Skip to main content
Log in

A Theory of Specular Surface Geometry

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

A theoretical framework is introduced for the perception of specular surface geometry. When an observer moves in three-dimensional space, real scene features such as surface markings remain stationary with respect to the surfaces they belong to. In contrast, a virtual feature which is the specular reflection of a real feature, travels on the surface. Based on the notion of caustics, a feature classification algorithm is developed that distinguishes real and virtual features from their image trajectories that result from observer motion. Next, using support functions of curves, a closed-form relation is derived between the image trajectory of a virtual feature and the geometry of the specular surface it travels on. It is shown that, in the 2D case, where camera motion and the surface profile are coplanar, the profile is uniquely recovered by tracking just two unknown virtual features. Finally, these results are generalized to the case of arbitrary 3D surface profiles that are traveled by virtual features when camera motion is not confined to a plane. This generalization includes a number of mathematical results that substantially enhance the present understanding of specular surface geometry. An algorithm is developed that uniquely recovers 3D surface profiles using a single virtual feature tracked from the occluding boundary of the object. All theoretical derivations and proposed algorithms are substantiated by experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bellver-Cebreros, C. and Rodriguez-Danta, M. 1992. Caustics and the Legendre transform. Optics Communications, 92(4-6):187- 192.

    Google Scholar 

  • Blake, A. 1985. Specular stereo. Proc. 9th IJCAI Conf., pp. 973-976.

  • Blake, A. and Brelstaff, G. 1988. Geometry from specularity. Proceedings of ICCV, Florida, pp. 394-403.

  • Blake, A. and Bulthoff, H. 1991. Shape from specularity: Computation and psychophysics. Phil. Trans. Royal Society London B., 331:237-252.

    Google Scholar 

  • Brelstaff, G. and Blake, A. 1988. Detecting specular reflections using Lambertian constraints. Proceedings of ICCV, Florida, pp. 297- 302.

  • Bruce, J. W. and Giblin, P. J. (Eds.) 1992. Curves and Singularities. Cambridge University Press, second edition.

  • Cornbleet, S. 1984. Microwave and Optical Ray Geometry. John Wiley and Sons.

  • do Carmo, M. P. 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall Inc.

  • Giblin, P. J. 1995. Personal communication.

  • Guggenheimer, H. W. 1963. Differential Geometry. McGraw-Hill Book Company, Reprinted by Dover Publications (1977).

  • Healey, G. and Binford, T. O. 1988. Local shape from specularity. CVGIP, pp. 62-86.

  • Horn, B. K. P. 1986. Robot Vision. The MIT Press.

  • Ikeuchi, K. 1981. Determining surface orientation of specular surfaces by using the photometric stereo method. IEEE Transactions on Pattern Analysis and Machine Intelligence, 3(6):661-669.

    Google Scholar 

  • Klinker, G. J., Shafer, S. A., and Kanade, T. 1988. The measurement of highlights in color images. International Journal of Computer Vision, 2:7-32.

    Google Scholar 

  • Koenderink, J. J. and van Doorn, A. J. 1980. Photometric invariants related to solid shapes. Optica Acta, 27(7):981-996.

    Google Scholar 

  • Lee, S. W. 1991. Understanding of surface reflections in computer vision by color and multiple views. Ph. D. Thesis, University of Pennsylvania.

  • Longuet-Higgens, M. S. 1960. Reflection and refraction at a random moving surface: (i) Patterns and paths of specular points. Journal of the Optical Society of America, 509:838-844.

    Google Scholar 

  • Nayar, S. K., Ikeuchi, K., and Kanade, T. 1990. Determining shape and reflectance of hybrid surfaces by photometric sampling. IEEE Transactions on Robotics and Automation, 6(4):418- 431.

    Google Scholar 

  • Nayar, S. K., Fang, X. S., and Boult, T. 1996. Separation of reflection components using color and polarization. International Journal of Computer Vision, To appear in 1996.

  • Sanderson, A. C., Weiss, L. E., and Nayar, S. K. 1988. Structured highlight inspection of specular surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 10(1):44-55.

    Google Scholar 

  • Schultz, H. 1994. Retrieving shape information from multiple images of a specular surface. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(2):195-201.

    Google Scholar 

  • Stavroudis, O. N. 1972. The Optics of Rays, Wavefronts, and Caustics. Academic Press.

  • Symosek, P. F. 1985. Parameter estimation and classification of machine parts based on specular or mirror-like image data. Ph. D. Thesis, Brown University.

  • Thrift, P. and Lee, C.-H. 1983. Using highlights to constrain object size and location. IEEE Transactions on Systems, Man, and Cybernetics, 13(3):426-431.

    Google Scholar 

  • Torrance, K. and Sparrow, E. 1967. Theory for off-specular reflection from rough surfaces. Journal of the Optical Society of America, 57:1105-1114.

    Google Scholar 

  • Ullman, S. 1976. On visual detection of light sources. Biological Cybernetics, 21:205-211.

    Google Scholar 

  • Waldon, S. and Dyer, C. R. 1993. Dynamic shading, motion parallax and qualitative shape. Proceedings of the IEEE Workshop on Qualitative Vision, pp. 61-70.

  • Wolff, L. B. and Boult, T. E. 1991. Constraining object features using a polarization reflectance model. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-13(6):635-657.

    Google Scholar 

  • Zheng, J. Y., Fugakawa, Y., and Abe, N. 1995. Shape and model from specular motion. Proceedings of ICCV, pp. 72- 79.

  • Zisserman, A., Giblin, P., and Blake, A. 1989. The information available to a moving observer from specularities. Image and Vision Computing, 7:287-291.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Oren, M., Nayar, S.K. A Theory of Specular Surface Geometry. International Journal of Computer Vision 24, 105–124 (1997). https://doi.org/10.1023/A:1007954719939

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1007954719939

Keywords

Navigation