A Comparison of Rotation Parameterisations for Bundle Adjustment

Abstract: Bundle Adjustment is an iterative process where 3D information is estimated from 2D image measurements. Typically, the position of object points are estimated simultaneously with the position and orientation of the cameras. While the object points and camera positions have a straightforward "natural" parameterisation, several possibilities exist for the rotation. In this thesis, seven parameterisation of the rotation were investigated; Euler angles (two variants), the Rodriguez representation, the axis-and-angle representation, unit quaternions, and two variants of the direction cosine matrix (DCM). The Euler and Rodriguez parameterisation are common in photogrammetry and each has three parameters. The other parameterisations have more parameters and one or more constraint between them. The parameterisations were analyzed with respect to singularities, i.e. well-defined rotations that do not have any bounded and/or unique set of parameters. Four bundle adjustment experiments were setup, each corresponding to a singularity for one or more parameterisations. A fitth, singularity-free, experiment was also added. The experiments were perturbation studies that investigated the convergence properties of each parameterisation. The unconstrained parameterisations were solved by a damped and undamped Gauss-Newton algorithm, whereas the parameterisations with constraints were solved using damped and undamped algorithms based on the Gauss-Helmert estimation model. As expected, the parameterisations corresponding to the constructed singularity had higher failure rates and required more iterations and execution time than the others when it did converge. Excluding their singular cases, the Euler xyz and Rodriguez representations were the fastest with about 37% of the dcm. Of the singularity-free parameterisation, the unit quaternion was the fastest with 79% of the dcm. Surprisingly, the undamped bundle algorithms converged more often and faster than the damped bundle algorithms, even close to singularities. However, the undamped convergence was to a higher degree associated with numerical warnings and convergence toward angular values outside the nominal 2 range. The results suggest that if singularities are not expected, the Euler xyz and Rodriguez representations are the best of the tested parameterisations. Otherwise, the unit quaternion is the best. As an alternative to the latter case, the switching algorithm by Singla may be used, at the expense of a more complex algorithm.