Multiple Interpretations of the Shape and Motion of Objects from Two Perspective Images

The following problem is investigated: given the position coordinates in two images of all points on an object obtained from two different camera positions, under what conditions can there be more than one interpretation for the shape of the object and the transformation between the coordinate systems at the two camera positions? It is shown that only certain hyperboloids of one sheet and their degeneracies, such as hyperbolic paraboloids, circular cylinders, and intersecting planes, that are viewed from a point on their surface can give rise to an ambiguity. In the case of hyperboloids of one sheet and hyperbolic paraboloids, there can be three possible solutions. In the case of circular cylinders and intersecting planes, there are at most two solutions. The author gives the relationship among the multiple interpretations and determines them all in closed form in terms of the true solution.