Elsevier

Computer-Aided Design

Volume 43, Issue 6, June 2011, Pages 598-604
Computer-Aided Design

Adaptive knot placement using a GMM-based continuous optimization algorithm in B-spline curve approximation

https://doi.org/10.1016/j.cad.2011.01.015Get rights and content

Abstract

One of the key problems in using B-splines successfully to approximate an object contour is to determine good knots. In this paper, the knots of a parametric B-spline curve were treated as variables, and the initial location of every knot was generated using the Monte Carlo method in its solution domain. The best km knot vectors among the initial candidates were searched according to the fitness. Based on the initial parameters estimated by an improved k-means algorithm, the Gaussian Mixture Model (GMM) for every knot was built according to the best km knot vectors. Then, the new generation of the population was generated according to the Gaussian mixture probabilistic models. An iterative procedure repeating these steps was carried out until a termination criterion was met. The GMM-based continuous optimization algorithm could determine the appropriate location of knots automatically. A set of experiments was then implemented to evaluate the performance of the new algorithm. The results show that the proposed method achieves better approximation accuracy than methods based on artificial immune system, genetic algorithm or squared distance minimization (SDM).

Highlights

► The locations of the knots of a parametric B-spline curve are treated as variables. ► We develop a GMM-based EDA to determine the appropriate locations of the knots. ► We initialize the parameters of GMM using a data-mass-based k-means algorithm. ► A point cloud representing a closed curve can be approximated successfully.

Section snippets

B-spline curve approximation

The equation of B-spline curve P(t) of order k defined by n+1 control points Pi is given by P(t)=∑i=0nPiNi,k(t),tmin≤t≤tmax where Ni,k is the B-spline function of order k defined on a knot vector T. The knot vector T consists of non-decreasing real-valued knots. The control points form the vertices of the control polygon.

The recursive relation of the B-spline basis is defined as: Ni,k(t)=(t−ti)Ni,k−1(t)ti+k−1−ti+(ti+k−t)Ni+1,k−1(t)ti+k−ti+1 where Ni,1={1,ti≤t≤ti+10,otherwise .

In this paper, we

Estimation of distribution algorithms

EDAs are an outgrowth of genetic algorithms and are able to estimate the probability distribution using the selected set ofsolutions itself and then employing this estimate to generate new solutions. In an EDA, a population of candidate solutions to a problem is maintained as part of the search for an optimal solution, just like in a genetic algorithm. However, the population is represented by a probability distribution that can prevent disruption of partial solutions contained in a chromosome

Automatic knot adjustment by GMM-based EDA

The B-spline curve-fitting problem is to produce a B-spline curve to approximate a target curve within a prespecified tolerance. After edge tracing and sampling, the image contour is defined in a 2D plane by a sequence of ordered dense data points. The first and the last points are superposed because the contour is a closed curve, thus an additional point occurs: qm+1=q0. In this paper, we adopt accumulated chord length to parameterize the data points, and we used GMM-based EDA to optimize the

Experimental results

To evaluate the proposed GMM-based automatic knot placement algorithm, four examples are used. Fig. 3 displays the initial contours of an intestine section, a cement particle, an SiC particle and a kidney section, which contain 418, 197, 323 and 294 points, respectively. The performance of our approach was then compared with that of a GA-based algorithm [12], an AIS-based algorithm proposed by Ulker [13] and the SDM algorithm [16]. For comparing the approximation error, the average error

Conclusions

To obtain a good B-spline curve model from a raw contour, knots are usually treated as variables. A curve is then modeled as a continuous, nonlinear and multivariate optimization problem with many local optima. Therefore, it is very difficult to reach a global optimum. To overcome this difficulty, we suggested a new method that solves this problem using a Gaussian mixture distribution based on the estimation of a distribution algorithm, and the initial parameters of the GMM were estimated by an

Acknowledgements

This research was supported by the National Natural Science Foundation of China under contract (Nos. 60873089 and 60933008), the Doctor Foundation of Shandong Province under Grant (Nos. 2007BS04018 and ZR2010FM047), the Postdoctoral Innovation Fund of Shandong Province under Grant (No. 200802026) and the Technology Plan Project of the Department of Education of Shandong Province under Grant (No. J07YJ23).

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