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

## 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 *y _{001}* represents a sequence after two consecutive
stages of lowpass filtering and one stage of highpass filtering. Accordingly,
the symbol

*y*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

_{000}*y*,

_{1}*y*, and

_{01}*y*, are the discrete wavelet transform that we need. The computational complexity of such a wavelet transform is

_{001}*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*(*N*log*N*) 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]).