Mathematical basis of Pyramid Generation

This method of pyramid formation as outlined in the previous section is equivalent to convolving an image g₀ with a weighting function h_l as :

$$\bold g_l = \bold h_l \otimes \bold g_0.$$ (6)

Here the size of the weighting function doubles from one level to the next as does the distance between the sample points. The shape of the weighting function infact converges rapidly to a characteristic form with successive higher levels of the pyramid. By characteristic form we mean the shape of the weighting function with a particular choice of the free parameter a (for example for a=0.4 it will approach a Gaussian distribution). The effect of convolving the image with one of the equvalent weighting functions, $\bold h_l$, is to low-pass filter the image.The pyramid algorithm mimics this low-pass filtering operation using a small compact two dimensional weighting function and uses a fast algorithm for generating different filtered versions of the original image.