D surfaces from 2-D contours,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 3–18 (1993).
[CrossRef]

Usher, M.

M. Usher, Y. Bonneh, D. Sagi, M. Herrmann, “Mechanisms for spatial integration in visual detection: a model based on lateral interactions,” Spatial Vision 12, 187–209 (1999).
[CrossRef] [PubMed]

van Doorn, A. J.

J. J. Koenderink, A. J. van Doorn, “The shape of smooth objects and the way contours end,” Perception 11, 129–137 (1982).
[CrossRef] [PubMed]

Wagemans, J.

J. Wagemans, “Perceptual use of nonaccidental properties,” Can. J. Psychol. 46, 236–279 (1992).
[CrossRef] [PubMed]

Waltz, D.

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

Weiss, I.

I. Weiss, “3D shape representation by contours,” Comput. Vis. Graph. Image Process. 41, 80–100 (1988).
[CrossRef]

Witkin, A. P.

A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intel. 17, 17–47 (1981).
[CrossRef]

Yuille, A. L.

M. Brady, A. L. Yuille, “An extremum principle for shape from contour,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 288–301 (1984).
[CrossRef]

Zaidi, Q.

A. Li, Q. Zaidi, “Shape from natural textures,” Invest. Ophthalmol. Visual Sci. 40, 2097 (1999).

Artif. Intel. (6)

M. B. Clowes, “On seeing things,” Artif. Intel. 2, 79–116 (1971).
[CrossRef]

A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intel. 17, 17–47 (1981).
[CrossRef]

T. Kanade, “Recovery of the three-dimensional shape of an object from a single view,” Artif. Intel. 17, 409–460 (1981).
[CrossRef]

K. A. Stevens, “The visual interpretation of surface contours,” Artif. Intel. 17, 47–73 (1981).
[CrossRef]

R. Horaud, M. Brady, “On the geometric interpretation of image contours,” Artif. Intel. 37, 333–353 (1988).
[CrossRef]

J. Garding, “Shape from texture and contour by weak isotropy,” Artif. Intel. 64, 243–297 (1993).
[CrossRef]

Can. J. Psychol. (1)

J. Wagemans, “Perceptual use of nonaccidental properties,” Can. J. Psychol. 46, 236–279 (1992).
[CrossRef] [PubMed]

Comput. Vis. Graph. Image Process. (1)

I. Weiss, “3D shape representation by contours,” Comput. Vis. Graph. Image Process. 41, 80–100 (1988).
[CrossRef]

Comput. Vision Graph. Image Process. (1)

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

IEEE Trans. Pattern Anal. Mach. Intell. (5)

F. Ulupinar, R. Nevatia, “Perception of 3-D surfaces from 2-D contours,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 3–18 (1993).
[CrossRef]

M. Brady, A. L. Yuille, “An extremum principle for shape from contour,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 288–301 (1984).
[CrossRef]

F. Ulupinar, R. Nevatia, “Shape form contour—straight homogeneous generalized cylinders and constant cross-section generalized cylinders,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 120–135 (1995).
[CrossRef]

B. Super, A. Bovik, “Shape from texture using local spectral moments,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 333–343 (1995).
[CrossRef]

A. R. Rao, R. C. Jain, “Computerized flow field analysis—oriented texture fields,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 693–709 (1992).
[CrossRef]

Int. J. Comput. Vision (1)

V. S. Nalwa, “Line-drawing interpretation: a mathematical framework,” Int. J. Comput. Vision 2(2), 103–124 (1988).
[CrossRef]

Invest. Ophthalmol. Visual Sci. (1)

A. Li, Q. Zaidi, “Shape from natural textures,” Invest. Ophthalmol. Visual Sci. 40, 2097 (1999).

J. Exp. Psychol. (1)

J. T. Todd, F. D. Reichel, “The visual perception of smoothly curved surfaces from double projected contour patterns,” J. Exp. Psychol. 16, 665–674 (1990).

J. Opt. Soc. Am. A (2)

Mach. Intell. (1)

D. A. Huffman, “Realizable configurations of lines in pictures of polyhedra,” Mach. Intell. 8, 493–509 (1971).

Perception (2)

J. J. Koenderink, A. J. van Doorn, “The shape of smooth objects and the way contours end,” Perception 11, 129–137 (1982).
[CrossRef] [PubMed]

J. J. Koenderink, “What does occluding contour tell us about solid shape,” Perception 13, 321–330 (1984).
[CrossRef]

Spatial Vision (1)

M. Usher, Y. Bonneh, D. Sagi, M. Herrmann, “Mechanisms for spatial integration in visual detection: a model based on lateral interactions,” Spatial Vision 12, 187–209 (1999).
[CrossRef] [PubMed]

Vision Res. (2)

Y. Bonneh, D. Sagi, “Effects of spatial configuration on contrast detection,” Vision Res. 38, 3541–3553 (1998).
[CrossRef]

P. Mamassian, M. S. Landy, “Observer biases in the 3D interpretation of line drawings,” Vision Res. 38, 2817–2832 (1998).
[CrossRef] [PubMed]

Other (8)

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

J. Malik, R. Rosenholtz, “Recovering surface curvature and orientation from texture distortion: a least squares algorithm and sensitivity analysis,” in Proceedings of the 3rd European Conference on Computer Vision, Vol. 800 of Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1995), pp. 353–364.

D. C. Knill, “From contour to texture: static texture flow is a strong cue to surface shape,” presented at the European Conference on Visual Perception, Helsinki, Finland, August 24–29, 1997.

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

A. Jepson, W. Richards, “What makes a good feature?” in Spatial Vision in Humans and Robots, L. Harris, M. Jenkin, eds. (Cambridge U. Press, Cambridge, UK, 1992).

Garding23took a different tack to show that Rosenholtz and Malik’s definition of homogeneity implies that surface texture orientations at neighboring points on a surface are equivalent under parallel transport. This result also implies that homogeneous, oriented textures flow along geodesics of developable surfaces.

J. Garding, “Surface orientation and curvature from differential texture distortion,” presented at the 5th International Conference on Computer Vision, Cambridge, Mass., June 1995.

A. Blake, C. Marinos, “Shape from texture: estimation, isotropy and moments,” (Oxford U. Press, Oxford, UK, 1989).

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Figures (22)

Fig. 1
Fig. 1

Examples of surface markings that generate (a) image contours and (b) image texture. Both forms of information are strong cues to surface shape.

Fig. 2
Fig. 2

Three examples of surface contours: (a) stripes on a circular cylinder that follow geodesics (but not lines of curvature), (b) pairs of geodesics that run in different directions on a cylindrical surface, and (c) the edges of a thin, developable surface that is folded and twisted in space. The contour pattern in (c) is not, technically, a surface contour but rather is a special kind of surface-orientation discontinuity, one formed by the edges of a thin surface. The pattern in (b) might also be formed from the edges of a thin surface such as a flag. The analysis presented in this paper applies to this class of contours as well as to surface contours. In fact, contours formed by the edges of thin surfaces are possibly the largest natural class of contours to which the analysis applies.

Fig. 3
Fig. 3

Two examples of homogeneous, oriented textures on developable surfaces. These are created by isomorphically mapping (wrapping) planar textures onto developable surfaces: (a) a texture mapped onto a one-dimensional ridge in such a way that the texture “flows” along lines of curvature (a special case of geodesics), (b) a continuous texture mapped onto a cone. In this case, the texture “flows” along geodesics of the cone but not along the lines of curvature.

Fig. 4
Fig. 4

Lines of parallelism between contours are the projections of surface rulings.

Fig. 5
Fig. 5

Examples of the three different types of parallel contours discussed in the text: (a) strictly parallel contours, (b) scaled parallel contours (the top and bottom elliptical arcs), and (c) general linear parallel contours.

Fig. 6
Fig. 6

(a) Parallel set of elliptical arcs, as might be projected from a segment of a circular cylinder slanted away from the front parallel by 30°. (b), (c), (d) Reconstructions of the surface shape from the contours in (a) for three different assumed surface slants: 15°, 30°, and 45°, respectively. Line of sight was along the z axis.

Fig. 7
Fig. 7

(a) Parallel set of sinusoidal arcs. (b), (c) Reconstructions of the surface shape from the contours in (a) for two different assumed orientations of the surface markings on the surface relative to the surface rulings; (b) shows the surface reconstructed for an assumed marking orientation of 90° (a line of maximal curvature on the surface); (c) shows the surface reconstructed for an assumed marking orientation of 70°. The global surface orientation was assumed to be 30°. Line of sight was along the z axis. The solid curved lines in (b) and (c) indicate the backprojections of the surface markings shown in (a).

Fig. 8
Fig. 8

The pair of contours in (a) ambiguously determine the shape of the underlying surface (under orthographic projection). Humans appear to resolve this ambiguity in part by assuming that the surface markings follow lines of curvature on a surface. (b), (c) Pairs of nonparallel geodesic contours uniquely determine the underlying surface shape. In these figures the upper contour has the same shape as the curves in (a). The second geodesic contour, however, disambiguates the shape, and we see the figures as all having different shapes, corresponding to cases in which the sinusoidal contour does not follow a line of surface curvature.

Fig. 9
Fig. 9

(a) Perspective projections of a pair of parallel geodesics on a sinuoidal surface viewed from above. (b) The surface was reconstructed by assuming that the ruling directions were fixed in the image at vertical (in this case, specified by the boundaries). The global orientation of the surface (0° out of the frontoparallel plane) and of the surface markings (90° form the rulings, i.e., lines of curvature) were correctly estimated from the image, as described in the text. Perspective projection supports the unambiguous estimation of surface shape.

Fig. 10
Fig. 10

(a) Oriented flow field composed of parallel line segments mapped under an isometry to parallel geodesics (in this case, lines of curvature) on a cylinder. (b) More natural oriented texture, mapped under an isometry to a cone. The apparent flow lines follow parallel geodesics of the cone.

Fig. 11
Fig. 11

Under spherical perspective, parallel lines in the world project to great circles on the view sphere that intersect at a common pole. Since the tangents of parallel texture flow lines along a surface ruling are parallel, their projections on the view sphere are tangent to great circles that intersect at a common pole.

Fig. 12
Fig. 12

Image of a texture composed of parallel ellipses with aspect ratio=0.5, mapped onto a Gaussian ridge. The solid line represents the measured flow line in the center of the image, computed by integrating the flow field given by the set of ellipse orientations in the image. The dotted line represents the projected flow line in the center of the image, that is, the projection of a flow line on the surface. The projected flow line is not directly available to an observer. The difference between the measured flow line and the projected flow line reflects the error that would be induced in one’s shape estimate should one apply the geodesic-constraint equation to the measured flow lines.

Fig. 13
Fig. 13

Illustration of the effects of projection on ellipse orientation. Ellipses rotated out of the frontoparallel plane projected to ellipses in the image. The solid arrows on the left figures represent the orientation of the ellipses on the surface. The solid arrows on the right represent the projections of the orientation vectors, and the dashed arrows represent the orientations of the projected ellipses. The difference between the two illustrates the fundamental problem in directly applying the shape-from-contour theory to weakly oriented texture flows: The image texture flow as given by the texture-orientation field in the image deviates from the projection of the texture flow on the surface.

Fig. 14
Fig. 14

Contour plot of the angular difference between the orientation vector of a projected ellipse and the projected orientation vector of an ellipse in the world. The error represents the orientation error induced by treating local image texture orientation as equivalent to the projection of the local orientation vector of a texture on a surface. Differences for textures with (a) an average aspect ratio=0.125, (b) aspect ratio=0.25, and (c) aspect ratio=0.5.

Fig. 15
Fig. 15

Proportional difference between the curvature of the image texture-flow field and the true projected flow field for textures with three different orientations relative to the direction of maximal curvature: (a) 0° (flow is oriented along a line of curvature), (b) 30° (middle figure) and (c) 60°. The aspect ratio of the texture elements for the texture was fixed at 0.125. Proportional error is given by error=|kmeasured-kprojected|/kprojected, where kmeasured is the curvature of the image texture flow and kprojected is the curvature of the projected surface texture flow field. Both slant and the direction of maximal surface curvature are given in degrees.

Fig. 16
Fig. 16

Same as Fig. 15, but with a texture aspect ratio= 0.25.

Fig. 17
Fig. 17

Same as Fig. 15, but with a texture aspect ratio= 0.5.

Fig. 18
Fig. 18

Images of two texture fields of ellipses mapped onto cylinders oriented at (a) 30° and (b) 60° away from the frontoparallel plane. The texture field on the surface is composed of ellipses with an aspect ratio of 0.5, all oriented along the line of maximal curvature of the cylinder. The solid line shows the idealized image texture flow line through the center of the image (computed with a much denser array of ellipses than are shown in the figure), and the dashed line shows an idealized representation of the average projected texture-flow line. As described in the text, the image texture flow is computed from an orientation field derived from the ellipses in the image. The projected flow is computed by projecting the flow lines on the surface into the image. Only the former is directly available to an observer.

Fig. 19
Fig. 19

Images of two texture fields of ellipses mapped onto cylinders oriented (a) in the frontoparallel plane and (b) 45° away from the frontoparallel plane. For both images the texture field contains ellipses with an aspect ratio=0.25, all oriented at an angle of 30° away from the lines of maximal curvature of the cylinders. The solid line shows an idealized representation of the image texture flow line through the center of the image, and the dashed line shows an idealized representation of the average projected texture flow line.

Fig. 20
Fig. 20

Images of two texture fields of ellipses mapped onto cylinders oriented (a) in the frontoparallel plane and (b) 45° away from the frontoparallel plane. For both images the texture field contains ellipses with an aspect ratio=0.5, all oriented at an angle of 30° away from the lines of maximal curvature of the cylinders. The solid line shows an idealized representation of the image texture flow line through the center of the image, and the dashed line shows an idealized representation of the average projected texture flow line.

Fig. 21
Fig. 21

Two oriented textures mapped onto a Gaussian ridge. (a) oriented along lines of maximal curvature on the surface, (b) oriented 37° away from the lines of maximal curvature. The shapes of the two surfaces appear markedly different. The texture in (b) appears to be interpreted as if the texture flowed along lines of curvature of a surface.

Fig. 22
Fig. 22

Texture flow that does not follow lines of curvature on a cylindrical surface provides perceptually salient information about the curvature of the surface. (a) A homogeneous texture oriented 30° away from the line of maximal curvature on a flattened cylinder with aspect ratio=0.66. (b) A similar texture pattern on an elongated cylinder with aspect ratio=2.0. The surface in (b) clearly appears more elongated than the one on (a). In this case the virtual occluding contours on either side of the cylinder preclude a line-of-curvature interpretation for the texture.

Equations (44)

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κnS(s)=|[n(s)  NS(s)]  V(s)|n(s)  V(s), n(s)  NS(s)NS(s)  V(s), n(s)  V(s)|n(s)  NS(s)|2 κ(s);
κnS(s)=1ρ(s)|[n(s)  NS(s)]  V(s)|n(s)  V(s), n(s)  NS(s)NS(s)  V(s), n(s)  V(s)|n(s)  NS(s)|2κg(s),
NS(s)s=f(NS(s);r(s), n(s), κ(s), V),
NS(s)s=κnS(s)sin[θ(s)],
sin[θ(s)]=[1-RS(s), TS(s)2]1/2.
NS(s)s=κnS(s)[1-RS(s), TS(s)2]1/2.
NS(s)/s|NS(s)/s|=[RS(s)  NS(s)] : κnS(s)0-[RS(s)  NS(s)] : κnS(s)<0;
NS(s)s=κnS(s) RS(s)  NS(s)[1-RS(s), TS(s)2]1/2.
NS(s)s=κnS(s) RS(s)  NS(s)[1-RS(s), TS(s)2]1/2ss.
ss=1t, T,
NS(s)s=κnS(s) RS(s)  NS(s)[1-RS(s), TS(s)2]1/21t(s), T(s).
RS(s)=[r(s)  V]  NS(s) 1|(r(s)  V)  NS(s)|,
TS(s)=[t(s)  V]  NS(s) 1|(t(s)  V)  NS(s)|,
=n(s)  NS(s) 1|n(s)  NS(s)|.
NSs=κ(s) {[r(s)  V]  NS(s)}  NS(s)|[r(s)  V]  NS(s)|×|[n(s)  NS(s)]  V|[NS(s)  V]  V, n(s)|n(s)  NS(s)|×11-[r(s)  V]  NS(s)|[r(s)  V]  N(s)|, n(s)  NS(s)|n(s)  NS(s)|21/2.
r : s1s2 ; s2=f(s1, r(s1)),
NSs1=κ1(s1) {[r(s1)  V]  NS(s1)}  NS(s1)|[r(s1)  V]  NS(s1)||[n1(s1)  NS(s1)]  V|[NS(s1)  V]  V, n1(s1)|n1(s1)  NS(s1)|×11-[r(s1)  V]  NS(s1)|[r(s1)  V]  NS(s1)|, n1(s1)  NS(s1)|n1(s1)  NS(s1)|21/2.
NSs2=κ2(s2) {[r(s2)  V]  NS(s2)}  NS(s2)|[r(s2)  V]  NS(s2)||[n2(s2)  NS(s2)]  V|[NS(s2)  V]  V, n2(s2)|n2(s2)  NS(s2)|×11-[r(s2)  V]  NS(s2)|[r(s2)  V]  NS(s2)|, n2(s2)  NS(s2)|n2(s2)  NS(s2)|21/2,
NSs1=κ2(s1) {[r(s1)  V]  NS(s1)}  NS(s1)|[r(s1)  V]  NS(s1)||[n2(s1)  NS(s1)]  V|[NS(s1)  V]  V, n2(s1)|n2(s1)  NS(s1)|×11-[r(s1)  V]  NS(s1)|[r(s1)  V]  NS(s1)|, n2(s1)  NS(s1)|n2(s1)  NS(s1)|21/2s2s1,
NS(s)S=κg(s) {[r(s)  V(s)]  NS(s)}  NS(s)|[r(s)  V(s)]  NS(s)||[n(s)  NS(s)]  V(s)|[NS(s)  V(s)]  V(s), n(s)|n(s)  NS(s)|×11-[r(s)  V(s)]  NS(s)|[r(s)  V(s)]  NS(s)|, n(s)  NS(s)|n(s)  NS(s)|21/2.
w(u, v)=[cos θ(u, v), sin θ(u, v)]T.
X=XuXv=x/ux/vy/uy/vz/uz/v.
E[T(u, v)]=X(u, v)[E[w]].
t(θ, ω)=[T(θ, ω)  V(θ, ω)]  V(θ, ω)|T(θ, ω)  V(θ, ω)|,
M(x, y)=mxx(x, y)mxy(x, y)mxy(x, y)myy(x, y)=Ω(x,y)2L(x, y)x2Ω(x,y)2L(x, y)xyΩ(x,y)2L(x, y)xyΩ(x,y)2L(x, y)y2,
MI=P-1TMSP-1,
n(s)=T(s)  NS(s)|T(s)  NS(s)|.
κ(s)=n(s)s, t(s),
n(s)s=T(s)/s  V|T(s)  V|-T(s)/s  V, T(s)  VT(s)  V|T(s)  V|3,
κ(s)=T(s)/s  V, t(s)|T(s)  V|-T(s)s  V, T(s)  VT(s)  V, t(s)|T(s)  V|3.
κ(s)=T(s)/s  V, t(s)|T(s)  V|.
T(s)s=T(s)sss=κnS(s)NS(s) ss.
κ(s)=κnSNS(s)  V, t(s)|T(s)  V|ss.
TS(s)=[t(s)  V]  NS(s) 1|[t(s)  V]  NS(s)|,
TS(s)=n(s)  NS(s) 1|n(s)  NS(s)|,
κ(s)=NS(s)  V(s), n(s)  V(s)|n(s)  NS(s)|2|[n(s)  NS(s)]  V(s)|n(s)  V(s), n(s)  NS(s) κnS(s).
κnS(s)=|[n(s)  NS(s)]  V(s)|n(s)  V(s), n(s)  NS(s)NS(s)  V(s), n(s)  V(s)|n(s)  NS(s)|2 κ(s).
w(s)=t(s)  V(s).
κg(s)=t(s)s, w(s).
κg(s)=w(s)s, t(s).
w(s)=T(s)  NS(s)|T(s)  NS(s)|.
κg(s)=T(s)/s  V(s), t(s)|T(s)  V(s)|+T(s)  V(s)/s, t(s)|T(s)  V(s)|-T(s)/s  V+T(s)  V(s)/s, T(s)  VT(s)  V, t(s)|T(s)  V|3.
κg(s)=T(s)s  V(s), t(s)|T(s)  V(s)|.
κnS(s)=1ρ(s)|[n(s)  NS(s)]  V(s)|n(s)  V(s), n(s)  NS(s)NS(s)  V(s), n(s)  V(s)|n(s)  NS(s)|2 κg(s).

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