Three New Algorithms for Projective Bundle Adjustment with Minimum Parameters

Bundle adjustment is a technique used to compute the maximum likelihood estimate of structure and motion from image feature correspondences. It practice, large non-linear systems have to be solved, most of the time using an iterative optimization process starting from a sub-optimal solution obtained by using linear methods. The behaviour, in terms of convergence, and the computational cost of this process depend on the parameterization used to represent the problem, i.e. of structure and motion. In this paper, we address the problem of finding a minimal parameterization of projective structure and motion, i.e. when camera calibration is not available. Most of the existing parameterizations are either sub-optimal, in the sense that they do change the cost function, or quite complicated to implement, requiring different closed-form expressions, also called maps, to model every case. Without loss of generality, we restrict the problem to the minimal parameterization of the two-view projective motion, equivalent to the fundamental matrix. We propose to classify existing ways allowing to obtain a minimal set of parameters into three categories and present three new algorithms, one for each category. These algorithms are simple to implement and yield minimal parameterizations. We also address the problem of minimally parameterizing homogeneous entities. We present what we call mapped coordinates, a tool allowing to optimize homogeneous entities over a minimal number of parameters. We make extensive use of this tool for both structure and motion parameterization. We compare these algorithms with existing ones using both simulated data and real images. In the light of these experiments, it appears that the new algorithms perform better than existing ones in terms of computational cost while achieving equivalent performances in terms of convergence.