Vector Subdivision Schemes for Arbitrary Matrix Masks

TL;DR

The paper develops a unified framework for vector subdivision schemes with general matrix masks by defining a vector subdivision scheme through an order $m+1$ matching filter and proving its convergence is equivalent to the convergence of the associated vector cascade algorithm. It introduces the key quantity $\operatorname{sm}_p(a)$ to characterize convergence rates and connects this to refinable vector functions, providing a robust bridge between subdivision, cascade algorithms, and multiwavelet refinability. The results generalize and strengthen classical Lagrange and Hermite subdivision theories, showing that these are special cases within the broader matrix-masked setting and offering fast convergence rates and practical guidelines. Through detailed proofs and illustrative examples, the work demonstrates how to analyze, construct, and accelerate vector subdivision schemes for arbitrary matrix masks, with implications for wavelet methods in numerical PDEs and CAGD. The framework thus enables efficient computation of refinable vector functions and their derivatives across a wide range of matrix masks.

Abstract

Employing a matrix mask, a vector subdivision scheme is a fast iterative averaging algorithm to compute refinable vector functions for wavelet methods in numerical PDEs and to produce smooth curves in CAGD. In sharp contrast to the well-studied scalar subdivision schemes, vector subdivision schemes are much less well understood, e.g., Lagrange and (generalized) Hermite subdivision schemes are the only studied vector subdivision schemes in the literature. Because many wavelets used in numerical PDEs are derived from refinable vector functions whose matrix masks are not from Hermite subdivision schemes, it is necessary to introduce and study vector subdivision schemes for any general matrix masks in order to compute wavelets and refinable vector functions efficiently. For a general matrix mask, we show that there is only one meaningful way of defining a vector subdivision scheme. Motivated by vector cascade algorithms and recent study on Hermite subdivision schemes, we shall define a vector subdivision scheme for any arbitrary matrix mask and then we prove that the convergence of the newly defined vector subdivision scheme is equivalent to the convergence of its associated vector cascade algorithm. We also study convergence rates of vector subdivision schemes. The results of this paper not only bridge the gaps and establish intrinsic links between vector subdivision schemes and vector cascade algorithms but also strengthen and generalize current known results on Lagrange and (generalized) Hermite subdivision schemes. Several examples are provided to illustrate the results in this paper on various types of vector subdivision schemes with convergence rates.

Figures (3)

• Figure 1: The refinable vector function $\phi=[\phi_1,\phi_2]^\mathsf{T}$ and $\phi"$ in \ref{['ex2']} using mask $a_2$ with $(t_1,t_2)=(1,-1)$. Solid lines for $\phi_1$ and dashed lines for $\phi_2$. (c) The convergence rates are the slopes of $-\log_2 E_{u_j}(n)$ for the subdivision level $n=1,\ldots,10$ with $j=1,2,3$ for computing $\phi$. (d) The convergence rates are the slopes of $-\log_2 E_{u_j}(n)$ for $n=1,\ldots,10$ with $j=4,5$ for computing $\phi"$.
• Figure 2: The refinable vector function $\phi=[\phi_1,\phi_2]^\mathsf{T}$ and $\phi'=[\phi_1',\phi_2']^\mathsf{T}$ in \ref{['ex3']} using mask $a$ with $(t_1,t_2)=(\frac{1}{128},-\frac{13}{512})$. Solid lines for $\phi_1$ and dashed lines for $\phi_2$. (c) The convergence rates are the slopes of $-\log_2 E_{u_j}(n)$ for the subdivision level $n=1,\ldots,10$ with $j=1,2$ for computing $\phi$. (d) The convergence rates are the slopes of $-\log_2 E_{u_3}(n)$ for $n=1,\ldots,10$ for computing $\phi'$.
• Figure 3: The refinable vector function $\phi=[\phi_1,\phi_2]^\mathsf{T}$ and $\phi'$ in \ref{['ex4']} using mask $a$ with $(t_1, t_2,t_3)=(-\frac{1}{64}, -\frac{1}{32},-\frac{1}{128})$. Solid lines for $\phi_1$ and dashed lines for $\phi_2$. (c) The convergence rates are the slopes of $-\log_2 E_{u_j}(n)$ for the subdivision level $1\leqslant n\leqslant 10$ with $j=1,2$ for computing $\phi$. (d) The convergence rates are the slopes of $-\log_2 E_{u_j}(n)$ with $j=3,4$ for $1\leqslant n\leqslant 10$ for computing $\phi'$.

Theorems & Definitions (27)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 4
• Theorem 5
• Theorem 6
• proof
• proof : Proof of \ref{['thm:mfilter']}
• Lemma 7