N

The tensor C controls the filter adaptation by weighting the components in the outer product description of the Fourier domain U according to its "shape". For a 3D signal this shape can be thought of as an ellipsoid with principal axes êk. Similar to the 2D case in Eq. (37), where 6 describes the main orientation of the local spectrum, the tensor C should describe the major axes of the local spectrum in the Fourier domain. The consequence of the inner product in Eq. (41) is a reduction of the high-frequency components in directions where the local spectrum is weak. In those directions the adaptive filter will mainly act as the stationary Wiener component H, which has lowpass characteristics.

5.3 Adaptation Process

Before we describe the adaptation process, we will introduce a set of fixed filters

where nk define the directions of the filters and û = A. k M

It turns out that the adaptive filter in Eq. (39) can be written as a linear combination of these fixed filters and the stationary

Wiener solution where the weights are defined by inner products of a control tensor C and the filter associated dual tensors Mk.

where the tensors Mk define the so-called dual tensor basis to the one defined by the outer product of the filter directions. The dual tensors Mk are defined by

Spatial domain: real part

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