Six-Point Subdivision Schemes with Cubic Precision

Abstract

This paper presents 6-point subdivision schemes with cubic precision. We first derive a relation between the 4-point interpolatory subdivision and the quintic B-spline refinement. By using the relation, we further propose the counterparts of cubic and quintic B-spline refinements based on 6-point interpolatory subdivision schemes. It is proved that the new family of 6-point combined subdivision schemes has higher smoothness and better polynomial reproduction property than the B-spline counterparts. It is also showed that, both having cubic precision, the well-known Hormann-Sabin's family increase the degree of polynomial generation and smoothness in exchange of the increase of the support width, while the new family can keep the support width unchanged and maintain higher degree of polynomial generation and smoothness.

A Class of Refinement Schemes With Two Shape Control Parameters

“…Example 25: Figure 3 shows the interpolatory behaviors of the scheme up to three refinement steps whereas the doted polygon is the initial sketch which is made by six initial control points (x, y) = (5, 0), (10, 0), (20,10), (10,20), (5,20), (−5, 10) denoted by red solid circles. Here we fix the values of parameters as α = 0 and β = −0.05.…”

“…In 2012, Pan et al [19] presented a C 2 continuous combined ternary approximating and interpolating schemes. The new family of ternary 6-point combined schemes with C 3 -continuity was derived by Shi et al [20] in 2018.…”

“…Moreover, the difference between the shapes with order of continuities 4 and 5 can not be observed by our naked eyes so one extra order of continuity in the shapes with high cost of complexity has no use. Furthermore, our schemes produce higher order continuous shape then ternary combined family of schemes [19], [20]. Our interpolating and exiting schemes in the combined family produce the same order of continuous shapes.…”

A Family of Binary Univariate Nonstationary Quasi‐Interpolatory Subdivision Reproducing Exponential Polynomials

“…, it actually becomes the exponential pseudospline scheme generating EP ΓΛ 2 (22) and reproducing EP ΓΛ 1 [33]. Here, we denote by e k ΓΛ 1 ,1 (z) and e k ΓΛ 1 ,2 (z) the k-level symbols of the nonstationary interpolatory 4-point scheme [25] and the aforementioned exponential pseudospline scheme, which can be written as 4 8…”

“…Luo and Qi [20] analyzed the interpolatory subdivision obtained by this connection systematically in the univariate case. For other references on this connection, see also [21][22][23] and the references therein. Yet, unlike the polynomial correction, most of the works related to the push-back operation, except the work in [24], are restricted to the stationary case, and the reproduction property of the obtained interpolatory schemes depends largely on that of the original approximating schemes [20].…”