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In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. Such a scheme is obtained by generalizing the cubic exponential B-spline scheme based on an iteration from the generation of exponential polynomials and a suitable function with two parameters

Subdivision schemes are efficient tools to generate smooth curves/surfaces from a set of initial control points and they play an important role in computer graphics, wavelets, and other fields like biomedical imaging [

Since the nonstationary schemes can generate richer function spaces and more kinds of curves, there have been continuous works on the construction and analysis of nonstationary schemes. The exponential B-spline schemes [

Due to nonstationary schemes’s ability in curve design, as illustrated above, in this paper, we aim to propose a new nonstationary subdivision scheme, which can generate curves with considerable variations of shapes and exponential polynomials as well. The inspiration comes from the works in [

The rest of this paper is organized as follows. In Section

In this section, we recall some basic definitions and known results about subdivision schemes and iterations to form the basis of the rest of this paper. Let

By attaching

In order to investigate the convergence and smoothness of nonstationary subdivision schemes, let us recall some definitions and results as follows.

A nonstationary subdivision scheme

let

Assume that the nonstationary subdivision scheme

Now we recall some knowledge about the generation of exponential polynomials.

Let

The exponential polynomial space

Let

We say the subdivision scheme

Now let us now recall some known definitions and results about fixed point iterations.

We say

The following result gives sufficient conditions for the existence and uniqueness of a fixed point and how to approximate it.

Let

In this section, we construct the generalized cubic exponential B-spline scheme and then investigate its convergence and smoothness.

Before we construct the generalized cubic exponential B-spline scheme, we briefly review the cubic exponential B-spline scheme [

As is known, the cubic exponential B-spline scheme (

In fact, from the cubic exponential B-spline scheme (

Now based on this observation, let us construct the new generalized cubic exponential B-spline scheme. In fact, to obtain our new scheme, we replace the function

Note that when one of the parameters

In fact, we only modified the existing elements in the masks of the cubic exponential B-spline scheme. Thus, the support of the new scheme

Now let us investigate the convergence and smoothness of the generalized cubic exponential B-spline scheme

For

We show that the generalized cubic exponential B-spline scheme

To show that the scheme

Let

When

Therefore, by combining (

Note that

In this section, we present several numerical examples and compare it with some existing subdivision schemes to illustrate the performance of the scheme

Figure

Curves generated by the scheme

Curves generated by the scheme

Curves generated by

Curves generated by

Curves generated by

Curves generated by

Figure

Figure

Curves generated by the cubic B-spline scheme (left column) cubic exponential B-spline (middle column), and the scheme

Recall that the nonstationary subdivision schemes in [

Comparison between different schemes.

Schemes | | | |
---|---|---|---|

support | | [-2.5,2.5] | |

smoothness order | 2 | 2 | 3 |

type | approximating | interpolatory | approximating |

p-ary | binary | ternary | binary |

Note that the scheme

Recall that the iteration (

Specifically speaking, we take

Similar to the iteration

Now we investigate the convergence and smoothness of the newly obtained scheme

The new scheme

Denote by

In fact, similar to the sequence

Figures

Curves generated the scheme

Curves generated the scheme

In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. The key ingredients of the construction and analysis is the iteration (

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This is paper is supported by the National Science Foundation for Young Scientists of China (Grant No. 61801389).

_{2}ternary non-stationary subdivision scheme with tension control