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

Abstract

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.

Acknowledgments

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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Appendix

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.

lAICBICSRMAICBICSRM
1001286121212
1001177131111
1001076111111
100888111111
100888111111
2501077121212
2501077131312
250777161515
2501277151215
250997151315
5001176161311
5001266161111
500776131211
5001299141411
5001177121212
10001177121212
1000976161616
100012106121211
10001186131313
10001276121212
200012126131313
20001277161212
20001266151411
20001176161211
20001066161611
500011116161411
5000776121212
5000876151515
500011117121212
5000777141411
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.

lAICBICSRMAICBICSRM
1001266111111
100966111111
100996111111
1001266111111
1001266111111
2501266111111
2501266111111
2501266111111
2501266111111
250666121212
50012126121211
5001266121212
50012126121212
50012126111111
50012126111111
10001266121212
1000666121212
10001266121212
10001266121212
10001266121212
20001266121212
200012126121212
200012126121212
2000666121212
20001266121212
50009126121212
50001266121212
500012126121212
50001266121212
50001296121212
Received: 2016-3-15
Accepted: 2016-7-13
Published Online: 2016-9-9
Published in Print: 2016-9-1

© 2016 Walter de Gruyter GmbH, Berlin/Munich/Boston