A non-stationary binary three-point approximating subdivision scheme

A Generalized Cubic Exponential B-Spline Scheme with Shape Control

“…Note that ∑ ∞ =0 |V − 1| < ∞ can also be derived from the fact that |V − 1| ≤ 3 2 − with 3 being a constant independent of . Here, we used the technique of fixed point iteration, which, we point out that, can also be used in the analysis of other nonstationary subdivision schemes, such as the ones in [10,11]. This will be shown in Section 5.…”

“…For this new scheme, by changing the values of and ], we can change the sensitivity of the shape of the obtained curve to the initial control value V 0 , and different kinds of curves, including the conics, can then be obtained by adjusting V 0 . We point out that compared with the cubic exponential B-spline scheme, this newly obtained one can generate curves with more kinds of shapes and, compared with the schemes in [10,11], this new one enjoys the advantages like shorter support and generation of exponential polynomials. For such a new scheme, we show that, with any admissible choice of and ], it keeps the same smoothness order and the support as the cubic exponential B-spline scheme.…”

“…Beccari et al [10] proposed a ternary 4-point nonstationary interpolatory scheme, whose nonstationarity can be seen as the result of an iteration, and this scheme can generate curves with considerable variations of shapes. Similarly, Tan et al [11] proposed a 3-point nonstationary approximating subdivision, which can be seen as constructed based on a different iteration and can also generate a wide variety of curves.…”

“…Due to nonstationary schemes's ability in curve design, as illustrated above, in this paper, we aim to propose a 2 Mathematical Problems in Engineering 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 [10,11]. In fact, we start from the 2 -convergent cubic exponential B-spline scheme [12], which generates the conics.…”

Family of odd point non-stationary subdivision schemes and their applications

“…Mustafa et al [19] introduced a subdivision-regularization framework for preventing over-fitting of data by a model in 2013. In 2016, Salam et al [23] presented two non-stationary forms of Chaikin perturbation SS, and Tan et al [27] derived a 3-point approximating non-stationary SS. For more recent work on SSs, one may be referred to [1,15,17,18,21].…”