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  1. 1 Two-Channel Perfect Reconstruction Filter Banks
  2. 2 Orthogonal Filter Banks
  3. 3 General Tree-Structure Filter Banks and Wavelet Packets
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6.3 General Tree-Structure Filter Banks and Wavelet Packets

Once the two-channel perfect reconstruction filter banks have been determined, based on the result obtained in Section 5.4, we can readily compute the discrete wavelet transform through the tree of filter banks, as shown in Figure 6-9. While the subscript "1" indicates a highpass, the subscript "0" corresponds to a lowpass filter. Hence, the symbol y001 represents a sequence after two consecutive stages of lowpass filtering and one stage of highpass filtering. Accordingly, the symbol y000 represents a sequence after three consecutive stages of lowpass filtering. To compute the discrete wavelet transform, we repeatedly split, filter, and decimate the lowpass bands. The outputs of the highpass filters, such as y1, y01, and y001, are the discrete wavelet transform that we need. The computational complexity of such a wavelet transform is O(N).

Figure 6-9 The tree of filter banks for computing the discrete wavelet transform

Intuitively, the tree of filter banks illustrated in Figure 6-9 is not the only possible decomposition scheme. For instance, the signal can also be decomposed by the system depicted in Figure 6-10. Particularly, if we split both the lowpass and highpass bands at all stages, the resulting filter bank structure becomes a full binary tree, as shown in Figure 6-11. This type of tree takes O(NlogN) calculations and results in a completely evenly spaced frequency resolution. The result of the full binary tree is similar to that calculated by the STFT algorithm.

Figure 6-10 An alternative three-stage decomposition

Figure 6-11 The full binary tree results in a completely evenly spaced frequency resolution, which is similar to that calculated by the STFT.

At this point, the question arises as to whether one can generate alternative decomposition trees that allow branches on the highpass bands, and whether such a decomposition has an interpretation in terms of basis functions for linear vector spaces of functions. The answer is yes. The reader can find a corresponding mathematical proof in [27] or [40]. The resulting decomposition is called wavelet packets by Coifman and Wickerhauser [261]. The main advantage of the wavelet packets is the flexibility it offers, which allows adaptation to particular signals. The potential of wavelet packets lies in the capacity to offer a rich menu of orthonormal bases from which the "best" one can be chosen ("best" according to a particular criterion).

It can be easily verified that for a given number of stages k, the number of different decompositions is

(6.44)

The number of possible choices grows dramatically as the stage k (or scale) increases. A successful application of the wavelet packets includes the FBI standard for fingerprint image compression (see [248] and [250]).

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