Pose determination from angles and relative line lengths using spherical trigonometry

This article describes a two-step approach to pose determination using spherical trigonometry. Line directions are initially determined in the eye co-ordinate system, and then used to determine the pose of the eye system as a Perspective-4-Points problem. Assuming correspondence of points in 2D image and 3D object model are known, constraint equations based on the invariant parameters of intersect angles and relative line lengths are developed. A closed form solution independent of viewing distance is obtained when three angles and three line lengths are known. An iterative solution is obtained when only three angles are known. An orthoperspective projection is assumed and an error analysis identifies the main causes of poor performance. Spherical trigonometry is shown to give a simple second order solution and requires the use of fewer parameters than solutions using cartesian trigonometry. Results are presented for a variety of synthetic image projections of blocksworld objects, with and without noise, and for a real world scene. Good accuracy is demonstrated, with errors around 5°, provided the projection distance between the object and camera is much larger than the size of the object being viewed, the point of convergence of the lines is near the optical axis, and the projection is not from an extreme position. Although robust to input noise caused by poor low level edge detection, the results show that errors of around 20° can occur when the underpinning assumptions are violated.