Mirror uncertainty and uniqueness conditions for determining shape and motion from orthographic projection

This paper presents new forms of necessary and sufficient conditions for determining shape and motion to within a mirror uncertainty from monocular orthographic projections of any number of point trajectories over any number of views. The new forms of conditions use image data only and can therefore be employed in any practical algorithms for shape and motion estimation. We prove that the mirror uncertainty for the three view problem also exists for a long sequence: if shape S is a solution, so is its mirror image S′ which is symmetric to S about the image plane. The necessary and sufficient conditions for determining the two sets of solutions are associated with the rank of the measurement matrix W. If the rank of W is 3, then the original 3D scene points cannot be coplanar and the shape and motion can be determined to within a mirror uncertainty from the image data if and only if there are three distinct views. This condition is different from Ullman's theorem (which states that three distinct views of four noncoplanar points suffice to determine the shape and motion up to a reflection) in two aspects: (1) it is expressed in terms of image data; (2) it applies to a long image sequence in a homogeneous way. If the rank of W is 2 and the image points in at least one view are not colinear in the image plane, then there are two possibilities: either the motion is around the optical axis, or the 3-D points all lie on the same plane. In the first case, the motion can be determined uniquely but the shape is not determined. In the second case, a necessary and sufficient condition is to be satisfied and at least 3 point trajectories over at least 3 distinct views are needed to determine the shape in each view to within a mirror uncertainty, and the number of motion solutions is equal to the combinatorial number of the possible positions of the plane in different views. The necessary and sufficient condition is associated with the rank of a matrix C: if C has a rank of 1, the plane is undetermined; if C has a rank of 2 (implying there are exactly 3 distinct views), then a necessary and sufficient condition, whose physical meaning is not completely clear, is to be satisfied to determine the plane to within 2 sets; if C has a rank of 3 (implying there are 4 or more distinct views), then the plane can always be determined to within two sets. If the rank of W is 2 or 1 and the image points in each view are colinear in the image plane, then the three dimensional motion problem reduces to a two dimensional motion problem. In this case, the uniqueness condition is associated with the rank of the reduced measurement matrix Ψ. If Ψ has a rank of 2, then the original 3D points cannot be colinear in the space and the shape and motion can be determined to within two sets if and only if three or more views are distinct. If Ψ has a rank of 1, there are two possibilities: if the rows of Ψ are identical, then either the original 3D points are not colinear and the motion is zero, or the points are colinear and possibly move between two mirror symmetric positions; if the rows of Ψ are not identical, then the motion is not determined. All proofs are constructive and thus define an algorithm for determining the uniqueness of solution as well as for estimating shape and motion from point trajectories.