Shading without shape
Authors:
Michael J. Brooks, Wojciech Chojnacki and Ryszard Kozera
Journal:
Quart. Appl. Math. 50 (1992), 27-38
MSC:
Primary 68U10; Secondary 35F10, 78A05
DOI:
https://doi.org/10.1090/qam/1146621
MathSciNet review:
MR1146621
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Abstract: A smooth object depicted in a monochrome image will often exhibit brightness variation, or shading. Of interest in computer vision is the inverse problem of how object shape may be recovered from such an image. This is referred to as the shape-from-shading problem. When the imaging conditions are such that an overhead point-source illuminates a smooth Lambertian surface, the problem may be formulated mathematically as that of finding a solution to an eikonal equation. In this paper, we seek images for which there are no corresponding object shapes. We are therefore concerned with the nonexistence of (bounded) solutions to certain eikonal equations. Specifically, we give a necessary and sufficient condition for a circularly-symmetric eikonal equation to admit exclusively unbounded solutions. In addition, we give a sufficient condition for an eikonal equation to have no solution. Examples are presented that elucidate the significance of these results for computer vision.
M. J. Brooks, Two results concerning ambiguity in shape from shading, Proceedings of the National Conference on Artificial Intelligence, Washington, D.C., August 22–26, 1983, The American Association for Artificial Intelligence, sponsor, pp. 36–39
- Anna R. Bruss, The eikonal equation: some results applicable to computer vision, J. Math. Phys. 23 (1982), no. 5, 890–896. MR 655907, DOI https://doi.org/10.1063/1.525441
- Percy Deift and John Sylvester, Some remarks on the shape-from-shading problem in computer vision, J. Math. Anal. Appl. 84 (1981), no. 1, 235–248. MR 639534, DOI https://doi.org/10.1016/0022-247X%2881%2990161-X
- Philip Hartman, Ordinary differential equations, S. M. Hartman, Baltimore, Md., 1973. Corrected reprint. MR 0344555
- Berthold Horn, Obtaining shape from shading information, The psychology of computer vision, McGraw-Hill, New York, 1975, pp. 115–155. MR 0416135
B. K. P. Horn, private communication, 1987
B. K. P. Horn, R. Szeliski, and A. Yuille, private communication, 1989
- Berthold K. P. Horn and Michael J. Brooks (eds.), Shape from shading, MIT Press Series in Artificial Intelligence, MIT Press, Cambridge, MA, 1989. MR 1062877
M. J. Brooks, Two results concerning ambiguity in shape from shading, Proceedings of the National Conference on Artificial Intelligence, Washington, D.C., August 22–26, 1983, The American Association for Artificial Intelligence, sponsor, pp. 36–39
A. R. Bruss, The eikonal equation: some results applicable to computer vision, J. Math. Phys. 23, 890–896 (1982)
P. Deift and J. Sylvester, Some remarks on the shape-from-shading problem in computer vision, J. Math. Anal. Appl. 84, 235–248 (1981)
P. Hartman, Ordinary Differential Equations, Wiley, New York, 1973
B. K. P. Horn, Obtaining shape from shading information, The Psychology of Computer Vision, P. H. Winston, ed., McGraw-Hill, New York, 1975
B. K. P. Horn, private communication, 1987
B. K. P. Horn, R. Szeliski, and A. Yuille, private communication, 1989
B. K. P. Horn and M. J. Brooks, eds., Shape from Shading, MIT Press, Cambridge, Mass., 1989
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Article copyright:
© Copyright 1992
American Mathematical Society
