引言

近年来，基于多分辨分析(MRA)的小波紧框架的构造和应用成为小波分析的热点研究内容，主要是由于它集框架和MRA的优点于一身，具有对分段光滑函数的稀疏逼近性、保证框架分解和重构的快速算法等特点，在实际应用中有着良好的效果^[1-3]。国内外有关小波框架的研究很多，Chui等^[4]利用Kronecker积构造了多元小波紧框架；Li等^[5]利用周期化方法，构造了多元周期性双小波框架；Jiang等^[6]构造了高度对称的双小波框架，用于三角形表面网格的多分辨率处理；随后他们又构造了具有理想高通滤波器的六边形二元紧小波框架滤波器组^[7]，然后对该结果进行了推广，提出规范滤波器的概念，构造了具有规范滤波器的低维小波紧框架系统^[8]。

文献[8]的思想是对给定的低维样条函数，利用酉扩展原理，首先求出构成紧小波框架系统所需高通滤波器的最小数目，然后根据滤波器和多相矩阵之间的关系，给出具体的滤波器构造方案。本文在文献[8]的工作基础上做了进一步的研究，将其构造思想推广到双小波框架，利用混合酉扩展原理，重点研究了一类具有规范滤波器的二元双小波框架的构造方案，证明生成半规范双小波框架至少需要7对高通滤波器，并给出了所需构造的滤波器的3种具体形式。

1 二元半规范双小波框架的相关概念

{V_j}_j∈Z是伸缩因子为2的加细函数ψ⁽⁰⁾(x)生成的L₂(R²)的MRA，且Ψ={ψ⁽¹⁾，ψ⁽²⁾，…，ψ^(L)}⊂V₁，那么对实值序列q_k^(l), ψ^(l)(l=0, 1, …, L)满足

$ {\psi ^{\left( l \right)}}\left( x \right) = {2^2}\sum\limits_{k \in {Z^2}} {q_k^{\left( l \right)}{\psi ^{\left( l \right)}}\left( {2x - k} \right)} ,x \in {R^2} $

或等价于

$ {{\hat \psi }^{\left( l \right)}}\left( {2\omega } \right) = {q^{\left( l \right)}}\left( \omega \right){{\hat \psi }^{\left( l \right)}}\left( \omega \right),\omega \in {R^2} $

其中，q^(l)(ω)=$\sum\limits_{k \in {Z^2}} {q_k^{\left( l \right)}{{\rm{e}}^{-{\rm{i}}k\omega }}} $，q⁽⁰⁾称作细化面具或低通滤波器, q^(l)(l=1, 2, …, L)称作小波面具或高通滤波器。当q⁽⁰⁾={q_k⁽⁰⁾}_k∈Z²含有有限个非零项时，称q⁽⁰⁾有一个紧支集，并称q⁽⁰⁾(ω)为有限脉冲反应滤波器(FIR)。

对Ψ={ψ⁽¹⁾, ψ⁽²⁾, …, ψ^(L)}⊂L₂(R²)，定义小波系统X(Ψ)={ψ_{j, k}^(l), 1≤l≤L; j∈Z, k∈Z²}，其中，ψ_{j, k}^(l)=2^j/2ψ^(l)(2^jx-k)，j∈Z, k∈Z²。

定义1  对任意f(x)∈L₂(R²)，若f(x)可表示成

$ f = \sum\limits_{l = 1}^L {\sum\limits_{j \in Z} {\sum\limits_{k \in {Z^2}} {\left\langle {f,\tilde \psi _{j,k}^{\left( l \right)}} \right\rangle \psi _{j,k}^{\left( l \right)}} } } $

其中，ψ_{j, k}^(l)∈X(Ψ)，$\tilde \psi _{j, k}^{\left( l \right)} \in X\left( {\mathit{\tilde \Psi }} \right)$，则称X(Ψ)、X(Ψ)构成一组双小波框架。

定义2  已知一对FIR滤波器q⁽⁰⁾(ω)、${\tilde q^{\left( 0 \right)}}\left( \omega \right)$，假设ψ⁽⁰⁾(x)和${\tilde \psi ^{\left( 0 \right)}}\left( x \right)$在R²上关于c=(c₁, c₂)对称，其中c_j=0或1/2(j=1, 2)，对l=1, 2, 3，令

$ {q^{\left( l \right)}}\left( \omega \right) = \left\{ \begin{array}{l} {{\rm{e}}^{{\rm{i}}\rho \left( {{\eta _l}} \right)\omega }}{{\tilde q}^{\left( 0 \right)}}\left( {\omega + {\rm{\pi }}{\eta _l}} \right)\;\;\;\;若2c{\eta _l}是奇数\\ {{\rm{e}}^{{\rm{i}}\rho \left( {{\eta _l}} \right)\omega }}\overline {{{\tilde q}^{\left( 0 \right)}}\left( {\omega + {\rm{\pi }}{\eta _l}} \right)} \;\;\;\;若2c{\eta _l}是偶数 \end{array} \right. $

(1)

$ {{\tilde q}^{\left( l \right)}}\left( \omega \right) = \left\{ \begin{array}{l} {{\rm{e}}^{{\rm{i}}\rho \left( {{\eta _l}} \right)\omega }}{q^{\left( 0 \right)}}\left( {\omega + {\rm{\pi }}{\eta _l}} \right)\;\;\;\;若2c{\eta _l}是奇数\\ {{\rm{e}}^{{\rm{i}}\rho \left( {{\eta _l}} \right)\omega }}\overline {{q^{\left( 0 \right)}}\left( {\omega + {\rm{\pi }}{\eta _l}} \right)} \;\;\;\;若2c{\eta _l}是偶数 \end{array} \right. $

(2)

其中，η₀=(0, 0), η₁=(1, 0), η₂=(1, 1), η₃=(0, 1);ρ(η₀)=(0, 0), ρ(η₁)=(1, 1), ρ(η₂)=(1, 0), ρ(η₄)=(0, 1)。此时，称由式(1)、(2) 定义的q^(l)(ω)、${\tilde q^{\left( l \right)}}\left( \omega \right)$(l=1, 2, 3) 为规范滤波器对。

一般情况下，对给定的加细函数ψ⁽⁰⁾、ψ⁽⁰⁾，由式(1) 和(2) 中的高通滤波器定义的ψ^(l)、ψ^(l)(l=1, 2, 3) 所生成的小波系统X(Ψ)和X(Ψ)不一定能构成L₂(R²)的双小波框架。本文考虑利用混合酉扩展原理，在原有系统中添加滤波器q^(l)、q^(l)(l=1, 4, …, L)，使其构成双小波框架滤波器组。

定义3  若{q⁽¹⁾(ω), …, q⁽³⁾(ω)}和{q⁽¹⁾(ω), …，q⁽³⁾(ω)}是按照式(1) 和(2) 定义的，则称双小波框架滤波器组{q⁽⁰⁾(ω), …, q^(L)(ω)}和{q⁽⁰⁾(ω)，…，q^(L)(ω)}(L≥4) 为半规范双小波框架滤波器组。

对滤波器组{q⁽⁰⁾(ω), q⁽¹⁾(ω), …, q^(L)(ω)}，M_{q⁽⁰⁾, …, q^(L)}(ω)和W_{q⁽⁰⁾, …, q^(L)}(ω)分别表示它的模矩阵和多相矩阵，具体形式如下:

$ \begin{array}{l} {\mathit{\boldsymbol{M}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( \omega \right) = \\ \left[ {\begin{array}{*{20}{c}} {{q^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _0}} \right)}&{{q^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _1}} \right)}& \cdots &{{q^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _3}} \right)}\\ {{q^{\left( 1 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _0}} \right)}&{{q^{\left( 1 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _1}} \right)}& \cdots &{{q^{\left( 1 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _3}} \right)}\\ \vdots & \vdots & \vdots & \vdots \\ {{q^{\left( L \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _0}} \right)}&{{q^{\left( L \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _1}} \right)}& \cdots &{{q^{\left( L \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _3}} \right)} \end{array}} \right] \end{array} $

$ \begin{array}{l} {\mathit{\boldsymbol{W}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( \omega \right) = \\ \left[ {\begin{array}{*{20}{c}} {q_0^{\left( 0 \right)}\left( \omega \right)}&{q_1^{\left( 0 \right)}\left( \omega \right)}& \cdots &{q_3^{\left( 0 \right)}\left( \omega \right)}\\ {q_0^{\left( 1 \right)}\left( \omega \right)}&{q_1^{\left( 1 \right)}\left( \omega \right)}& \cdots &{q_3^{\left( 1 \right)}\left( \omega \right)}\\ \vdots & \vdots & \vdots & \vdots \\ {q_0^{\left( L \right)}\left( \omega \right)}&{q_1^{\left( L \right)}\left( \omega \right)}& \cdots &{q_3^{\left( L \right)}\left( \omega \right)} \end{array}} \right] \end{array} $

模矩阵和多相矩阵满足

$ {\mathit{\boldsymbol{M}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( \omega \right) = {\mathit{\boldsymbol{W}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( {2\omega } \right)\mathit{\boldsymbol{U}}\left( \omega \right), $

其中U(ω)=$\frac{1}{2}{\left[{{{\rm{e}}^{-{\rm{i}}{\eta _k}\left( {\omega + \pi {\eta _j}} \right)}}} \right]_{0 \le k < 4, 0 \le j < 4'}}$。

与之对应的混合酉扩展原理如下。

引理1  对滤波器组{q⁽⁰⁾(ω), …, q^(L)(ω)}和$\left\{ {{{\tilde q}^{\left( 0 \right)}}\left( \omega \right), \cdots, {{\tilde q}^{\left( L \right)}}\left( \omega \right)} \right\}$，X(Ψ)和$X\left( {\mathit{\tilde \Psi }} \right)$构成双小波框架，只需满足下列条件之一：

(a)模矩阵M_{q⁽⁰⁾, …, q^(L)}(ω)和${\mathit{\boldsymbol{\tilde M}}_{{{\tilde q}^{\left( 0 \right)}}, \cdots, {{\tilde q}^{\left( L \right)}}}}\left( \omega \right)$满足

$ {\mathit{\boldsymbol{M}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( \omega \right) * {{\mathit{\boldsymbol{\tilde M}}}_{{{\tilde q}^{\left( 0 \right)}}, \cdots ,{{\tilde q}^{\left( L \right)}}}}\left( \omega \right) = {\mathit{\boldsymbol{I}}_4} $

(3)

(b)多相矩阵W_{q⁽⁰⁾, …, q^(L)}(ω)和${\mathit{\boldsymbol{\tilde W}}_{{{\tilde q}^{\left( 0 \right)}}, \cdots, {{\tilde q}^{\left( L \right)}}}}\left( \omega \right)$满足

$ {\mathit{\boldsymbol{W}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( \omega \right) * {{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 0 \right)}}, \cdots ,{{\tilde q}^{\left( L \right)}}}}\left( \omega \right) = {\mathit{\boldsymbol{I}}_4} $

(4)

2 二元半规范双小波框架滤波器的构造

为利用混合酉扩展原理构造双小波框架，本文假设加细函数ψ⁽⁰⁾、${\tilde \psi ^{\left( 0 \right)}}$对应的细化面具满足

$ \sum\limits_{k = 0}^3 {\overline {{q^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _k}} \right)} \;{{\tilde q}^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _k}} \right) < 1} $

(5)

下面讨论构成半规范双小波框架滤波器组至少有7对高通滤波器。

定理1  已知一对FIR低通滤波器q⁽⁰⁾(ω)、$\tilde q$⁽⁰⁾(ω)满足式(5)，若滤波器组{q⁽⁰⁾(ω), …, q^(L)(ω)}和{$\tilde q$⁽⁰⁾(ω), …, $\tilde q$^(L)(ω)}是一对半规范双小波框架滤波器组，其中q⁽¹⁾(ω), q⁽²⁾(ω), q⁽³⁾(ω)和$\tilde q$⁽¹⁾(ω)，$\tilde q$⁽²⁾(ω), $\tilde q$⁽³⁾(ω)是按照式(1) 和(2) 定义的，则L≥7。

证明  令M_{q⁽⁰⁾, …, q⁽³⁾}(ω)和分别表示{q⁽⁰⁾(ω), …, q⁽³⁾(ω)}和{q⁽⁰⁾(ω)，…，q⁽³⁾(ω)}的模矩阵。那么

$ \begin{array}{l} {\mathit{\boldsymbol{M}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( 3 \right)}}}}\left( \omega \right) * {{\mathit{\boldsymbol{\tilde M}}}_{{{\tilde q}^{\left( 0 \right)}}, \cdots ,{{\tilde q}^{\left( 3 \right)}}}}\left( \omega \right) = \left[ {\sum\limits_{k = 0}^3 {} } \right.\\ \left. {\overline {{q^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _k}} \right)} \;{{\tilde q}^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _k}} \right)} \right]{\mathit{\boldsymbol{I}}_4}。\end{array} $

取M_{q⁽⁰⁾, …, q^(L)}(ω)和${\boldsymbol{\tilde M}_{{{\tilde q}^{\left( 0 \right)}}, \cdots, {{\tilde q}^{\left( L \right)}}}}$的上半部分，可得到

$ \begin{array}{l} {\mathit{\boldsymbol{M}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( 3 \right)}}}}\left( \omega \right) = {\mathit{\boldsymbol{W}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( 3 \right)}}}}\left( {2\omega } \right)U\left( \omega \right)\\ {{\mathit{\boldsymbol{\tilde M}}}_{{{\tilde q}^{\left( 0 \right)}}, \cdots ,{{\tilde q}^{\left( 3 \right)}}}}\left( \omega \right) = {{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 0 \right)}}, \cdots ,{{\tilde q}^{\left( 3 \right)}}}}\left( {2\omega } \right)U\left( \omega \right) \end{array} $

令W_{q⁽⁰⁾, …, q⁽³⁾}(ω)和W_{q⁽⁴⁾, …, q^(L)}(ω)分别表示{q⁽⁰⁾(ω), …, q⁽³⁾(ω)}和{q⁽⁴⁾(ω), …, q^(L)(ω)}的多相矩阵，同理${\boldsymbol{\tilde W}_{{{\tilde q}^{\left( 0 \right)}}, \cdots, {{\tilde q}^{\left( 0 \right)}}}}\left( \omega \right)$和${\boldsymbol{\tilde W}_{{{\tilde q}^{\left( 4 \right)}}, \cdots, {{\tilde q}^{\left( L \right)}}}}\left( \omega \right)$分别表示{$\tilde q$⁽⁰⁾(ω), …, $\tilde q$⁽³⁾(ω)}和{$\tilde q$⁽⁴⁾(ω), …, $\tilde q$^(L)(ω)}的多相矩阵，那么

$ \begin{array}{l} {\mathit{\boldsymbol{M}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( 3 \right)}}}}\left( {2\omega } \right) * {{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 0 \right)}}, \cdots ,{{\tilde q}^{\left( 3 \right)}}}}\left( {2\omega } \right) = \left[ {\sum\limits_{k = 0}^3 {} } \right.\\ \left. {\overline {{q^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _k}} \right)} \;{{\tilde q}^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _k}} \right)} \right]{\mathit{\boldsymbol{I}}_4} \end{array} $

(6)

由式(4) 可知

$ \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{W}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( 3 \right)}}}}\left( {2\omega } \right)}\\ {{\mathit{\boldsymbol{W}}_{{q^{\left( 4 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( {2\omega } \right)} \end{array}} \right] * \left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 0 \right)}}, \cdots ,{{\tilde q}^{\left( 3 \right)}}}}\left( {2\omega } \right)}\\ {{{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 4 \right)}}, \cdots ,{{\tilde q}^{\left( L \right)}}}}\left( {2\omega } \right)} \end{array}} \right] = {\mathit{\boldsymbol{I}}_4}, $

则

$ \begin{array}{l} {\mathit{\boldsymbol{W}}_{{q^{\left( 0 \right)}}, \cdots ,{q^{\left( 3 \right)}}}}\left( {2\omega } \right) * {{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 0 \right)}}, \cdots ,{{\tilde q}^{\left( 3 \right)}}}}\left( {2\omega } \right) + \\ {\mathit{\boldsymbol{W}}_{{q^{\left( 4 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( {2\omega } \right) * {{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 4 \right)}}, \cdots ,{{\tilde q}^{\left( L \right)}}}}\left( {2\omega } \right) = {\mathit{\boldsymbol{I}}_4} \end{array} $

结合式(6)，可得

$ \begin{array}{l} {\mathit{\boldsymbol{W}}_{{q^{\left( 4 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( {2\omega } \right) * {{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 4 \right)}}, \cdots ,{{\tilde q}^{\left( L \right)}}}}\left( {2\omega } \right) = \left[ {1 - \sum\limits_{k = 0}^3 {} } \right.\\ \left. {\overline {{q^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _k}} \right)} \;{{\tilde q}^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _k}} \right)} \right]{\mathit{\boldsymbol{I}}_4} \end{array} $

(7)

又由式(5)

$ 1 - \sum\limits_{k = 0}^3 {\overline {{q^{\left( 0 \right)}}\left( {\omega + {\rm{\pi }}{\eta _k}} \right)} \;{{\tilde q}^{\left( 0 \right)}}\left( {\omega + {\rm{\pi }}{\eta _k}} \right)} < 0 $

那么

$ \begin{array}{l} rank\left[ {{\mathit{\boldsymbol{W}}_{{q^{\left( 4 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( {2\omega } \right)} \right] = rank\left[ {{{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 4 \right)}}, \cdots ,{{\tilde q}^{\left( L \right)}}}}} \right.\\ \left. {\left( {2\omega } \right)} \right] = rank\left[ {{\mathit{\boldsymbol{W}}_{{q^{\left( 4 \right)}}, \cdots ,{q^{\left( L \right)}}}}\left( {2\omega } \right) * {{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 4 \right)}}, \cdots ,{{\tilde q}^{\left( L \right)}}}}\left( {2\omega } \right)} \right] = 4。\end{array} $

所以L-4+1≥4，即L≥7。证毕。

于是，要构造有最少高通滤波器数目的半规范双小波框架，只需构造{q⁽⁴⁾, q⁽⁵⁾, q⁽⁶⁾, q⁽⁷⁾}和{$\tilde q$⁽⁴⁾, $\tilde q$⁽⁵⁾, $\tilde q$⁽⁶⁾, $\tilde q$⁽⁷⁾}满足式(7) 即可。或者说找到一对4×4的三角多相矩阵W_{q⁽⁴⁾, …, q⁽⁷⁾}(2ω)和${\mathit{\boldsymbol{\tilde W}}_{{{\tilde q}^{\left( 4 \right)}}, \cdots, {{\tilde q}^{\left( 7 \right)}}}}\left( {2\omega } \right)$满足

$ \begin{array}{l} {\mathit{\boldsymbol{W}}_{{q^{\left( 4 \right)}}, \cdots ,{q^{\left( 7 \right)}}}}\left( {2\omega } \right) * {{\mathit{\boldsymbol{\tilde W}}}_{{{\tilde q}^{\left( 4 \right)}}, \cdots ,{{\tilde q}^{\left( 7 \right)}}}}\left( {2\omega } \right) = \left[ {1 - \sum\limits_{k = 0}^3 {} } \right.\\ \left. {\overline {{q^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _k}} \right)} \;{{\tilde q}^{\left( 0 \right)}}\left( {\omega + {\rm{ \mathsf{ π} }}{\eta _k}} \right)} \right]{\mathit{\boldsymbol{I}}_4}。\end{array} $

若找到这样一对矩阵，那么

$ \left[ {\begin{array}{*{20}{c}} {{q^{\left( 4 \right)}}\left( \omega \right)}\\ {{q^{\left( 5 \right)}}\left( \omega \right)}\\ {{q^{\left( 6 \right)}}\left( \omega \right)}\\ {{q^{\left( 7 \right)}}\left( \omega \right)} \end{array}} \right] = \frac{1}{2}\mathit{\boldsymbol{W}}\left( {2\omega } \right)\left[ {\begin{array}{*{20}{c}} {{{\rm{e}}^{ - {\rm{i}}{\eta _0}\omega }}}\\ {{{\rm{e}}^{ - {\rm{i}}{\eta _1}\omega }}}\\ {{{\rm{e}}^{ - {\rm{i}}{\eta _2}\omega }}}\\ {{{\rm{e}}^{ - {\rm{i}}{\eta _3}\omega }}} \end{array}} \right] $

(8)

$ \left[ {\begin{array}{*{20}{c}} {{{\tilde q}^{\left( 4 \right)}}\left( \omega \right)}\\ {{{\tilde q}^{\left( 5 \right)}}\left( \omega \right)}\\ {{{\tilde q}^{\left( 6 \right)}}\left( \omega \right)}\\ {{{\tilde q}^{\left( 7 \right)}}\left( \omega \right)} \end{array}} \right] = \frac{1}{2}\mathit{\boldsymbol{\tilde W}}\left( {2\omega } \right)\left[ {\begin{array}{*{20}{c}} {{{\rm{e}}^{ - {\rm{i}}{\eta _0}\omega }}}\\ {{{\rm{e}}^{ - {\rm{i}}{\eta _1}\omega }}}\\ {{{\rm{e}}^{ - {\rm{i}}{\eta _2}\omega }}}\\ {{{\rm{e}}^{ - {\rm{i}}{\eta _3}\omega }}} \end{array}} \right] $

(9)

令

$ R\left( \omega \right) = 1 - \sum\limits_{k = 0}^3 {\overline {{q^{\left( 0 \right)}}\left( {\omega /2 + {\rm{\pi }}{\eta _k}} \right)} \;{{\tilde q}^{\left( 0 \right)}}\left( {\omega /2 + {\rm{\pi }}{\eta _k}} \right)} , $

则R(ω)可写成

$ R\left( \omega \right) = \overline {{g_1}\left( \omega \right)} {{\tilde g}_1}\left( \omega \right) + \overline {{g_2}\left( \omega \right)} {{\tilde g}_2}\left( \omega \right) + \overline {{g_3}\left( \omega \right)} {{\tilde g}_3}\left( \omega \right)。$

定义z₁=e^-iω₁，z₂=e^-iω₂，用K表示矩阵中每行非零元个数，则所需构造的高通滤波器有下面3种情况：

(1) K=1时

$ \mathit{\boldsymbol{W}}\left( \omega \right) = \left[ {\begin{array}{*{20}{c}} {{g_1}\left( \omega \right)}&0&0&0\\ 0&{{g_1}\left( \omega \right)}&0&0\\ 0&0&{{g_1}\left( \omega \right)}&0\\ 0&0&0&{{g_1}\left( \omega \right)} \end{array}} \right] $

$ \mathit{\boldsymbol{\tilde W}}\left( \omega \right) = \left[ {\begin{array}{*{20}{c}} {{{\tilde g}_1}\left( \omega \right)}&0&0&0\\ 0&{{{\tilde g}_1}\left( \omega \right)}&0&0\\ 0&0&{{{\tilde g}_1}\left( \omega \right)}&0\\ 0&0&0&{{{\tilde g}_1}\left( \omega \right)} \end{array}} \right] $

根据式(8) 和(9)

$ {q^{\left( 4 \right)}}\left( \omega \right) = \frac{1}{2}{g_1}\left( {2\omega } \right), $

$ {q^{\left( 5 \right)}}\left( \omega \right) = \frac{1}{2}{g_1}\left( {2\omega } \right){{\rm{e}}^{ - {\rm{i}}\left( {{\omega _1} + {\omega _2}} \right)}} = \frac{1}{2}{g_1}\left( {2\omega } \right){z_1}{z_2} $

$ {q^{\left( 6 \right)}}\left( \omega \right) = \frac{1}{2}{g_1}\left( {2\omega } \right){{\rm{e}}^{ - {\rm{i}}{\omega _1}}} = \frac{1}{2}{g_1}\left( {2\omega } \right){z_1}, $

$ {q^{\left( 7 \right)}}\left( \omega \right) = \frac{1}{2}{g_1}\left( {2\omega } \right){{\rm{e}}^{ - {\rm{i}}{\omega _2}}} = \frac{1}{2}{g_1}\left( {2\omega } \right){z_2}, $

$ {{\tilde q}^{\left( 4 \right)}}\left( \omega \right) = \frac{1}{2}{{\tilde g}_1}\left( {2\omega } \right), $

$ {{\tilde q}^{\left( 5 \right)}}\left( \omega \right) = \frac{1}{2}{{\tilde g}_1}\left( {2\omega } \right){{\rm{e}}^{ - {\rm{i}}\left( {{\omega _1} + {\omega _2}} \right)}} = \frac{1}{2}{{\tilde g}_1}\left( {2\omega } \right){z_1}{z_2} $

$ {{\tilde q}^{\left( 6 \right)}}\left( \omega \right) = \frac{1}{2}{{\tilde g}_1}\left( {2\omega } \right){{\rm{e}}^{ - {\rm{i}}{\omega _1}}} = \frac{1}{2}{{\tilde g}_1}\left( {2\omega } \right){z_1}, $

$ {{\tilde q}^{\left( 7 \right)}}\left( \omega \right) = \frac{1}{2}{{\tilde g}_1}\left( {2\omega } \right){{\rm{e}}^{ - {\rm{i}}{\omega _2}}} = \frac{1}{2}{{\tilde g}_1}\left( {2\omega } \right){z_2}; $

(2) K=2时

$ \mathit{\boldsymbol{W}}\left( \omega \right) = \left[ {\begin{array}{*{20}{c}} 0&{ - {g_1}\left( \omega \right)}&{{g_2}\left( \omega \right)}&0\\ { - {g_1}\left( \omega \right)}&0&0&{{g_2}\left( \omega \right)}\\ {\overline {{{\tilde g}_2}\left( \omega \right)} }&0&0&{\overline {{{\tilde g}_1}\left( \omega \right)} }\\ 0&{\overline {{{\tilde g}_2}\left( \omega \right)} }&{\overline {{{\tilde g}_1}\left( \omega \right)} }&0 \end{array}} \right] $

$ \mathit{\boldsymbol{\tilde W}}\left( \omega \right) = \left[ {\begin{array}{*{20}{c}} 0&{ - {{\tilde g}_1}\left( \omega \right)}&{{{\tilde g}_2}\left( \omega \right)}&0\\ { - {{\tilde g}_1}\left( \omega \right)}&0&0&{{{\tilde g}_2}\left( \omega \right)}\\ {\overline {{g_2}\left( \omega \right)} }&0&0&{\overline {{g_1}\left( \omega \right)} }\\ 0&{\overline {{g_2}\left( \omega \right)} }&{\overline {{g_1}\left( \omega \right)} }&0 \end{array}} \right] $

根据式(8) 和(9)

$ {q^{\left( 4 \right)}}\left( \omega \right) = \frac{1}{2}\left[ { - {g_1}\left( {2\omega } \right){z_1} + {g_2}\left( {2\omega } \right){z_1}{z_w}} \right], $

$ {q^{\left( 5 \right)}}\left( \omega \right) = \frac{1}{2}\left[ { - {g_1}\left( {2\omega } \right) + {g_2}\left( {2\omega } \right){z_2}} \right], $

$ {q^{\left( 6 \right)}}\left( \omega \right) = \frac{1}{2}\left[ {\overline {{{\tilde g}_2}\left( {2\omega } \right)} + \overline {{{\tilde g}_1}\left( {2\omega } \right)} \;{z_2}} \right], $

$ {q^{\left( 7 \right)}}\left( \omega \right) = \frac{1}{2}\left[ {\overline {{{\tilde g}_2}\left( {2\omega } \right)} \;{z_1} + \overline {{{\tilde g}_1}\left( {2\omega } \right)} \;{z_1}{z_2}} \right], $

$ {{\tilde q}^{\left( 4 \right)}}\left( \omega \right) = \frac{1}{2}\left[ { - {{\tilde g}_1}\left( {2\omega } \right){z_1} + {{\tilde g}_2}\left( {2\omega } \right){z_1}{z_w}} \right], $

$ {{\tilde q}^{\left( 5 \right)}}\left( \omega \right) = \frac{1}{2}\left[ { - {{\tilde g}_1}\left( {2\omega } \right) + {{\tilde g}_2}\left( {2\omega } \right){z_2}} \right], $

$ {{\tilde q}^{\left( 6 \right)}}\left( \omega \right) = \frac{1}{2}\left[ {\overline {{g_2}\left( {2\omega } \right)} + \overline {{g_1}\left( {2\omega } \right)} \;{z_2}} \right], $

$ {{\tilde q}^{\left( 7 \right)}}\left( \omega \right) = \frac{1}{2}\left[ {\overline {{g_2}\left( {2\omega } \right)} \;{z_1} + \overline {{g_1}\left( {2\omega } \right)} \;{z_1}{z_2}} \right]; $

(3) K=3时

$ \mathit{\boldsymbol{W}}\left( \omega \right) = \left[ {\begin{array}{*{20}{c}} 0&{{g_1}\left( \omega \right)}&{{g_2}\left( \omega \right)}&{{g_3}\left( \omega \right)}\\ { - {g_1}\left( \omega \right)}&0&{ - \overline {{{\tilde g}_3}\left( \omega \right)} }&{\overline {{{\tilde g}_2}\left( \omega \right)} }\\ { - {g_2}\left( \omega \right)}&{\overline {{{\tilde g}_3}\left( \omega \right)} }&0&{ - \overline {{{\tilde g}_1}\left( \omega \right)} }\\ { - {g_3}\left( \omega \right)}&{ - \overline {{{\tilde g}_2}\left( \omega \right)} }&{\overline {{{\tilde g}_1}\left( \omega \right)} }&0 \end{array}} \right] $

$ \mathit{\boldsymbol{\tilde W}}\left( \omega \right) = \left[ {\begin{array}{*{20}{c}} 0&{{{\tilde g}_1}\left( \omega \right)}&{{{\tilde g}_2}\left( \omega \right)}&{{{\tilde g}_3}\left( \omega \right)}\\ { - {{\tilde g}_1}\left( \omega \right)}&0&{ - \overline {{g_3}\left( \omega \right)} }&{\overline {{g_2}\left( \omega \right)} }\\ { - {{\tilde g}_2}\left( \omega \right)}&{\overline {{g_3}\left( \omega \right)} }&0&{ - \overline {{g_1}\left( \omega \right)} }\\ { - {{\tilde g}_3}\left( \omega \right)}&{ - \overline {{g_2}\left( \omega \right)} }&{\overline {{g_1}\left( \omega \right)} }&0 \end{array}} \right] $

根据式(8) 和(9)

$ {q^{\left( 4 \right)}}\left( \omega \right) = \frac{1}{2}\left[ {{g_1}\left( {2\omega } \right) + {g_2}\left( {2\omega } \right){z_1}{z_2} + {g_3}\left( {2\omega } \right){z_2}} \right], $

$ {q^{\left( 5 \right)}}\left( \omega \right) = \frac{1}{2}\left[ { - {g_1}\left( {2\omega } \right) - \overline {{{\tilde g}_3}\left( {2\omega } \right)} {z_1}{z_2} + \overline {{{\tilde g}_2}\left( {2\omega } \right)} {z_2}} \right], $

$ {q^{\left( 6 \right)}}\left( \omega \right) = \frac{1}{2}\left[ { - {g_2}\left( {2\omega } \right) + \overline {{{\tilde g}_3}\left( {2\omega } \right)} {z_1} - \overline {{{\tilde g}_1}\left( {2\omega } \right)} {z_2}} \right], $

$ {q^{\left( 7 \right)}}\left( \omega \right) = \frac{1}{2}\left[ { - {g_3}\left( {2\omega } \right) - \overline {{{\tilde g}_2}\left( {2\omega } \right)} {z_1} + \overline {{{\tilde g}_1}\left( {2\omega } \right)} {z_1}{z_2}} \right], $

$ {{\tilde q}^{\left( 4 \right)}}\left( \omega \right) = \frac{1}{2}\left[ {{{\tilde g}_1}\left( {2\omega } \right) + {{\tilde g}_2}\left( {2\omega } \right){z_1}{z_2} + {{\tilde g}_3}\left( {2\omega } \right){z_2}} \right], $

$ {{\tilde q}^{\left( 5 \right)}}\left( \omega \right) = \frac{1}{2}\left[ { - {{\tilde g}_1}\left( {2\omega } \right) - \overline {{g_3}\left( {2\omega } \right)} {z_1}{z_2} + \overline {{g_2}\left( {2\omega } \right)} {z_2}} \right], $

$ {{\tilde q}^{\left( 6 \right)}}\left( \omega \right) = \frac{1}{2}\left[ { - {{\tilde g}_2}\left( {2\omega } \right) + \overline {{g_3}\left( {2\omega } \right)} {z_1} - \overline {{g_1}\left( {2\omega } \right)} {z_2}} \right], $

$ {{\tilde q}^{\left( 7 \right)}}\left( \omega \right) = \frac{1}{2}\left[ { - {{\tilde g}_3}\left( {2\omega } \right) - \overline {{g_2}\left( {2\omega } \right)} {z_1} + \overline {{g_1}\left( {2\omega } \right)} {z_1}{z_2}} \right]。$

3 结束语

本文利用混合酉扩展原理，证明了对给定的一对低通滤波器，生成二元半规范双小波框架所需的高通滤波器对的最小数目为7。由于{q⁽¹⁾, q⁽²⁾, q⁽³⁾}和{$\tilde q$⁽¹⁾，$\tilde q$⁽²⁾, $\tilde q$⁽³⁾}已知，半规范双小波框架滤波器的构造问题就归结为求{q⁽⁴⁾，q⁽⁵⁾，q⁽⁶⁾，q⁽⁷⁾}和{$\tilde q$⁽⁴⁾, $\tilde q$⁽⁵⁾, $\tilde q$⁽⁶⁾, $\tilde q$⁽⁷⁾}。根据滤波器和多相矩阵之间的关系，可构造出所求滤波器的3种具体形式。