Wavelet Toolbox

Organizing the Wavelet Packets

The set of functions W_j,n = (W_j,n,k(x), ) is the (j,n) wavelet packet. For positive values of integers j and n, wavelet packets are organized in trees. The tree in Figure 6-40 is created to give a maximum level decomposition equal to 3. For each scale j, the possible values of parameter n are 0, 1, ..., 2j -1.

Figure 6-40: Wavelet Packets Organized in a Tree; Scale j Defines Depth and Frequency n Defines Position in the Tree

The notation W_j,n, where j denotes scale parameter and n the frequency parameter, is consistent with the usual depth-position tree labeling.

We have , and .

It turns out that the library of wavelet packet bases contains the wavelet basis and also several other bases. Let us have a look at some of those bases. More precisely, let V₀ denote the space (spanned by the family W_0,0) in which the signal to be analyzed lies; then (W_d,1; d 1) is an orthogonal basis of V₀.

For every strictly positive integer D, (W_D,0, (W_d,1; 1 d D)) is an orthogonal basis of V₀.

We also know that the family of functions {(W_j+1,2n),(W_j+1,2n+1)} is an orthogonal basis of the space spanned by W_j,n, which is split into two subspaces: W_j+1,2n spans the first subspace, and W_j+1,2n+1 the second one.

This last property gives a precise interpretation of splitting in the wavelet packet organization tree, because all the developed nodes are of the form shown in the figure below.

Figure 6-41: Wavelet Packet Tree: Split and Merge

It follows that the leaves of every connected binary subtree of the complete tree correspond to an orthogonal basis of the initial space.

For a finite energy signal belonging to V₀, any wavelet packet basis will provide exact reconstruction and offer a specific way of coding the signal, using information allocation in frequency scale subbands.

Wavelet Packet Atoms

Choosing the Optimal Decomposition

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