Point-wise weighted solution for the similarity transformation parameters

Abstract

AbstractIn the paper, we consider solutions to the similarity transformation under different adjustment scenarios. We consider asymmetric cases, i.e., random errors included either in a target system (Gauss–Markov model, the most common case) or in a source system and a symmetric case when both systems are considered to be contaminated by random errors (Gauss–Helmert or errors-in-variables model). The presented solutions are based on Procrustes approach and all cases use point-wise weighting scheme. The asymmetric cases have closed-form solutions but the symmetric one, in the most general case considered herein, does not. In the latter case, weight matrices for a source and target systems are combined into one equivalent weight matrix which depends on a scale factor. This case is solved through a simple iterative process but there are some special instances which lead to closed-form solutions. The paper discusses the behavior of transformation parameters (rotations, translations, and a scale) under different adjustment scenarios.