Closed-form least squares solution for the planar three-parameter transformation with a general weight matrix

Closed-form solutions of non-linear least squares problems avoid the risk of divergence or drifting to a local minimum as in the iterative methods. The unique solution is obtained with low numerical effort. In this contribution, a closed-form solution of the non-symmetric planar three-parameter transformation problem is derived in detail. The presented solution considers a fully populated weight matrix and is not restricted to infinitesimal rotation angles. The analysis of the objective function demonstrates that there may also be a secondary local minimum. Applying iterative approaches have the risk of getting stuck in the local minimum of the objective function.