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Unformatted text preview: Recovering Baseline and Orientation from Essential Matrix Berthold K.P. Horn January 1990 Abstract: 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 op- erations. The two solutions for the baseline b can be obtained from the equality bb T = 1 Trace ( EE T ) I − EE T , 2 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 recov- ering 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 [ ( ) 2 surface shape. The same is true of the use of image feature correspon- dences in long-range motion vision (but not in the case of short-range motion vision, where motion can be treated as infinitesimal and rotations can conveniently be represented by vectors [Horn & Weldon 88].) Let b be the baseline (translation of the right center of projection with respect to the left center of projection), while and r are the rays from the left and right centers of projection to a given point in the scene. These vectors are all measured in the coordinate system of the left camera. The coplanarity condition expresses the fact that the rays from the left and the right centers of projection meet, that is, that for some α and β , α = b + β r . ( 1 ) The well-known triple product form of the coplanarity condition, b r ] = , ( 2 ) is obtained by simply taking the dot-product of both sides of this equation with b × r . Since the triple product is the volume of the parallelepiped formed by the three vectors, we see that this is equivalent to requiring that the vectors be coplanar [Zeller 52] [Thompson 59]....
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This note was uploaded on 10/06/2009 for the course ECE Vision taught by Professor Bertholdhorn during the Spring '04 term at MIT.
- Spring '04