Analysis of 3-D Rigid Motion

Abstract In this chapter, we study general 3-D rigid motion with special emphasis on the robustness of computation. The uniqueness of the solution is also an important issue. First, we study 3-D motion of a planar surface, which induces a collineation on the image plane. A noise robust least-squares scheme is presented for determining the collineation from both point correspondence and line correspondence. Then, we show that the 3-D surface position and motion can be computed analytically. For an object of general shape, the analysis is divided into two types: point correspondence analysis and line correspondence analysis. For point correspondence analysis, the problem is first formulated as nonlinear least-squares optimization, and then a noise robust analytical procedure is derived from the epipolar equation expressed in terms of the essential matrix. Line correspondence analysis is also formulated as nonlinear least-squares optimization, and then a noise robust analytical procedure is derived from the motion parallax equations expressed in terms of the essential tensor. In both cases, the "optimal resolution" introduced in Chapter 5 plays an essential role. Finally, we study ambiguities of 3-D interpretation. For point correspondence, multiple interpretations arise if and only if all feature points are on a special type of quadric surface called the critical surface. The condition for the critical surface to degenerate into two planar surfaces is also given. For line correspondence, ambiguities occur if and only if all feature lines are on a special type of algebraic set of dimension 2, called the critical line congruence.