Pros and cons of constrained and unconstrained formulation of the bundle adjustment problem

Résumé 0

Two implementations of the bundle adjustment problem were applied to a subset of the Z ̈urich City Hall reference data set. One implementation used the standard Euler angle parameterisation of the rotation matrix. The second implemen- tation used all nine elements of the rotation matrix as unknowns and six functional constraints. The second formulation was constructed to reduce the non-linearity of the optimisation problem. The hypothesis was that a lower degree of non-linearity would lead to faster convergence. Furthermore, each implementation could optionally use the line search damping technique known from optimisation theory. The algorithms were used to solve the relative orientation problem for a varying number of homologous points from 33 different camera pairs. The results show that the constrained formulation has marginally better convergence properties, with or without damping. However, damping alone halves the number of convergence failures at a minor computational cost. The conclusion is that except to avoid the singularities associated with the Euler angles, the preferred use of the constrained formulation remains an open question. However, the results strongly suggest that the line search damping technique should be included in standard implementations of the bundle adjustment algorithm.