The projective geometry of ambiguous surfaces

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Abstract
The projective geometry underlying the ambiguous case of scene reconstruction from image correspondences is developed. The am biguous case arises when reconstruction yields two or more essentially different surfaces in space, each capable of giving rise to the image correspondences. Such surfaces naturally occur in complementary pairs. Ambiguous surfaces are examples of rectangular hyperboloids. Complementary ambiguous surfaces intersect in a space curve of degree four, which splits into two components, namely a twisted cubic (space curve of degree three), and a straight line. For each ambiguous surface compatible with a given set of image correspondences, a complementary surface compatible with the same image correspondences can always be found such that both the original surface and the twisted cubic contained in the intersection of the two surfaces are invariant under the same rotation through 180°. In consequence, each ambiguous surface is subject to a cubic polynomial constraint. This constraint is the basis of a new proof of the known result that there are, in general, exactly ten scene reconstructions compatible with five given image correspondences. Ambiguity also arises in reconstruction based on image velocities rather than on image correspondences. The two types of ambiguity have m any sim ilarities because image velocities are obtained from image correspondences as a limit, when the distances between corresponding points become small. It is shown that the amount of similarity is restricted, in that when passing from image correspondences to image velocities, some of the detailed geometry of the ambiguous case is lost.

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