Recovering Baseline and Orientation from `Essential' Matrix

Certain approaches to the problem of relative orientation in binocular stereo (as well as long-range motion vision) lead to an encoding of the baseline (translation) and orientation (rotation) in a single 3 × 3 matrix called the “essential” matrix. The essential matrix is defined by E = BR, where B is the skew-symmetric matrix that satisfies Bv = b × v for any vector v, with b being the baseline and R the orientation. Shown here is a simple method for recovering the two solutions for the baseline and the orientation from a given essential matrix using elementary matrix operations. The two solutions for the baseline b can be obtained from the equality bbT = 1 2 Trace(EET )I− EET , where I is the 3× 3 identity matrix. The two solutions for the orientation can be found using (b · b)R = Cofactors(E)T − BE, where Cofactors(E) is the matrix of cofactors of E. There is no need to perform a singular value decomposition, to transform coordinates, or to use circular functions, and it is easy to see that there are exactly two solutions given a particular essential matrix. If the sign of E is reversed, an additional pair of solutions is obtained that are related the two already found in a simple fashion. This helps shed some light on the question of how many solutions a given relative orientation problem can have. 1. Coplanarity Condition in Relative Orientation Relative orientation is the well-known photogrammetric problem of recovering the position and orientation of one camera relative to another from five or more pairs of corresponding ray directions [Zeller 52] [Ghosh 72] [Slama et al. 80] [Wolf 83] [Horn 86, 87b]. Relative orientation has to be determined before binocular stereo information can be used to recover