Weighted Total Least Squares (WTLS) Solutions for Straight Line Fitting to 3D Point Data

Georgios Malissiovas, Frank Neitzel, Sven Weisbrich and Svetozar Petrovic
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Georgios Malissiovas: Institute of Geodesy and Geoinformation Science, Technische Universität Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany
Frank Neitzel: Institute of Geodesy and Geoinformation Science, Technische Universität Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany
Sven Weisbrich: Institute of Geodesy and Geoinformation Science, Technische Universität Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany
Svetozar Petrovic: Institute of Geodesy and Geoinformation Science, Technische Universität Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany

Mathematics, 2020, vol. 8, issue 9, 1-19

Abstract: In this contribution the fitting of a straight line to 3D point data is considered, with Cartesian coordinates x i , y i , z i as observations subject to random errors. A direct solution for the case of equally weighted and uncorrelated coordinate components was already presented almost forty years ago. For more general weighting cases, iterative algorithms, e.g., by means of an iteratively linearized Gauss–Helmert (GH) model, have been proposed in the literature. In this investigation, a new direct solution for the case of pointwise weights is derived. In the terminology of total least squares (TLS), this solution is a direct weighted total least squares (WTLS) approach. For the most general weighting case, considering a full dispersion matrix of the observations that can even be singular to some extent, a new iterative solution based on the ordinary iteration method is developed. The latter is a new iterative WTLS algorithm, since no linearization of the problem by Taylor series is performed at any step. Using a numerical example it is demonstrated how the newly developed WTLS approaches can be applied for 3D straight line fitting considering different weighting cases. The solutions are compared with results from the literature and with those obtained from an iteratively linearized GH model.

Keywords: 3D straight line fitting; total least squares (TLS); weighted total least squares (WTLS); nonlinear least squares adjustment; direct solution; singular dispersion matrix; laser scanning data (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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