Pose determination from angles and relative, line lengths using spherical trigonometry

This article describes a two-step approach to pose determination using sphe rical trigonometry. Line directions are initially determined in the eye co- ordinate system, and then used to determine the pose of the eye system als a Perspective-4-Points problem. Assuming correspondence of points in 2D ima ge and 3D object model are known, constraint equations based on the invaria nt parameters of intersect angles and relative line lengths are developed. A closed form solution independent of viewing distance is obtained when thr ee angles and three line lengths are known. An iterative solution is obtain ed when only three angles are known. An orthoperspective projection is assu med and an error analysis identifies the main causes of poor performance. S pherical trigonometry is shown to give a simple second order solution and r equires the use of fewer parameters than solutions using cartesian trigonom etry. Results are presented for a variety of synthetic image projections of blocksworld objects, with and without noise, and for a real world scene. G ood accuracy is demonstrated, with errors around 5 degrees, provided the pr ojection distance between the object and camera is much larger than the siz e 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. Alth ough robust to input noise caused by poor low level edge detection, the res ults show that errors of around 20 degrees can occur when the underpinning assumptions are violated. (C) 1999 Elsevier Science B.V. All rights reserve d.