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  Spline Approximation, Part 2: From Polynomials in the Monomial Basis to B-splines—A Derivation

Ezhov, N., Neitzel, F., Petrovic, S. (2021): Spline Approximation, Part 2: From Polynomials in the Monomial Basis to B-splines—A Derivation. - Mathematics, 9, 18, 2198.
https://doi.org/10.3390/math9182198

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Item Permalink: https://gfzpublic.gfz-potsdam.de/pubman/item/item_5007711 Version Permalink: https://gfzpublic.gfz-potsdam.de/pubman/item/item_5007711_1
Genre: Journal Article

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 Creators:
Ezhov, Nikolaj1, Author
Neitzel, Frank1, Author
Petrovic, S.2, Author              
Affiliations:
1External Organizations, ou_persistent22              
21.2 Global Geomonitoring and Gravity Field, 1.0 Geodesy, Departments, GFZ Publication Database, Deutsches GeoForschungsZentrum, ou_146026              

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Free keywords: spline, B-spline, polynomial, monomial, basis change, Lagrange, Bernstein, interpolation, approximation, least squares adjustment
 Abstract: In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3.

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Language(s): eng - English
 Dates: 2021-09-082021
 Publication Status: Finally published
 Pages: -
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 Table of Contents: -
 Rev. Type: -
 Identifiers: DOI: 10.3390/math9182198
GFZPOF: p4 T2 Ocean and Cryosphere
OATYPE: Gold Open Access
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Title: Mathematics
Source Genre: Journal, SCI, Scopus, oa
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Pages: - Volume / Issue: 9 (18) Sequence Number: 2198 Start / End Page: - Identifier: ISSN: 2227-7390
CoNE: https://gfzpublic.gfz-potsdam.de/cone/journals/resource/190704
Publisher: MDPI