Nonlinear analysis of the three-dimensional datum transformation[conformal group C7(3)]

TLDR: The problem of incorporating the stochasticity measures of both systems of coordinates involved in the seven parameter datum transformation problem [conformal group ℂ7(3)] which is free of linearization and any iteration procedure can be considered to be solved.

Abstract: The weighted Procrustes algorithm is presented as a very effective tool for solving the three-dimensional datum transformation problem In particular, the weighted Procrustes algorithm does not require any initial datum parameters for linearization or any iteration procedure As a closed-form algorithm it only requires the values of Cartesian coordinates in both systems of reference Where there is some prior information about the variance–covariance matrix of the two sets of Cartesian coordinates, also called pseudo-observations, the weighted Procrustes algorithm is able to incorporate such a quality property of the input data by means of a proper choice of weight matrix Such a choice is based on a properly designed criterion matrix which is discussed in detail Thanks to the weighted Procrustes algorithm, the problem of incorporating the stochasticity measures of both systems of coordinates involved in the seven parameter datum transformation problem [conformal group ℂ7(3)] which is free of linearization and any iterative procedure can be considered to be solved Illustrative examples are given