Total Least Squares Spline Approximation

dc.contributor.authorNeitzel, Frank
dc.contributor.authorEzhov, Nikolaj
dc.contributor.authorPetrovic, Svetozar
dc.date.accessioned2019-06-11T08:29:48Z
dc.date.available2019-06-11T08:29:48Z
dc.date.issued2019-05-22
dc.description.abstractSpline 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.sponsorshipDFG, 414044773, Open Access Publizieren 2019 - 2020 / Technische Universität Berlinen
dc.identifier.eissn2227-7390
dc.identifier.urihttps://depositonce.tu-berlin.de/handle/11303/9494
dc.identifier.urihttp://dx.doi.org/10.14279/depositonce-8548
dc.language.isoenen
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/en
dc.subject.ddc510 Mathematikde
dc.subject.otherspline approximationen
dc.subject.othertotal least squaresen
dc.subject.otherTLSen
dc.subject.otherconstrained weighted total least squaresen
dc.subject.otherCWTLSen
dc.subject.otherGauss-Helmert modelen
dc.subject.othersingular dispersion matrixen
dc.titleTotal Least Squares Spline Approximationen
dc.typeArticleen
dc.type.versionpublishedVersionen
dcterms.bibliographicCitation.articlenumber462en
dcterms.bibliographicCitation.doi10.3390/math7050462en
dcterms.bibliographicCitation.issue5en
dcterms.bibliographicCitation.journaltitleMathematicsen
dcterms.bibliographicCitation.originalpublishernameMDPIen
dcterms.bibliographicCitation.originalpublisherplaceBaselen
dcterms.bibliographicCitation.volume7en
tub.accessrights.dnbfreeen
tub.affiliationFak. 6 Planen Bauen Umwelt>Inst. Geodäsie und Geoinformationstechnik>FG Geodäsie und Ausgleichungsrechnungde
tub.affiliation.facultyFak. 6 Planen Bauen Umweltde
tub.affiliation.groupFG Geodäsie und Ausgleichungsrechnungde
tub.affiliation.instituteInst. Geodäsie und Geoinformationstechnikde
tub.publisher.universityorinstitutionTechnische Universität Berlinen
Files
Original bundle
Now showing 1 - 1 of 1
Loading…
Thumbnail Image
Name:
mathematics-07-00462-v2.pdf
Size:
1.51 MB
Format:
Adobe Portable Document Format
Description:
Collections