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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):

 (14.38)

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

 (14.39)

where is the signal reconstructed by the following operation (see figure 14.8):
 (14.40)

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

 (14.41)

and is:
 (14.42)

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).

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

Next: Pyramidal Algorithm with one Up: Pyramidal Algorithm Previous: Pyramidal Algorithm
Petra Nass
1999-06-15