The idea behind image pyramids is to generate a number of homogeneous parameters that represent the response of a bank of filters at different scales and possibly different orientations. There are many different types of filters that can be used for this purpose.

One of the simplest types of filters is the Laplacian, also referred to as the ``Mexican hat'' because of its physical shape. It is a symmetrical filter, consisting of a highly positive center region and a negative ``surround.'' It is over-complete, however, meaning that some of the information encoded is redundant, and therefore we cannot recover the original signal from the representation.

Gabor filters are topologically similar to Laplacian filters, but they are elliptical rather than round, and they are highly eccentric. As such, they can be set at different orientations within the same scale, allowing possible detection of rotated versions of the same texture.

Wavelets are another type of filter family. They are constructed from a mother wavelet, which can be any admissible function (a function with no DC component; the Fourier transform of that function evaluated at 0 is 0). A family is constructed by dilating and translating the mother wavelet by different amounts. The advantage over simply using the Fourier transform for frequency analysis is that wavelets respond better to discontinuities (that is, edges) and spikes. Prominent examples of wavelet families are the Daubechies, Coiflet, Haar, and Symmlet. The Daubechies family is of particular interest because it is fractal in nature, as is the Haar because it is shaped like a simple square wave.

The most flexible of the image pyramids, however, are the steerable pyramids. Steerable pyramids can be thought of as an over-complete wavelet transform. They are called steerable because, by using only a small number of filters corresponding to a few directions, the output of a filter in any direction can easily be computed as a weighted sum of the filters that have already been calculated. In addition, steerable pyramids mostly avoid the problem of aliasing, or the confusion of low and high frequency components.

Fri Sep 5 16:40:07 PDT 1997