Splitting a Matrix of Laurent Polynomials with Symmetry and its Application to Symmetric Framelet Filter Banks

Bin Han and Qun Mo


Let $M$ be a $2\times 2$ matrix of Laurent polynomials with real coefficients and symmetry. In this paper, we obtain a necessary and sufficient condition for the existence of four Laurent polynomials (or FIR filters) $u_1, u_2, v_1, v_2$ with real coefficients and symmetry such that $$ \left[ \begin{matrix} u_1(z) &v_1(z)\\ u_2(z) &v_2(z) \end{matrix}\right] \left[ \begin{matrix} u_1(1/z) &u_2(1/z)\\ v_1(1/z) &v_2(1/z)\end{matrix}\right]=M(z) \qquad \forall\; z\in \CC \bs \{0 \} $$ and $[Su_1](z)[Sv_2](z)=[Su_2](z)[Sv_1](z)$, where $[Sp](z)=p(z)/p(1/z)$ for a nonzero Laurent polynomial $p$. Our criterion can be easily checked and a step-by-step algorithm will be given to construct the symmetric filters $u_1, u_2, v_1, v_2$. As an application of this result to symmetric framelet filter banks, we present a necessary and sufficient condition for the construction of a symmetric MRA tight wavelet frame with two compactly supported generators derived from a given symmetric refinable function. Once such a necessary and sufficient condition is satisfied, an algorithm will be used to construct a symmetric framelet filter bank with two high-pass filters which is of interest in applications such as signal de-noising and image processing. As an illustration of our results and algorithms in this paper, we give several examples of symmetric framelet filter banks with two high-pass filters which have good vanishing moments and are derived from various symmetric low-pass filters including some $B$-spline filters.

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