Impossible and ambiguous shading patterns
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A smooth object depicted in a monochrome image will often exhibit brightness variation, or shading. A problem much studied in computer vision has been that of how object shape may be recovered from image shading. When the imaging conditions are such that an overhead point-source illuminates a smooth Lambertian surface, the problem may be formulated as that of finding a solution to an eikonal equation. This article will focus on the existence and uniqueness of such solutions, reporting recent results obtained. With regard to existence, shading patterns are exhibited for which there is no corresponding object shape. Specifically, a necessary and sufficient condition is presented for a circularly symmetric eikonal equation to admit exclusively unbounded solutions; additionally, a sufficient condition is given for an eikonal equation to have no solution whatsoever. In connection with uniqueness, we consider eikonal equations, defined over a disc, such that the Euclidean norm of the gradient of any solution is circularly symmetric, vanishes exactly at the disc center, and diverges to infinity as the circumference of the disc is approached. Contrary to earlier influential work, a class of such eikonal equations is shown to possess simultaneously circularly symmetric and noncircularly symmetric bounded smooth solutions.
KeywordsImage Processing Computer Vision Computer Image Recent Result Imaging Condition
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- Brooks, M.J. 1983. Two results concerning ambiguity in shape from shading. Proc. Nat. Conf. Artif. Intell. American Association for Artificial Intelligence (sponsor). Washington, D.C., August 22–26, pp. 36–39.Google Scholar
- BrooksM.J., ChojnackiW., and KozeraR. 1992a. Shading without shape. Quart. Appl. Math. 50: 27–38.Google Scholar
- BrookM.J., ChojnackiQ., and KozeraR. 1992b. Circularlysymmetric eikonal equations and non-uniqueness in computer vision. J. Math. Anal. Appl. 165: 192–215.Google Scholar
- BrussA.R. 1982. The eikonal equation: some results applicable to computer vision. J. Math. Phys. (5) 23: 890–896.Google Scholar
- DeiftP., and SylvesterJ. 1981. Some remarks on the shape-from-shading problem in computer vision J. Math. Anal. Appl. (5) 34: 235–248.Google Scholar
- HornB.K.P. 1975. Obtaining shape from shading information. The Psychology of Computer Vision. P.H.Winston (ed.). McGraw Hill: New York, pp. 115–155.Google Scholar
- Horn, B.K.P. 1987. Private communication.Google Scholar
- HornB.K.P., and BrooksM.J. (eds.) 1989. Shape from Shading. MIT Press: Cambridge, Mass.Google Scholar
- Horn, B.K.P., Szeliski, R., and Yuille, A. 1989. Private communication.Google Scholar