MATH 414 Lecture 30

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Wavelet Decomposition and Reconstruction

$\begin{array}{rl}{f}_j(x)& =\sum _{k\in \mathbb{\mathbb{Z}}}{a}_k^j\varphi (2^{j}x-k)\\ f_{j-1}(x)& ={\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}_{V_{j-1}}(f_j)=\sum _{k\in \mathbb{\mathbb{Z}}}{a}_k^{j-1}\varphi (2^{j-1}x-k)\\ w_{j-1}(x)& ={\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}_{W_{j-1}}(f_j)=\sum _{k\in \mathbb{\mathbb{Z}}}{b}_k^{j-1}\psi (2^{j-1}x-k)\\ a_k^{j-1}& =\frac{1}{2}(a_{2k}^j+a_{2k+1}^j)\\ b_k^{j-1}& =\frac{1}{2}(a_{2k}^j-a_{2k+1}^j)\end{array}$

Decomposition

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

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

Downsampling:

1. Form $\frac{1}{2}(x_k+x_{k+1})=y_k$
2. Downsample by discarding odd $k$
3. Call the resulting sequence $a_k^{j-1}$

$\begin{array}{rl}y& =\ell *x\\ y_k& ={(\ell *x)}_k\\ (Dy{)}_{2k}& =y_{2k}\\ (Dy{)}_{2k+1}& =0\end{array}$

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

$\begin{array}{rl}{a}_{2k}^j& =a_k^{j-1}+b_k^{j-1}\\ a_{2k+1}^j& =a_k^{j-1}-b_k^{j-1}\end{array}$

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

New low pass filter $\tilde{\ell }=(\dots ,0,0,0,\underset{k=0}{\underbrace{1}},\underset{k=1}{\underbrace{1}},0,0,0,\dots )$

${(\tilde{\ell }*z)}_k=z_k+z_{k-1}$

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

${(\tilde{h}*z)}_k=z_k-z_{k-1}$

Upsampling:

1. Form $x=(\dots ,x_{-1},x_0,x_1,\dots )$
2. $Ux$ inserts zeroes at every odd term: $(\dots ,0,x_{-1},0,x_0,0,x_1,0,\dots )$

Summary