MATH 414 Lecture 30

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Wavelet Decomposition and Reconstruction

${\begin{aligned}f_{j}(x)&=\sum _{k\in \mathbb {Z} }a_{k}^{j}\,\phi (2^{j}\,x-k)\\f_{j-1}(x)&=\mathrm {proj} _{V_{j-1}}{(f_{j})}=\sum _{k\in \mathbb {Z} }a_{k}^{j-1}\,\phi (2^{j-1}\,x-k)\\w_{j-1}(x)&=\mathrm {proj} _{W_{j-1}}{(f_{j})}=\sum _{k\in \mathbb {Z} }b_{k}^{j-1}\,\psi (2^{j-1}\,x-k)\\a_{k}^{j-1}&={\frac {1}{2}}\,\left(a_{2k}^{j}+a_{2k+1}^{j}\right)\\b_{k}^{j-1}&={\frac {1}{2}}\,\left(a_{2k}^{j}-a_{2k+1}^{j}\right)\end{aligned}}$

Decomposition

Low pass filter $\ell =\left(\dots ,0,0,0,\underbrace {\frac {1}{2}} _{k=-1},\underbrace {\frac {1}{2}} _{k=0},0,0,0,\dots \right)$

High pass filter $h=\left(\dots ,0,0,0,\underbrace {-{\frac {1}{2}}} _{k=-1},\underbrace {\frac {1}{2}} _{k=0},0,0,0,\dots \right)$

Downsampling:

1. Form ${\frac {1}{2}}\left(x_{k}+x_{k+1}\right)=y_{k}$
2. Downsample by discarding odd $k$
3. Call the resulting sequence $a_{k}^{j-1}$

${\begin{aligned}y&=\ell *x\\y_{k}&=\left(\ell *x\right)_{k}\\(Dy)_{2k}&=y_{2k}\\(Dy)_{2k+1}&=0\end{aligned}}$

Thus $a^{j-1}=D(\ell *a^{j})$

Similarly, $b^{j-1}=D(h*a^{j})$

Downsampling, high pass $h$, nor low pass $\ell$ depend on the level $j$!:

Reconstruction

${\begin{aligned}a_{2k}^{j}&=a_{k}^{j-1}+b_{k}^{j-1}\\a_{2k+1}^{j}&=a_{k}^{j-1}-b_{k}^{j-1}\end{aligned}}$

To handle the even and odd cases in filter form, we need a few more operations:

New low pass filter ${\tilde {\ell }}=\left(\ldots ,0,0,0,\underbrace {1} _{k=0},\underbrace {1} _{k=1},0,0,0,\dots \right)$

$\left({\tilde {\ell }}*z\right)_{k}=z_{k}+z_{k-1}$

New high pass filter ${\tilde {h}}=\left(\ldots ,0,0,0,\underbrace {1} _{k=0},\underbrace {-1} _{k=1},0,0,0,\dots \right)$

$\left({\tilde {h}}*z\right)_{k}=z_{k}-z_{k-1}$

Upsampling:

1. Form $x=\left(\ldots ,x_{-1},x_{0},x_{1},\dots \right)$
2. $Ux$ inserts zeroes at every odd term: $\left(\ldots ,0,x_{-1},0,x_{0},0,x_{1},0,\dots \right)$

Summary