Noniterative Approach for Solving the Indirect Problems of Linear Reference Frame Transformations

Abstract

A linear transformation, including a similarity model and an affine model, plays an important role in reference frame transformation problems. It establishes geometric relations between different reference frames by adopting physically interpretable parameters, and allows one to integrate positional coordinates across various systems. In this study, an analytical noniterative approach for solving the indirect problem (i.e., parameter estimation) of a linear transformation model is proposed. It gives a closed-form solution for the transformation without any prerequisite on the magnitude of parameters. Consequently, neither good initial values nor iterations are required during the parameter estimation process. Through numerical examples, it is illustrated that the proposed approach is capable of giving accurate parameter estimates as a classic least-squares technique does, but with much higher computational efficiency. In a reference frame transformation problem of 1,000 control points, the proposed approach takes less than 0.1 s to produce an accurate solution, while an ordinary least-squares approach takes 10–17 s to finish the same analysis. With the above advantages, this approach would especially benefit the cases in which a real-time application or large network is involved in an analysis.