Accessible Requires Authentication Published by De Gruyter September 9, 2016

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

Corinna Harmening and Hans Neuner


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.


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).


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Complete Results of the Determination of the Optimal Number of Control Points

In the tables 5 and 6 the complete results of the estimation of the optimal number of control points for the Bézier and the B-spline curve are listed.

Table 5

Optimal number of control points of two simulated Bézier curves with n1 + 1 = 6 and n2 + 1 = 12 according to AIC, BIC and SRM. Correct choices are highlighted in bold characters.

Table 6

Optimal number of control points of two simulated B-spline curves with n + 1 = 6 and two different knot vectors U1 and U2 according to AIC, BIC and SRM. Correct choices are highlighted in bold characters.

Received: 2016-3-15
Accepted: 2016-7-13
Published Online: 2016-9-9
Published in Print: 2016-9-1

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