Many nonlinear frameworks can be related to the classic Relative Orientation problem proposed by Horn [29] [30]. The technique is a two-frame one with a perspective projection camera model. Structure and motion (but not camera internal geometry) are recovered by minimizing a nonlinear cost function.
The technique begins with a setup similar to the one in Figure 4 without any of the epipolar details. In addition, the focal length f is assumed to be given. Assume that the point P is actually represented as unknown 3D coordinates P_{1} and P_{2} in two different coordinate systems (one for each COP). These 3D coordinates are directly related to their 2D projections (u_{1},v_{1}) and (u_{2},v_{2}) by perspective projection as in Equation 7.
The projection of P in one image plane can be defined as a translation and rotation of the 3D point in the coordinate system of the other. Thus, the 3D point P_{1} in COP_{1} can be computed from the 3D point P_{2} in COP_{2} as in Equation 8. Here, rotation R and translation define the relative 3D motion between the two cameras.
Each corresponded pair of points increases the system by three more equations with two more unknowns (the Z_{1} and Z_{2}). The system can undergo an arbitrary scaling so a normality constraint is applied to translation to force a unique solution. In addition, there are orthonormality constraints on the rotation matrix (which only has 3 true degrees of freedom). Thus, the 5 unknowns for the relative 3D motion can be solved using 5 point correspondences. However, a least-squares version with more points is preferred for accuracy. The solution minimizes error using iterative nonlinear optimization. In the process, the values of Z_{1} and Z_{2} are solved for each corresponded point and 3D structure is recovered.