Necessary and Sufficient Conditions for a Unique Solution of Plane Motion and Structure

First, we show an uncertain situation for tile two-view plane motion problem in which an infinite number of solutions may result. Then, we present necessary and sufficient conditions Ior a unique solution of plane motion and structure from any number of views. These algorithm-independent conditions enhance our understanding about the problem of estimating plane motion and structure from image sequences and may be incorporated in any practical algorithms for estimating the motion and slructme of a planar surface. 1 I n t r o d u c t i o n Although many plane motion algorithms have been developed (e.g., [9], [11], [6], [3], [7], and [1] ), the conditions obtained for a unique solution of motion and structure of a planar surface (1911, 1171, I11], [1], 113]) are restricted to only two and three views. Even for two and three views, the existing conditions are not complete in the sense that some rare sitttalions have been ignored, as will be discussed here. This paper presents necessary attd sufficient conditions for a unique solution of the motion and structure of a planar surface from any number of views under perspective projection. First, we show an uncertain situation in the two-view problem in which an infinite number of solutions result. Though this uncertain situation rarely occurs for opaque planes, it has to be considered when some of the images are observed through a mirror. The knowledge of this uncertain situation therefore help analyze mirror-reflected images. Then, we present necessary and sufficient conditions for a unique solution of the motion and structure from multiple images. These conditions apply to any number of images and are algorithm-independent. 2 T h e U n c e r t a i n S ih la l ion in Ihe Two-View P r o b l e m A plane can be described by an equation as follows: NIX=[nt 112 l l 3 ] X = 1, (2-1) where X = [X, Y, Z] r is a poinl on the plane, and N is the plane parameters which are used to represent the plane. A degenerate case occurs when a plane has an equation of NTX = 0. In this case, the projeclion of the plane in the image plane is a line. Degenerate views can be removed from the image sequence, since it is difficult to obtain point correspondences when the projection of a plane is a line. Therefore; in The support of National Science Foundation and Defense Advanced Research Projects Agency under grant IRI-89-02728, and lhe US Army Advanced Construction Technology Center under granl 1)AAL (/3-87 K--001), is gratefully acknowledged .