## Total Least Squares Spline Approximation

 dc.contributor.author Neitzel, Frank dc.contributor.author Ezhov, Nikolaj dc.contributor.author Petrovic, Svetozar dc.date.accessioned 2019-06-11T08:29:48Z dc.date.available 2019-06-11T08:29:48Z dc.date.issued 2019-05-22 dc.description.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. en dc.description.sponsorship DFG, 414044773, Open Access Publizieren 2019 - 2020 / Technische Universität Berlin en dc.identifier.eissn 2227-7390 dc.identifier.uri https://depositonce.tu-berlin.de/handle/11303/9494 dc.identifier.uri http://dx.doi.org/10.14279/depositonce-8548 dc.language.iso en en dc.rights.uri https://creativecommons.org/licenses/by/4.0/ en dc.subject.ddc 510 Mathematik de dc.subject.other spline approximation en dc.subject.other total least squares en dc.subject.other TLS en dc.subject.other constrained weighted total least squares en dc.subject.other CWTLS en dc.subject.other Gauss-Helmert model en dc.subject.other singular dispersion matrix en dc.title Total Least Squares Spline Approximation en dc.type Article en dc.type.version publishedVersion en dcterms.bibliographicCitation.articlenumber 462 en dcterms.bibliographicCitation.doi 10.3390/math7050462 en dcterms.bibliographicCitation.issue 5 en dcterms.bibliographicCitation.journaltitle Mathematics en dcterms.bibliographicCitation.originalpublishername MDPI en dcterms.bibliographicCitation.originalpublisherplace Basel en dcterms.bibliographicCitation.volume 7 en tub.accessrights.dnb free en tub.affiliation Fak. 6 Planen Bauen Umwelt>Inst. Geodäsie und Geoinformationstechnik>FG Geodäsie und Ausgleichungsrechnung de tub.affiliation.faculty Fak. 6 Planen Bauen Umwelt de tub.affiliation.group FG Geodäsie und Ausgleichungsrechnung de tub.affiliation.institute Inst. Geodäsie und Geoinformationstechnik de tub.publisher.universityorinstitution Technische Universität Berlin en
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