Quasi-stationary Subdivision Schemes in Arbitrary Dimensions

TL;DR

The notion of a multivariate quasi-stationary subdivision scheme is introduced and its convergence and smoothness are fully characterized to demonstrate the advantage of quasi-stationary subdivision schemes, which can circumvent the difficulty with stationary subdivision schemes.

Abstract

Stationary subdivision schemes have been extensively studied and have numerous applications in CAGD and wavelet analysis. To have high-order smoothness of the scheme, it is usually inevitable to enlarge the support of the mask that is used, which is a major difficulty with stationary subdivision schemes due to complicated implementation and dramatically increased special subdivision rules at extraordinary vertices. In this paper, we introduce the notion of a multivariate quasi-stationary subdivision scheme and fully characterize its convergence and smoothness. We will also discuss the general procedure of designing interpolatory masks with short support that yields smooth quasi-stationary subdivision schemes. Specifically, using the dyadic dilation of both triangular and quadrilateral meshes, for each smoothness exponent $m=1,2$, we obtain examples of $C^m$-convergent quasi-stationary $2I_2$-subdivision schemes with bivariate symmetric masks having at most $m$-ring stencils. Our examples demonstrate the advantage of quasi-stationary subdivision schemes, which can circumvent the difficulty above with stationary subdivision schemes.

Figures (4)

• Figure 1: (A) is the graph of the interpolating $4I_2$-refinable function $\phi\in C^1(\mathbb{R}^2)$ in Example \ref{['ex3']} and (B) is its contour. (C) is the graph of the partial derivative $\frac{\partial \phi}{\partial x} \in C^0(\mathbb{R}^2)$, and (D) is the contour of $\frac{\partial \phi}{\partial x}$.
• Figure 2: (A) is the graph of the interpolating $4I_2$-refinable function $\phi\in C^1(\mathbb{R}^2)$ in Example \ref{['ex1']} and (B) is its contour. (C) is the graph of the partial derivative $\phi_{x}=\frac{\partial \phi}{\partial x} \in C^0(\mathbb{R}^2)$, and (D) is the contour of $\frac{\partial \phi}{\partial x}$.
• Figure 3: (A) is the graph of the interpolating $4I_2$-refinable function $\phi\in C^2(\mathbb{R}^2)$ in Example \ref{['ex4']} and (B) is its contour. (C) is the graph of the partial derivative $\frac{\partial^2 \phi}{\partial x^2} \in C^0(\mathbb{R}^2)$, and (D) is the contour of $\frac{\partial^2 \phi}{\partial x^2}$.
• Figure 4: The first row is for the first choice in Example \ref{['ex2']}: (A) is the graph of the interpolating $4I_2$-refinable function $\phi\in C^2(\mathbb{R}^2)$ and (B) is its contour. (C) is the graph of the partial derivative $\frac{\partial^2 \phi}{\partial x^2} \in C^0(\mathbb{R}^2)$, and (D) is the contour of $\frac{\partial^2 \phi}{\partial x^2}$. The second row is for the second choice in Example \ref{['ex2']}: (E) is the graph of the interpolating $4I_2$-refinable function $\phi\in C^2(\mathbb{R}^2)$ and (F) is its contour. (G) is the graph of the partial derivative $\frac{\partial^2 \phi}{\partial x^2} \in C^0(\mathbb{R}^2)$, and (H) is the contour of $\frac{\partial^2 \phi}{\partial x^2}$.

Theorems & Definitions (15)

• Definition 1
• Theorem 1
• Lemma 2
• proof
• Remark 3
• Lemma 4
• proof
• Remark 5
• proof : Proof of Theorem \ref{['thm:qss']}
• Theorem 6