## 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. [16] 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 [16]

(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 [26].

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:

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