逆小波转换

逆小波转换(inverse wavelet transform)为小波转换的反函数，小波转换大致分为三类

连续小波转换
离散变数连续小波转换
离散小波转换

分别介绍此三种的反函数

连续小波转换反函数

已知

$X_{w}(a,b)={\frac {1}{\sqrt {|(b)|}}}\int _{-\infty }^{\infty }x(t)\psi ({\frac {t-a}{b}})\,dt$

则逆转换为

$x(t)={\frac {1}{C}}_{\psi }\int _{0}^{\infty }\int _{-\infty }^{\infty }{\frac {1}{b^{5/2}}}X_{w}(a,b)\psi ({\frac {t-a}{b}})\,dadb$

其中

$C_{\psi }=\int _{0}^{\infty }{\frac {|{\boldsymbol {\psi }}(f)|^{2}}{|f|}}df<\infty$

证明:

由于 $X_{w}(a,b)=x(t)*{\frac {1}{\sqrt[{}]{b}}}\psi ({\frac {-t}{b}})$

假设 $y(t)={\frac {1}{C}}_{\psi }\int _{0}^{\infty }\int _{-\infty }^{\infty }{\frac {1}{b^{5/2}}}X_{w}(a,b)\psi ({\frac {t-a}{b}})\,dadb$

则 $y(t)={\frac {1}{C}}_{\psi }\int _{0}^{\infty }x(t)*\psi ({\frac {-t}{b}})*\psi ({\frac {t}{b}}){\frac {db}{b^{3}}}$

经过傅立叶转换，原本的折积性质变为相乘

$Y(f)={\frac {1}{C}}_{\psi }\int _{0}^{\infty }X(f){\boldsymbol {\psi }}(-bf){\boldsymbol {\psi }}(bf){\frac {db}{b}}$

如果母小波为实函数，则其傅立叶转换有以下性质

${\boldsymbol {\psi }}(-f)={\boldsymbol {\psi }}^{*}(f),{\boldsymbol {\psi }}(-bf){\boldsymbol {\psi }}(bf)={\boldsymbol {\psi }}^{*}(f){\boldsymbol {\psi }}(bf)=|{\boldsymbol {\psi }}(bf)|^{2}$

$Y(f)=X(f){\frac {1}{C}}_{\psi }\int _{0}^{\infty }|{\boldsymbol {\psi }}(bf)|^{2}{\frac {db}{b}}$

$=X(f){\frac {1}{C}}_{\psi }\int _{0}^{\infty }|{\boldsymbol {\psi }}(f_{1})|^{2}{\frac {df_{1}}{bf}}$ (使用变数代换 $f_{1}=bf$ )

$=X(f){\frac {1}{C}}_{\psi }\int _{0}^{\infty }|{\boldsymbol {\psi }}(f_{1})|^{2}{\frac {df_{1}}{bf_{1}}}$

$=X(f)$

得证 $y(t)=x(t)$

离散变数连续小波转换反函数

$x(t)=\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }2^{m/2}\psi _{1}(2^{m}-n)X_{w}(n,m)$

$\psi _{1}(t)$ 为 $\psi (t)$ 的双效函数(dual function)，满足以下正交(orthogonal)特性

$\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }2^{m}\psi _{1}(2^{m}-n)\psi (2^{m}t_{1}-n)=\delta (t-t_{1})$

或是

$\int _{-\infty }^{\infty }2^{m}\psi _{1}(2^{m_{1}}t-n_{1})\psi (2^{m}t-n)\,dt=\delta (m-m_{1})\delta (n-n_{1})$

通常会设计成 $\psi _{1}(t)=\psi (t)$

因此离散变数连续小波转换能进行逆转换的条件为:

$\int _{-\infty }^{\infty }2^{m}\psi (2^{m_{1}}t-n_{1})\psi (2^{m}t-n)\,dt=\delta (m-m_{1})\delta (n-n_{1})$

离散小波转换反函数

在这里解释的是如何重建(reconstruction)一个经过离散小波转换的函数

以进行一阶离散小波转换，升降频倍率为2为例，可以得到右图的架构

DWT reconstruction

$g_{1}[n],h_{1}[n]$ 需要满足一些条件才能使 $x[n]=x_{0}[n]$

将此流程进行Z转换及化简可得到:

$X_{0}(z)={\frac {1}{2}}[G(z)G_{1}(z)+H(z)H_{1}(z)]X(z)+{\frac {1}{2}}[G(-z)G_{1}(z)+H(-z)H_{1}(z)]X(-z)$

因此为了得到 $X_{0}(z)=X(z)$ ，须满足以下二条件

$G(z)G_{1}(z)+H(z)H_{1}(z)=2$
$G(-z)G_{1}(z)+H(-z)H_{1}(z)=0$

可转换为 ${\begin{bmatrix}G(z)&H(z)\\G(-z)&H(-z)\end{bmatrix}}{\begin{bmatrix}G_{1}(z)\\H_{1}(z)\end{bmatrix}}={\begin{bmatrix}2\\0\end{bmatrix}}$

化简得到 ${\begin{bmatrix}G_{1}(z)\\H_{1}(z)\end{bmatrix}}={\frac {2}{det(H_{m}(z))}}{\begin{bmatrix}H(-z)\\-G(-z)\end{bmatrix}}$ 其中 $det(H_{m}(z))=G(z)H(-z)-H(z)G(-z)$