An introduction to a hybrid trigonometric box spline surface producing subdivision scheme

Abstract

Trigonometric box splines can be considered as the multivariate generalizations of univariate B-splines. This means multivariate trigonometric box splines can be constructed from univariate trigonometric B-splines by suitable adaptive methods. Mainly, this paper is concerned with the construction of a new class of bivariate trigonometric box spline functions with the help of trigonometric B-splines through directional convolution method. These are refinable functions and some of their properties are also investigated. Secondly, a new effective non-stationary subdivision scheme is derived from the two-scale relationship of a particular trigonometric box spline. The non-stationary subdivision scheme is capable of producing such trigonometric box spline surfaces in regular regions. Some important properties along with the interactive modeling capability of this subdivision scheme are described in detail.

Appendices

Appendix A

Here it is described that how the subdivision weights around the extraordinary vertices of valences 3 and 4 have been derived in Sect. 4. Basically, the mth-level subdivision weights around an extraordinary vertex of valence 3 are designed by using the regular subdivision weights as follows:

$$\begin{aligned} \begin{aligned}&{\widetilde{w}}_{0}^{(m)} = \frac{1}{2}w_{0}^{(m)} + \frac{3}{2}w_{1}^{(m)},\;\;\;{\widetilde{w}}_{1}^{(m)} = \frac{1}{3} (1-{\widetilde{w}}_{0}^{(m)})\\&{\widetilde{w}}_{3}^{(m)}=3 w_{2}^{(m)} - w_{3}^{(m)} - \frac{5}{2}w_{4}^{(m)} + \frac{1}{2}w_{5}^{(m)},\;\; {\widetilde{w}}_{5}^{(m)}=6 w_{4}^{(m)} - 4(w_{2}^{(m)}- w_{3}^{(m)}). \end{aligned} \end{aligned}$$

On the other hand, the mth-level subdivision weights around an extraordinary vertex of valence 4 are designed as given below:

$$\begin{aligned} \begin{aligned} \widetilde{\widetilde{w}}_{0}^{(m)}&= \frac{15}{16} w_{0}^{(m)} -\frac{3}{8} w_{1}^{(m)} - \frac{1}{24 \cos ^{4}\big (\frac{h}{2^{m+1}}\big )},\;\; \widetilde{\widetilde{w}}_{1}^{(m)} = \frac{1}{4} (1-\widetilde{\widetilde{w}}_{0}^{(m)}),\\ \widetilde{\widetilde{w}}_{3}^{(m)}&= \frac{1}{2} w_{2}^{(m)} + w_{3}^{(m)},\;\;\widetilde{\widetilde{w}}_{4}^{(m)} = \frac{1}{4} w_{2}^{(m)} + w_{4}^{(m)},\\ \widetilde{\widetilde{w}}_{5}^{(m)}&= \frac{3}{4} w_{2}^{(m)} + w_{5}^{(m)},\; \widetilde{\widetilde{w}}_{7}^{(m)} = \frac{1}{32 \cos ^{4}\big (\frac{h}{2^{m+1}}\big )},\\ \text {and}\; \widetilde{\widetilde{w}}_{6}^{(m)}&= \frac{1}{2} - \frac{1}{24 \cos ^{4}\big (\frac{h}{2^{m+1}}\big )}. \end{aligned} \end{aligned}$$

All the subdivision weights around the extraordinary vertices are designed in such a way that, the sums of the weights in all mask maps in Figs. 4 and 5 are equal to 1.

Appendix B

Let us take $\mathbf {{Z}}$ and $D^{(\gamma _{1}, \gamma _{2})}$ as defined in Definition 8. Similarly, the mth-level symbol $a_{\star }^{(m)}(z_{1}, z_{2})$ of the non-stationary subdivision scheme is given in (24). Now if we consider the directional derivatives $D^{(\gamma _{1}, \gamma _{2})}$ of $a_{\star }^{(m)}(z_{1}, z_{2})$ such that $\gamma _{1}+\gamma _{2}=2$, then $(\gamma _{1}, \gamma _{2}) \in \{(1,1),\;(2,0),\;(0,2)\}$. After computation of different directional derivatives of $a_{\star }^{(m)}(z_{1}, z_{2})$, it is found that

$$\begin{aligned} \begin{aligned}&\displaystyle {\max _{\gamma _{1}+\gamma _{2}=2}\;\max _{(\xi _{1}, \xi _{2})\in \mathbf {{Z}}}} \vert D^{(\gamma _{1},\gamma _{2})}a_{\star }^{(m)}(\xi _{1}, \xi _{2})\vert \\&= \vert D^{(2,0)}a_{\star }^{(m)}(-1,1)\vert = \vert D^{(0,2)}a_{\star }^{(m)}(1,-1)\vert = 2\;\bigg (\frac{1-\cos (\frac{h}{2^{m}})}{1+\cos (\frac{h}{2^{m}})}\bigg )^{2} =: E_{1}^{(m)}. \end{aligned} \end{aligned}$$

Similarly, considering the directional derivatives $D^{(\gamma _{1}, \gamma _{2})}$ of $a_{\star }^{(m)}(z_{1}, z_{2})$ such that $\gamma _{1}+\gamma _{2}=3$, then $(\gamma _{1}, \gamma _{2}) \in \{(2,1),\;(1,2),\;(3,0),\;(0,3)\}$. It is obtained that

$$\begin{aligned} \begin{aligned}&\displaystyle \max _{\gamma _{1}+\gamma _{2}=3}\;\max _{(\xi _{1}, \xi _{2})\in \mathbf {{Z}}} \vert D^{(\gamma _{1},\gamma _{2})}a_{\star }^{(m)}(\xi _{1}, \xi _{2})\vert \\&= \vert D^{(3,0)}a_{\star }^{(m)}(-1,1)\vert = \vert D^{(0,3)}a_{\star }^{(m)}(1,-1)\vert = 6\;\bigg (\frac{1-\cos (\frac{h}{2^{m}})}{1+\cos (\frac{h}{2^{m}})}\bigg )^{2} =: E_{2}^{(m)}. \end{aligned} \end{aligned}$$

Finally, if we consider the directional derivatives $D^{(\gamma _{1}, \gamma _{2})}$ of $a_{\star }^{(m)}(z_{1}, z_{2})$ such that $\gamma _{1}+\gamma _{2}=4$, then $(\gamma _{1}, \gamma _{2}) \in \{(3,1),\;(1,3),\;(2,2),\;(4,0),\;(0,4)\}$. It is observed that

$$\begin{aligned} \begin{aligned}&\displaystyle {\max _{\gamma _{1}+\gamma _{2}=4}\;\max _{(\xi _{1}, \xi _{2})\in \textbf{Z}}} \vert D^{(\gamma _{1},\gamma _{2})}a_{\star }^{(m)}(\xi _{1}, \xi _{2})\vert = \vert D^{(4,0)}a_{\star }^{(m)}(-1,1)\vert = \vert D^{(0,4)}a_{\star }^{(m)}(1,-1)\vert \\&= \frac{24 \;\vert (1-\cos (\frac{h}{2^{m}}))(-2+\cos (\frac{h}{2^{m}}))\vert }{(1+\cos (\frac{h}{2^{m}}))^{2}} =: E_{3}^{(m)}. \end{aligned} \end{aligned}$$

Since $\frac{2^{-3m}E_{2}^{(m)}}{2^{-2m}E_{1}^{(m)}} = \frac{3}{2^{m}},\;\forall \;m\in \mathcal {N}_{0}$,

$$\begin{aligned} \max \{2^{-2m} E_{1}^{(m)},\;2^{-3m} E_{2}^{(m)}\} = {\left\{ \begin{array}{ll}2^{-3m} E_{2}^{(m)},\;\;&{}\text {for}\;m = 0,1, \\ 2^{-2m} E_{1}^{(m)}, &{}\text {for}\; m\ge 2.\\ \end{array}\right. } \end{aligned}$$

Now for $m = 0,1$, it can be observed that

$$\begin{aligned} \frac{2^{-4m}E_{3}^{(m)}}{2^{-3m}E_{2}^{(m)}} = 2^{-m}\;4\; \bigg \vert \frac{-2+\cos (\frac{h}{2^{m}})}{1-\cos (\frac{h}{2^{m}})}\bigg \vert > 1. \end{aligned}$$

Also for $m \ge 2$, we get that

$$\begin{aligned} \frac{2^{-4m}E_{3}^{(m)}}{2^{-2m}E_{1}^{(m)}} = 2^{-2m}\;12\; \bigg \vert \frac{-2+\cos (\frac{h}{2^{m}})}{1-\cos (\frac{h}{2^{m}})}\bigg \vert > 1. \end{aligned}$$

Thus, as a result

$$\begin{aligned} \begin{aligned}&\displaystyle {\max _{2\le \gamma _{1}+\gamma _{2} \le 4}\;\max _{(\xi _{1}, \xi _{2})\in \textbf{Z}} 2^{-m(\gamma _{1}+\gamma _{2})}} \vert D^{(\gamma _{1},\gamma _{2})}a_{\star }^{(m)}(\xi _{1}, \xi _{2})\vert \\&= \max \{2^{-2m} E_{1}^{(m)},\;2^{-3m} E_{2}^{(m)},\; 2^{-4m} E_{3}^{(m)}\}\\&= 2^{-4m} E_{3}^{(m)} = 2^{-4m}\;\frac{24 \;\vert (1-\cos (\frac{h}{2^{m}}))(-2+\cos (\frac{h}{2^{m}}))\vert }{(1+\cos (\frac{h}{2^{m}}))^{2}}. \end{aligned} \end{aligned}$$

Appendix C

We here describe how the bounds between the non-stationary and stationary weights for regular subdivision rules are obtained in the verification of Assumption (iii) of Theorem 5.2 while proving the Theorem 5.4. From (35), we can exploit the identity $\vert 1-\cos {\big (\frac{h}{2^{m}}\big )}\vert \le \frac{\kappa }{4^m}$ to compute the bounds, letting $\kappa$ be some positive constant. For different m, $\cos {\big (\frac{h}{2^{m}}\big )}$ is always positive since $0<h\le \frac{\pi }{3}$.

First we compute the bound of $w_{1}^{(m)}-\frac{3}{32}$. Note that,

$$\begin{aligned} w_{1}^{(m)}-\frac{3}{32} = \frac{1+2\cos {\big (\frac{h}{2^{m}}\big )}}{32 \cos ^{4}{\big (\frac{h}{2^{m+1}}\big )}} - \frac{3}{32} = \frac{2\cos {(\frac{h}{2^{m}})} + 1}{8 (1+\cos {(\frac{h}{2^{m}})})^2} - \frac{3}{32}. \end{aligned}$$

By making use of the fact that $2\cos {\big (\frac{h}{2^{m}}\big )} + 1 \le 3$, we get

$$\begin{aligned} \begin{aligned} \bigg \vert w_{1}^{(m)}-\frac{3}{32}\bigg \vert&\le \frac{3}{8}\bigg \vert \frac{1}{(1+\cos {(\frac{h}{2^{m}})})^2}-\frac{1}{4}\bigg \vert \\&=\frac{3}{8}\bigg \vert \bigg (\frac{1}{1+\cos {(\frac{h}{2^{m}})}}+\frac{1}{2}\bigg ) \bigg (\frac{1}{1+\cos {(\frac{h}{2^{m}})}}-\frac{1}{2}\bigg )\bigg \vert \\&\le \frac{3}{8}\big (\frac{1}{2} + 1\big ) \bigg \vert \frac{1-\cos {(\frac{h}{2^{m}})}}{1+\cos {(\frac{h}{2^{m}})}} \bigg \vert \le \frac{9}{16}\big \vert 1-\cos {\big (\frac{h}{2^{m}}\big )}\big \vert \\&\le \frac{9}{16}\; \kappa \; \frac{1}{4^{m}} = \frac{\mathcal {C}_{1}}{4^{m}},\;\;\; \text {where}\;\mathcal {C}_{1}=\frac{9}{16} \kappa . \end{aligned} \end{aligned}$$

From the regular vertex point subdivision rule we know that $w_{0}^{(m)} + 6 w_{1}^{(m)} = 1$. Using this fact the bound of $(w_{0}^{(m)}-\frac{7}{16})$ can be computed as follows. $\big \vert w_{0}^{(m)}-\frac{7}{16}\big \vert = \big \vert (1- 6 w_{1}^{(m)})-\big (1-\frac{9}{16}\big )\big \vert = 6\big \vert w_{1}^{(m)}-\frac{3}{32}\big \vert \le \frac{\mathcal {C}_{0}}{4^{m}}$, where $\mathcal {C}_{0} = 6\mathcal {C}_{1}$.

Now, $w_{2}^{(m)}-\frac{3}{128} = \frac{1+2\cos {(\frac{h}{2^{m}})}}{128 \cos ^{6}{(\frac{h}{2^{m+1}})}}= \frac{1}{16}\big (\frac{2\cos {(\frac{h}{2^{m}})} + 1}{(1+\cos {(\frac{h}{2^{m}})})^3} - \frac{3}{8}\big )$. Again using $2\cos {\big (\frac{h}{2^{m}}\big )} + 1 \le 3$, we get

$$\begin{aligned} \begin{aligned} \bigg \vert w_{2}^{(m)}-\frac{3}{128}\bigg \vert&\le \frac{3}{16}\bigg \vert \frac{1}{\big (1+\cos {(\frac{h}{2^{m}})}\big )^3}-\frac{1}{2^3} \bigg \vert \\&= \frac{3}{16}\bigg \vert \bigg ( \frac{1}{1+\cos {(\frac{h}{2^{m}})}}-\frac{1}{2}\bigg ) \bigg (\frac{1}{(1+\cos {(\frac{h}{2^{m}})})^2}+ \frac{1}{2(1+\cos {(\frac{h}{2^{m}})})}+\frac{1}{4}\bigg )\bigg \vert \\&\le \frac{3}{16}\big (1+\frac{1}{2}+\frac{1}{4}\big )\bigg \vert \frac{1-\cos {(\frac{h}{2^{m}})}}{2(1+\cos {(\frac{h}{2^{m}})})} \bigg \vert \le \frac{21}{128}\big \vert 1-\cos {\big (\frac{h}{2^{m}}\big )}\big \vert \\&\le \frac{21}{128}\; \kappa \; \frac{1}{4^{m}} = \frac{\mathcal {C}_{2}}{4^{m}},\;\;\; \text {where}\;\mathcal {C}_{1}=\frac{21}{128} \kappa . \end{aligned} \end{aligned}$$

In a similar manner, the bounds of $(w_{3}^{(m)}-\frac{9}{64})$, $(w_{4}^{(m)}-\frac{1}{128})$ and $(w_{5}^{(m)}-\frac{39}{128})$ can be obtained to show that $\big \vert w_{3}^{(m)}-\frac{9}{64}\big \vert \le \frac{\mathcal {C}_{3}}{4^{m}},\;\; \big \vert w_{4}^{(m)}-\frac{1}{128}\big \vert \le \frac{\mathcal {C}_{4}}{4^{m}},\;\; \big \vert w_{5}^{(m)}-\frac{39}{128}\big \vert \le \frac{\mathcal {C}_{5}}{4^{m}}$ with $\mathcal {C}_{3}$, $\mathcal {C}_{4}$, $\mathcal {C}_{5}$ being finite positive constants independent of m. Particularly in the derivations of the bounds of $\big (w_{3}^{(m)}-\frac{9}{64}\big )$ and $\big (w_{5}^{(m)}-\frac{39}{128}\big )$ the identities, respectively, $\big (1+2\cos {\big (\frac{h}{2^{m}}\big )}\big )^2 \le 9$ and $\big (5+6\cos \big (\frac{h}{2^{m}}\big )+2\cos \big (\frac{h}{2^{m-1}}\big )\big ) \le 13$ can be used.