2 D-3 D Pose Estimation

In this paper we describe a robust technique for solving the 3D to 2D perspective projection pose estimation problem given corresponding point sets. The technique has considerable advantage over the least squares technique in that twenty or thirty percent of the corresponding point pair matches can be completely incorrect and the robust technique is able to determine the correct pose almost as accurately as the least squares technique if the least squares technique were to be given only the seventy to eighty percent correctly matched corresponding point pairs. Evidently, when the least squares technique is given a corresponding point pair data set with twenty or thirty percent of the corresponding pairs being incorrect matches, the technique becomes one of least virtue in the sense that the answer determined by least squares becomes virtually meaningless. Since computer vision procedures for determining corresponding point matches are notorious for being errorful, techniques for robust estimation of pose are important to have in the computer vision toolbox. Our conclusion about the robust technique are supported by hundreds of thousands of controlled experiments. 1. The Pose Estimation Problem Let y1 , ••• , YN be the observed 3D model points in Euclidean 3-space. Let R be a rotation matrix and t be a translation vector. Let (unl,un2),n = 1, . ,N be the corresponding 2D perspective projection of the 3D points. Then, the relationship between the 3D model points and the 2D perspective projection points is given by Unl = Jr1Yn + t1 raYn + ta f r2Yn + t2 Un2 = raYn + ta t = (t1, t2, ta)' R= U~) where f, the focal length, is the distance of the image plane in front of the origin which is the center of perspectivity. CH2614-6/88/0000/0385$01.00 © 1988 IEEE 385 In the 3D coordinate system of the camera, the perspective projections are given by where Unl = fvnl and Un2 = fvn2· The problem of pose estimation is to determine the unknown rotation matrix R and the translation vector t given the 3D model points and the corresponding 2D perspective projection points on the image plane. This problem is known as the exterior orientation problem in the photogrammetry literature. The dissertation by Szczepanski (1958) surveys nearly 80 different solutions beginning with one given by Schrieber of Karlsruhe in the year 1879. The first robust solution in the computer vision literature was Fischler and Bolles (1981). Wrobel and Klemm (1984) discuss the fact that there are configurations of points for which the solution is unstable. 2. The Iterative Least Squares Solution This section describes iterative procedures for determining a least squares solution for R and t. In the following subsections, we use the superscript or subscript k to denote the values in the k1h iteration step. Let ( Xnl) (Ynl) Xn = Xn2 = R Yn2 + t Xna Yna be the rotated and translated point of Yn· Let dn be the estimated depth of each point Xn relative to the camera coordinate system. 2.1 Method 1 One iterative procedure for determining a least squares solution for R and t is (1) Choose initial reasonable values for the depth d~ of each point. The initial values could, for example, be the same constant for each point, the constant representing an initial guess of how far the object is from the perspective center. (2) Iterate. Suppose the depth values d~, n = 1, . , N are given. Define the depth values for the ( k + 1 )'h iteration by: ( 2.1) Find the rotation matrix Rk and the translation vector tk which minimizes N f~ =I: Wn! RkYn + tkd~vnl! 2 n=l where the {wn I n = 1, . , N} are non-negative weights reflecting the goodness of the observations. Rk and tk constitute the solution to the 3D-3D pose estimation problem (Arun, 1987; Haralick et al., 1987). (2.2) Define