The Gaussian Pyramid is a hierarchy of low-pass filtered versions of the original image, such that successive levels correspond to lower frequencies.
Figure 6.2: Gaussian pyramid
The low-pass filtering is done using convolution with a Gaussian filter kernel. Where the filter overhangs the image edges we reflect the image about its edge. Since the lowest frequencies have been removed, the full-size image contains redundant pixels. One may define the REDUCE operator which is a filtering followed by elimination of unnessesary pixels. For a filter kernel w[i,j] of dimension and reduction factor 4 we have:
There is a corresponding EXPAND operator which will reconstruct the low-pass filtered image by interpolating between pixels in the reduced image.
A Gaussian pyramid G(I) is defined on image I as: