2.7 Steerable Filters

While looking for a flexible tool for processing oriented data, Freeman and Adelson [170] introduced some time ago the concept of steerable filters. These were further developed by Perona [311] and Simoncelli et al. [342]. Here again one obtains a multiscale pyramid decomposition, which is quite efficient in a number of problems, mostly related to machine vision. Similar techniques have been used with the Gabor transform [234]. We will briefly describe this scheme and compare it to the directional wavelet packets of Section 2.6.

The basic idea is quite simple, and best illustrated on the example of a Gaussian kernel G( x, y). From the partial derivatives , one computes the derivative in the direction ?:

Since convolution is a linear operation, one may use G _? for filtering an image f in the direction ? by superposing the filterings in directions x and y:

This is the property of orientability. More generally, a filter f is orientable or steerable if any oriented version of it may be obtained from a finite number of basic orientations:

The weights { k _m( ?) , m = 1 ,... ,M} are called interpolation functions. (The notion of orientability may be extended to other transformations, such as scaling [311], but we will not consider these generalizations here.)

When the filter f admits a finite Fourier series (i.e., it is a real trigonometric polynomial),

Freeman...