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The Laplacian Pyramid

The Laplacian Pyramid has been developed by Burt and Adelson in 1981 [4] in order to compress images. After the filtering, only one sample out of two is kept. The number of pixels decreases by a factor two at each scale.

The convolution is done with the filter h by keeping one sample out of two (see figure 14.7):

$\displaystyle c_{j+1}(k) = \sum_l h(l-2k) c_j(l)$     (14.38)


  
Figure 14.7: Passage from c0 to c1, and from c1 to c2.
\begin{figure}
\centerline{
\hbox{
\psfig{figure=fig_shema_lap1.ps,bbllx=3.5cm,bblly=14.5cm,bburx=18cm,bbury=22cm,height=6cm,width=10cm,clip=}
}}
\end{figure}

To reconstruct cj from cj+1, we need to calculate the difference signal wj+1.

$\displaystyle w_{j+1}(k) = c_j(k) - \tilde{c}_j(k)$     (14.39)

where $\tilde{c}_j$ is the signal reconstructed by the following operation (see figure 14.8):
$\displaystyle \tilde{c}_j(k) = 2 \sum_l h(k-2l) c_j(k)$     (14.40)


  
Figure 14.8: Passage from C1 to C0.
\begin{figure}
\centerline{
\hbox{
\psfig{figure=fig_shema_lap2.ps,bbllx=3.5cm,bblly=7cm,bburx=18cm,bbury=12cm,height=4cm,width=10cm,clip=}
}}
\end{figure}

In two dimensions, the method is similar. The convolution is done by keeping one sample out of two in the two directions. We have:

$\displaystyle c_{j+1}(n,m) = \sum_{k,l} h(k-2n,l-2m) c_j(k,l)$     (14.41)

and $\tilde{c}_j$ is:
$\displaystyle \tilde{c}_j(n,m) = 2 \sum_{k,l} h(n-2l,m-2l) c_{j+1}(k,l)$     (14.42)

The number of samples is divided by four. If the image size is $N\times N$, then the pyramid size is $\frac{4}{3}N^2$. We get a pyramidal structure (see figure 14.9).


  
Figure 14.9: Pyramidal Structure
\begin{figure}
\centerline{
\hbox{
\psfig{figure=fig_struc_pyr.ps,bbllx=4cm,bblly=7.5cm,bburx=17cm,bbury=23cm,height=8cm,width=6.5cm,clip=}
}}
\end{figure}

The laplacian pyramid leads to an analysis with four wavelets [3] and there is no invariance to translation.


next up previous contents
Next: Pyramidal Algorithm with one Up: Pyramidal Algorithm Previous: Pyramidal Algorithm
Petra Nass
1999-06-15