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):

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

  

Figure 14.7: Passage from c₀ to c₁, and from c₁ to c₂.

To reconstruct c_j from c_j+1, we need to calculate the difference signal w_j+1.

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

where $\tilde{c}_j$ is the signal reconstructed by the following operation (see figure 14.8):

$$\tilde{c}_j(k) = 2 \sum_l h(k-2l) c_j(k)$$ (14.40)

  

Figure 14.8: Passage from C₁ to C₀.

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

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

and $\tilde{c}_j$ is:

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

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