On the Non-Linear optimization of projective motion using minimal parameters

Abstract

I address the problem of optimizing projective motion over a minimal set of parameters. Most of the existingworks overparameterize the problem. While this can simplify the estimation process and may ensure well-conditioning of the parameters, this also increases the computational cost since more unknowns than necessary are involved. I propose a method whose key feature is that the number of parameters employed is minimal. The method requires singular value decomposition and minor algebraic manipulations and is therefore straightforward to implement. It can be plugged into most of the optimization algorithms such as Levenberg-Marquardt as well as the corresponding sparse versions. The method relies on the orthonormal camera motion representation that I introduce here. This representation can be locally updated using minimal parameters. I give a detailled description for the implementation of the two-view case within a bundle adjustment framework, which corresponds to the maximum likelihood estimation of the fundamental matrix and scene structure. Extending the algorithm to the multiple-view case is straightforward. Experimental results using simulated and real data show that algorithms based on minimal parameters perform better than the others in terms of the computational cost, i.e. their convergence is faster, while achieving comparable results in terms of convergence to a local optimum. An implementation of the method will be made available.