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):

  

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.

where is the signal reconstructed by the following operation (see figure 14.8):

  

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:

and is:

The number of samples is divided by four. If the image size is , then the pyramid size is . 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.