Non-stationary subdivision schemes generalizing exponential B-splines

Abstract

An important capability for a subdivision scheme is the reproducing property of circular shapes or parts of conics that are important analytical shapes in geometrical modeling. In this regards, this thesis first provides necessary and sufficient conditions for a non-stationary subdivision to have the reproducing property of exponential polynomials. Then, the approximation order of such non-stationary schemes is discussed to quantify their approximation power. Based on these results, we see that the exponential B-spline generates exponential polynomials in the associated spaces, but it may not reproduce any exponential polynomials. Thus, we present {\em normalized} exponential B-splines that reproduce certain sets of exponential polynomials. One interesting feature is that depending on the normalization factor, the set of exponential polynomials to be reproduced is varied. This provides us with the necessary accuracy and flexibility in designing target curves and surfaces. Some numerical results are presented to support the advantages of the normalized scheme by comparing them to the results without normalization.

In fact, the dimension of the space of exponential polynomials reproduced by the normalized exponential B-spline is limited to $2$, which leads to the limitation of the approximation order. By sacrificing the support of the subdivision mask, the defect can be overcome. We present the exponential {\em quasi}-spline with the unlimited reproduction capability by generalizing the exponential B-spline. Its built-in parameters enable us to control the tension of the limit curve so that the artifact of the interpolatory schemes occurred in highly irregular region of control points can be avoided. Depending on the asymptotic equivalence of (non-)stationary schemes, we analyze the smoothness of the exponential quasi-spline and provide the actual computation for it. Numerical examples which show fruitful benefits obtained from the exponential quasi-spline are...