The Kernel of a Direct Closed-form Solution to General Relative Orientation

Recovering the relative geometry of two perspective stereo images is a classical problem in photogrammetry and also a basic problem in computer vision. General relative orientation refers to recovering ve relative orientation parameters and two principal distances from pure image measurements. This paper exposes the kernel of a new direct closed-form solution to this problem while the general frame of this solution was rst published in 1995 Pan et al, 1995a, 1995b]. The approach recasts the explicit stereo coplanarity equation into an implicit form whose coeecients, which can be put in a coplanarity matrix, are solvable via a direct solution from the image measurements. The two principal distances (focal lengths) are then fully determined by sixth-order algebraic equations whose coeecients are fourth-order polynomials of coplanarity matrix elements. Fortunately, these sixth-order algebraic equations are solvable in closed-form. After the two principal distances have been solved, the relative orientation (5 parameters or, rotation matrix and baseline vector) are then solved successively also in closed-form. This direct closed-form solution is useful not only in real-time implementation of stereo vision, but also for providing the approximate values of unknown geometric parameters to high-precision photogrammetry, as well as for direct functional analysis of error propagation.