10.6084/m9.figshare.5117485.v1 Orhan Kurt Orhan Kurt An integrated solution for reducing ill-conditioning and testing the results in non-linear 3D similarity transformations Taylor & Francis Group 2017 Similarity transformation Linearized Least Squares direct least squares ill-conditioning testing and quality analysis 65F05 65F08 65F10 65F15 65F20 65F22 65F35 65F40 62F03 62F25 2017-06-19 10:33:40 Journal contribution https://tandf.figshare.com/articles/journal_contribution/An_integrated_solution_for_reducing_ill-conditioning_and_testing_the_results_in_non-linear_3D_similarity_transformations/5117485 <p>To find 3D similarity transformation parameters (a scale, three rotational angles, and three translation elements) between two orthogonal coordinate systems in 3D is an ill-posed non-linear inverse problem by means of common points (their Cartesian components are known in the both systems). The problem can be solved via Linearized Least Squares (LLS) or Direct (non-iterative) Least Squares (DLS). Since the parameters in LLS take different quantities (and units) from each other, the condition problems can arise during the solution of normal equations. In this paper, we propose a combined solution to reducing ill-conditioning and to perform precision analysis and global outlier test in LLS accordingly. The way is based on column norming and uses the normalized unknowns instead of the original ones at the solution stage of the normal equations. While the global outlier test is fulfilled on the normalized unknowns, the original unknowns and their precisions obtained using the normalized matrix with linear transformation to the normalized unknowns and by the error propagation law to their variance-covariance matrix. A direct method (Direct Least Squares, DLS) is used to compute the initial values of LLS for reducing iteration number (to 1–3 cycles) with a 1.E-7 threshold in the paper. And, when the solution of 3D-ST having large rotational angles and uncertainties, it is also shown that DLS goes to a worse solution than LLS by means of a simulated transformation problem.</p>