Linear Recovery of the Exterior Orientation Parameters in a Planar Object Space

The Direct Linear Transformation (DLT) model requires 3-D object space control points to estimate the full set of its parameters that can be used to recover exterior orientation parameters. The use of DLT in a planar object space leads to a rank deficient model. This rank deficient model leaves the DLT defined up to a 2-D projective transformation, which makes the explicit solution of the exterior orientation parameters or a single photo-resection represented by the camera position and orientations, a non-trivial task. This paper presents a new closed form solution to a single photo-resection in a planar object space based on homogenous coordinate representation and matrix factorization. Homogenous coordinate representation offers a direct matrix correspondence between the parameters of the 2-D projective transformation and the collinearity model. This correspondence lends itself to a direct matrix factorization to solve the photo-resection problem. The matrix factorization starts by recovering the elements of the rotation matrix and then solving for the camera position. It will be shown that an incomplete representation of the rotation matrix is captured by the 2-D projective parameters but the actual physical parameters of the rotation matrix still can be recovered explicitly and without any ambiguity. These elements were used to build a complete rotation matrix, which in turn facilitates the correct solution of the camera position. The developed solution can serve as a complementary companion to the classical DLT model. In this paper, a detailed derivation of the proposed solution will be presented, accommodated with two simulated examples to demonstrate its validity. The first example simulates aerial photography and the second one simulates close range photogrammetric application.