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Journal of Applied Geodesy

Editor-in-Chief: Kahmen, Heribert / Rizos, Chris


CiteScore 2018: 1.61

SCImago Journal Rank (SJR) 2018: 0.532
Source Normalized Impact per Paper (SNIP) 2018: 1.064

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1862-9024
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Volume 11, Issue 1

Issues

Choosing the optimal number of B-spline control points (Part 2: Approximation of surfaces and applications)

Corinna Harmening
  • Corresponding author
  • Department of Geodesy and Geoinformation, TU Wien, Gusshausstr. 25-29 / E120, 1040 Vienna, Austria
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  • Other articles by this author:
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/ Hans Neuner
Published Online: 2017-02-13 | DOI: https://doi.org/10.1515/jag-2016-0036

Abstract

Freeform surfaces like B-splines have proven to be a suitable tool to model laser scanner point clouds and to form the basis for an areal data analysis, for example an areal deformation analysis.

A variety of parameters determine the B-spline's appearance, the B-spline's complexity being mostly determined by the number of control points. Usually, this parameter type is chosen by intuitive trial-and-error-procedures.

In [10] the problem of finding an alternative to these trial-and-error-procedures was addressed for the case of B-spline curves: The task of choosing the optimal number of control points was interpreted as a model selection problem. Two model selection criteria, the Akaike and the Bayesian Information Criterion, were used to identify the B-spline curve with the optimal number of control points from a set of candidate B-spline models. In order to overcome the drawbacks of the information criteria, an alternative approach based on statistical learning theory was developed. The criteria were evaluated by means of simulated data sets.

The present paper continues these investigations. If necessary, the methods proposed in [10] are extended to areal approaches so that they can be used to determine the optimal number of B-spline surface control points. Furthermore, the methods are evaluated by means of real laser scanner data sets rather than by simulated ones.

The application of those methods to B-spline surfaces reveals the datum problem of those surfaces, meaning that location and number of control points of two B-splines surfaces are only comparable if they are based on the same parameterization. First investigations to solve this problem are presented.

Keywords: AIC; BIC; B-spline Surface; Structural Risk Minimization; VC-dimension

References

  • [1]

    Hirotogu Akaike, Information Theory and an Extension of the Maximum Likelihood Principle, in: Selected Papers of Hirotugu Akaike (Emanuel Parzen, Kunio Tanabe and Genshiro Kitagawa, eds.) Springer Series in Statistics, Springer, New York, 1998, pp. 199–213.Google Scholar

  • [2]

    Mario Alba, Luigi Fregonese, Federico Prandi, Marco Scaioni and Paolo Valgoi, Structural Monitoring of a Large Dam by Terrestrial Laserscanning, The ISPRS International Archives of Photogrammetry, Remote Sensing and Spatial. Information Sciences, Dresden, Deutschland (2006).Google Scholar

  • [3]

    Carl de Boor, On calculating with B-splines, Journal of Approximation Theory 6 (1972), pp. 50–62.Google Scholar

  • [4]

    Johannes Bureick, Hamza Alkhatib and Ingo Neumann, Robust Spatial Approximation of Laser Scanner Point Clouds by Means of Free-form Curve Approaches in Deformation Analysis, Journal of Applied Geodesy 10 (2016), pp. 27–35.Google Scholar

  • [5]

    Kenneth P. Burnham and David R. Anderson, Multimodel Inference: Understanding AIC and BIC in Model Selection, Sociological Methods & Research 33 (2004), pp. 261–304.Google Scholar

  • [6]

    Vladimir S. Cherkassky and Filip Mulier, Learning from data: Concepts, theory, and methods, 2nd ed., IEEE Press and Wiley-Interscience, Hoboken, N.J., 2007.Google Scholar

  • [7]

    Gerda Claeskens and Nils Lid Hjort, Model selection and model averaging, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge and New York, 2008.Google Scholar

  • [8]

    Maurice G. Cox, The Numerical Evaluation of B-Splines, IMA Journal of Applied Mathematics 10 (1972), pp. 134–149.Google Scholar

  • [9]

    Corinna Harmening and Hans Neuner, A constraint-based parameterization technique for B-spline surfaces, Journal of Applied Geodesy 9 (2015), pp. 143–161.Google Scholar

  • [10]

    Corinna Harmening and Hans Neuner, Choosing the Optimal Number of B-spline Control Points (Part 1: Methodology and Approximation of Curves), Journal of Applied Geodesy 10 (2016), pp. 139–157.Google Scholar

  • [11]

    Corinna Harmening and Hans Neuner, Detecting Rigid Body Movements from TLS-Based Areal Deformation Measurements, in: Proceedings of the FIG Working Week 2016 (International Federation of Surveyors, ed.), 2016.Google Scholar

  • [12]

    Otto Heunecke, Heiner Kuhlmann, Walter Welsch, Andreas Eichhorn and Hans Neuner, Handbuch Ingenieurgeodäsie: Auswertung geodätischer Überwachungsmessungen, 2nd ed., Wichmann, H, Heidelberg, Neckar, 2008.Google Scholar

  • [13]

    Christoph Holst and Heiner Kuhlmann, Mathematische Modelle zur flächenhaften Approximation punktweise gemessener Bodensenkungen auf Basis von Präzisionsnivellements, in: Geomonitoring (Wolfgang Busch, Wolfgang Niemeier and Ingo Neumann, eds.), 2015, pp. 189–206.Google Scholar

  • [14]

    Christoph Holst and Heiner Kuhlmann, Challenges and Present Fields of Action at Laser Scanner Based Deformation Analyses, Journal of Applied Geodesy 10 (2016), pp. 17–25.Google Scholar

  • [15]

    Karl-Rudolf Koch, NURBS surface with changing shape, in: Allgemeine Vermessungsnachrichten (2010), pp. 83–89.Google Scholar

  • [16]

    Weiyin Ma and Jean-Pierre Kruth, Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces, Computer-Aided Design 27 (1995), pp. 663–675.Google Scholar

  • [17]

    Hans Neuner, Claudius Schmitt and Ingo Neumann, Zur Bestimmung der verkehrsseitig verursachten Dehnung an einem Brückentragwerk mittels terrestrischem Laserscanning, in: Ingenieurvermessung 14: Beiträge zum 17. Internationalen Ingenieursvermessungskurs Zürich, 2014 (2014).Google Scholar

  • [18]

    Johannes Ohlmann-Lauber and Thomas Schäfer, Ansätze zur Ableitung von Deformationen aus TLS-Daten, in: Terrestrisches Laserscanning – TLS 2011 mit TLS-Challenge (2011), pp. 147–158.Google Scholar

  • [19]

    Les A. Piegl and Wayne Tiller, The NURBS book, 2nd ed., Monographs in Visual Communications, Springer, Berlin and New York, 1997.Google Scholar

  • [20]

    Michael Schmidt, Denise Dettmering and Florian Seitz, Using B-Spline Expansions for Ionosphere Modeling, in: Handbook of Geomathematics (Willi Freeden, M. Zuhair Nashed and Thomas Sonar, eds.), Springer, Berlin and Heidelberg, 2014, pp. 1–40.Google Scholar

  • [21]

    Claudius Schmitt and Hans Neuner, Knot estimation on B-Spline curves, Österreichische Zeitschrift für Vermessung und Geoinformation (VGI) 103 (2015), pp. 188–197.Google Scholar

  • [22]

    Gideon Schwarz, Estimating the Dimension of a Model, The Annals of Statistics 6 (1978), pp. 461–464.Google Scholar

  • [23]

    Rinske van Gosliga, Roderik Lindenbergh and Norbert Pfeifer, Deformation Analysis of a bored tunnel by means of Terrestrial Laserscaning, in: Proceedings on ISPRS Commission V Symposium, Dresden (2006).Google Scholar

  • [24]

    Vladimir Vapnik, Esther Levin and Yann Le Cun, Measuring the VC-Dimension of a Learning Machine, Neural Computation 6 (1994), pp. 851–876.Google Scholar

  • [25]

    Vladimir N. Vapnik, Statistical learning theory, Adaptive and Learning Systems for Signal Processing, Communications, and Control, Wiley, New York, 1998.Google Scholar

About the article

Received: 2016-08-03

Accepted: 2016-11-22

Published Online: 2017-02-13

Published in Print: 2017-03-01


Funding Source: Austrian Science Fund

Award identifier / Grant number: 1706-N29

The presented paper shows results developed during the research project “Integrierte raumzeitliche Modellierung unter Nutzung korrelierter Messgrößen zur Ableitung von Aufnahmekonfiguationen und Beschreibung von Deformationsvorgängen” (IMKAD) (1706-N29), which is funded by the Austrian Science Fund (FWF).


Citation Information: Journal of Applied Geodesy, Volume 11, Issue 1, Pages 43–52, ISSN (Online) 1862-9024, ISSN (Print) 1862-9016, DOI: https://doi.org/10.1515/jag-2016-0036.

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