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Journal of Applied Geodesy

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

4 Issues per year

CiteScore 2017: 1.23

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Volume 10, Issue 3


Choosing the Optimal Number of B-spline Control Points (Part 1: Methodology and Approximation of Curves)

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:
  • De Gruyter OnlineGoogle Scholar
/ Hans Neuner
Published Online: 2016-09-09 | DOI: https://doi.org/10.1515/jag-2016-0003


Due to the establishment of terrestrial laser scanner, the analysis strategies in engineering geodesy change from pointwise approaches to areal ones. These areal analysis strategies are commonly built on the modelling of the acquired point clouds.

Freeform curves and surfaces like B-spline curves/surfaces are one possible approach to obtain space continuous information. A variety of parameters determines the B-spline’s appearance; the B-spline’s complexity is mostly determined by the number of control points. Usually, this number of control points is chosen quite arbitrarily by intuitive trial-and-error-procedures. In this paper, the Akaike Information Criterion and the Bayesian Information Criterion are investigated with regard to a justified and reproducible choice of the optimal number of control points of B-spline curves. Additionally, we develop a method which is based on the structural risk minimization of the statistical learning theory. Unlike the Akaike and the Bayesian Information Criteria this method doesn’t use the number of parameters as complexity measure of the approximating functions but their Vapnik-Chervonenkis-dimension. Furthermore, it is also valid for non-linear models. Thus, the three methods differ in their target function to be minimized and consequently in their definition of optimality.

The present paper will be continued by a second paper dealing with the choice of the optimal number of control points of B-spline surfaces.

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


  • [1]

    Ken Aho, DeWayne Derryberry and Teri Peterson, Model selection for ecologists: the worldviews of AIC and BIC, Ecology 95 (2014), pp. 631–636.Google Scholar

  • [2]

    Hirotogu Akaike, Information Theory and an Extension of the Maximum Likelihood Principle, 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

  • [3]

    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

  • [4]

    Oliver Baur, Michael Kuhn and Will E. Featherstone, GRACE-Derived Linear and Non-linear Secular Mass Variations Over Greenland, VII Hotine-Marussi Symposium on Mathematical Geodesy (Nico Sneeuw, Pavel Novák, Mattia Crespi and Fernando Sansò, eds.), International Association of Geodesy Symposia 137, Springer Berlin Heidelberg, Berlin, Heidelberg, 2012, pp. 381–386.Google Scholar

  • [5]

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

  • [6]

    George E. P. Box, Science and Statistics, Journal of the American Statistical Association 71 (1976), pp. 791–799.Google Scholar

  • [7]

    Stephen P. Boyd and Lieven Vandenberghe, Convex optimization, Cambridge University Press, Cambridge, UK and New York, 2004.Google Scholar

  • [8]

    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

  • [9]

    Kenneth P. Burnham and David R. Anderson, Model selection and multimodel inference: A practical information-theoretic approach, 2nd ed, Springer, New York, 2002.Google Scholar

  • [10]

    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

  • [11]

    Joseph Cavanaugh and Andrew Neath, Generalizing the derivation of the Schwarz information criterion, Communications in Statistics - Theory and Methods 28 (1999), pp. 49–66.Google Scholar

  • [12]

    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

  • [13]

    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

  • [14]

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

  • [15]

    Jan Dupuis, Christoph Holst and Heiner Kuhlmann, Laser Scanning Based Growth Analysis of Plants as a New Challenge for Deformation Monitoring, Journal of Applied Geodesy 10 (2016), pp. 37–44.Google Scholar

  • [16]

    Gerald E. Farin, Curves and surfaces for CAGD: A practical guide, 5th ed, The Morgan Kaufmann series in computer graphics and geometric modeling, Morgan Kaufmann and Academic Press, San Francisco, CA and London, 2002.Google Scholar

  • [17]

    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

  • [18]

    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

  • [19]

    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.), pp. 189–206, 2015.Google Scholar

  • [20]

    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

  • [21]

    Clifford M. Hurvich and Chih-Ling Tsai, Bias of the corrected AIC criterion for underfitted regression and time series models, Biometrika 78 (1991), pp. 499–509.Google Scholar

  • [22]

    Stephanie Kauker and Volker Schwieger, Approach for a Synthetic Covariance Matrix for Terrestrial Laser Scanner, Proceedings of the 2nd International Workshop: Integration of point- and area-wise geodetic monitoring for structures and natural objects, March 23–24, 2015, Stuttgart (2015).Google Scholar

  • [23]

    Karl-Rudolf Koch, Parameterschätzung und Hypothesentests in linearen Modellen, 3rd ed, Dümmlerbuch 7892, Dümmler, Bonn, 1997.Google Scholar

  • [24]

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

  • [25]

    Jouni Kuha, AIC and BIC: Comparisons of Assumptions and Performance, Sociological Methods & Research 33 (2004), pp. 188–229.Google Scholar

  • [26]

    Allan D. R. McQuarrie and Chih-Ling Tsai, Regression and time series model selection, World Scientific, Singapore and River Edge, N. J., 1998.Google Scholar

  • [27]

    Hans Neuner, Model selection for system identification by means of artificial neural networks, Journal of Applied Geodesy 6 (2012), pp. 117–124.Google Scholar

  • [28]

    Hans Neuner, Claudius Schmitt and Ingo Neumann, Zur Bestimmung der verkehrsseitig verursachten Dehnung an einem Brückentragwerk mittels terrestrischem Laserscanning, Ingenieurvermessung 14: Beiträge zum 17. Internationalen Ingenieursvermessungskurs Zürich, 2014 (2013).Google Scholar

  • [29]

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

  • [30]

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

  • [31]

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

  • [32]

    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

  • [33]

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

  • [34]

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

  • [35]

    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

  • [36]

    Vladimir N. Vapnik, Principles of Risk Minimization for Learning Theory, Advances in Neural Information Processing Systems 4 (1992).Google Scholar

  • [37]

    Vladimir N. Vapnik, Statistical learning theory, Adaptive and learning systems for signal processing, communications, and control, Wiley, New York, 1998.Google Scholar

  • [38]

    Vladimir. N. Vapnik, An overview of statistical learning theory, IEEE transactions on neural networks / a publication of the IEEE Neural Networks Council 10 (1999), pp. 988–999.Google Scholar

  • [39]

    Vladimir N. Vapnik, Steven E. Golowich and Alex Smola, Support Vector Method for Function Approximation, Regression Estimation, and Signal Processing, Advances in Neural Information Processing Systems 9 (1996), pp. 281–287.Google Scholar

  • [40]

    Larry Wasserman, Bayesian Model Selection and Model Averaging, Journal of mathematical psychology 44 (2000), pp. 92–107.Google Scholar

About the article

Received: 2016-03-15

Accepted: 2016-07-13

Published Online: 2016-09-09

Published in Print: 2016-09-01

Citation Information: Journal of Applied Geodesy, Volume 10, Issue 3, Pages 139–157, ISSN (Online) 1862-9024, ISSN (Print) 1862-9016, DOI: https://doi.org/10.1515/jag-2016-0003.

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