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Steering Dual-Tree Complex Wavelets: is this possible?

One of the primary advantages of the Steerable Pyramid transform (see [1]) is that the filter response in an arbitrary direction can be obtained as a linear sum of the filter responses computed for a *fixed* number of orientations. Equivalently, by taking linear sums of the basis elements, an arbitrary rotated version of the main basis element can be obtained. This property is usually called *steerability*.

For the Dual-Tree complex wavelet transform (DT-CWT), it is known that the basis elements are *not* steerable, this is by the way the transform is designed. Nevertheless, the DT-CWT is still a directional transform with 6 orientations (15°;45°;75°;105°;135°;165°) and the basis elements can be steered *approximately*, as shown in the example below:

This is particularly of importance when considering inter-orientation dependencies between complex wavelet coefficients. In this example, we assume (for simplicity) that the wavelet filters for different orientations are rotationally similar. In practice, this is not the case for orientations at +/- 45°, because these filter responses are concentrated in higher radial frequencies than the others. More information on this topic can be found in [2].

#### References

- [1] W. T. Freeman and E. H. Adelson, "The design and use of steerable filters", IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 9, pp. 891--906, 1991.
- [2] N. Kingsbury, "Rotation-invariant local feature matching with complex wavelets," in Proc. European Conference on Signal Processing (EUSIPCO), Florence, 4-8 Sept 2006