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

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.

B-Spline-Based Sharp Feature Preserving Shape Reconstruction Approach for Electrical Impedance Tomography

“…Note that for cubic curve, at least four control points are required to define a cubic curve. It is worth remarking that there are some other alternative ways for choosing the optimal number of control points, e.g., in [32], the task of choosing the optimal number was interpreted as a model selection problem based on the Bayesian information criterion. However, choosing the optimal number of control point in B-spline is out of the scope of this paper, hence it was left as a future research topic.…”

Deformation Analysis Using B-Spline Surface with Correlated Terrestrial Laser Scanner Observations—A Bridge Under Load

“…Different methods can be used (e.g., uniform, cord length or centripetal parametrization; [40]). They all have shortcomings, which should not be underestimated for complicated geometries [41]. An exhaustive description of the parametrization is beyond the scope of the present paper.…”

Robust and automatic modeling of tunnel structures based on terrestrial laser scanning measurement

“…29 Some researchers determined the optimal number of control points, which was usually a model selection problem, by means of the Akaike Information Criterion, the Bayesian Information Criterion, or statistical learning theory. 30 The quality of the control point positions estimated was evaluated by the average estimated standard deviation, which focused on the quality of estimation procedure regarding datasets instead of the real shape of the object. 31 Minimizing energy function has recently been adopted to adjust control points which could be applied in smoothing and gap filling.…”