## Anisotropic Adaptive Filtering

5.1 Anisotropic Adaptive Filtering in Two Dimensions

On the basis of the characteristics of the human visual system, Knutsson et al.  argued that local anisotropy is an important property in images and introduced an anisotropic component in Abramatic and Silverman's model (Eq. (36))

Ha;i = H +(1 - a)(y + (1 - y) cos2(<p - 6))(1 - H), (37)

where the parameter y controls the level of anisotropy, ^ defines the angular direction of the filter coordinates, and 6 is the orientation of the local image structure. The specific choice of weighting function cos2(^ — 6) was imposed by its ideal interpolation properties, the directed anisotropy function could be implemented as a steerable filter from three fixed filters 

(these filters span the same space as the three filters 1, cos2(^), sin2(^), which also can be used). Freeman and Adelson later applied this concept to several problems in computer vision .

Knutsson et al. estimated the local orientation and the degree of anisotropy with three oriented Hilbert transform pairs, so-called quadrature filters, with the same angular profiles as the three basis functions describing the steerable weighting function. Figure 6 shows one of these Hilbert transform pairs. In areas of the image lacking a dominant orientation, y is set to 1, and Eq. (37) reverts to the isotropic Abramatic and Silverman solution. The more dominant the local orientation, the smaller the y value and the more anisotropic the filter.

### 5.2 Multidimensional Anisotropic Adaptive Filtering

The cos2(•) term in Eq. (37) can be viewed as a squared inner product of a vector defining the main directionality of the signal (6) and the frequency coordinate vectors. This main direction is denoted e:, and an orthogonal direction e2, Eq. (37) can be rewritten as

where u is the 2D frequency variable, and the parameters y1 and y2 define the anisotropy of the filter. For y: = y2 = 1 - a, the filter becomes isotropic (y = 1 in Eq. (37)), and for y: = 1 - a and y2 = 0 the filter becomes maximally anisotropic (y = 0 in Eq. (37)) and mainly favors signals oriented as e:.

The inner product notation in Eq. (38) allows for a direct extension to multiple dimensions:

0 0