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Main Title: Total Least Squares Spline Approximation
Author(s): Neitzel, Frank
Ezhov, Nikolaj
Petrovic, Svetozar
Type: Article
Language Code: en
Abstract: Spline approximation, using both values yi and xi as observations, is of vital importance for engineering geodesy, e.g., for approximation of profiles measured with terrestrial laser scanners, because it enables the consideration of arbitrary dispersion matrices for the observations. In the special case of equally weighted and uncorrelated observations, the resulting error vectors are orthogonal to the graph of the spline function and hence can be utilized for deformation monitoring purposes. Based on a functional model that uses cubic polynomials and constraints for continuity, smoothness and continuous curvature, the case of spline approximation with both the values yi and xi as observations is considered. In this case, some of the columns of the functional matrix contain observations and are thus subject to random errors. In the literature on mathematics and statistics this case is known as an errors-in-variables (EIV) model for which a so-called “total least squares” (TLS) solution can be computed. If weights for the observations and additional constraints for the unknowns are introduced, a “constrained weighted total least squares” (CWTLS) problem is obtained. In this contribution, it is shown that the solution for this problem can be obtained from a rigorous solution of an iteratively linearized Gauss-Helmert (GH) model. The advantage of this model is that it does not impose any restrictions on the form of the functional relationship between the involved quantities. Furthermore, dispersion matrices can be introduced without limitations, even the consideration of singular ones is possible. Therefore, the iteratively linearized GH model can be regarded as a generalized approach for solving CWTLS problems. Using a numerical example it is demonstrated how the GH model can be applied to obtain a spline approximation with orthogonal error vectors. The error vectors are compared with those derived from two least squares (LS) approaches.
Issue Date: 22-May-2019
Date Available: 11-Jun-2019
DDC Class: 510 Mathematik
Subject(s): spline approximation
total least squares
constrained weighted total least squares
Gauss-Helmert model
singular dispersion matrix
Sponsor/Funder: DFG, 414044773, Open Access Publizieren 2019 - 2020 / Technische Universität Berlin
Journal Title: Mathematics
Publisher: MDPI
Publisher Place: Basel
Volume: 7
Issue: 5
Article Number: 462
Publisher DOI: 10.3390/math7050462
EISSN: 2227-7390
Appears in Collections:FG Geodäsie und Ausgleichungsrechnung » Publications

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