Autocorrelation Wavelet Packet transform

The wavelet packet transform is a wavelet transform where both the approximation and detail coefficients are passed through low and high pass quadrature mirror filters to create a full binary tree. The main distinction between normal wavelet packets and the autocorrelation wavelet packets is that the autocorrelation transform is redundant. Therefore the number of wavelet coefficients will double at each subsequent level of the decomposition tree. To perform the autocorrelation wavelet transform, use the acwpt function.

acwpt(x, wt[, L=maxtransformlevels(x)])

acwpt(x, wavelet(WT.db4))

By design, the original signal is contained in the root node of the binary decomposition tree. However, One can reconstruct the signal from the wavelet coefficents, perhaps after pruning the tree, using the aciwpt function.

For a wavelet packet decomposition, it is interesting to find an optimal decomposition with respect to a convenient criterion. For a more detailed explanation on how an optimal decomposition is chosen for a wavelet packet decomposition, refer to the Choosing the Optimal Decomposition section of the Wavelet Packets documentation on the MathWorks website.

The acwpt_postorder_bb traverses the binary tree in an bottom-up order and is therefore the most democratic way of choosing the optimal wavelet coefficients. On the otherhand, acwpt_preorder_bb will traverse the binary tree in an top-bottom order and is therefore a "greedy" way of choosing the optimal wavelet coefficients. The two function will often return a different binary tree for the same decomposition.

We use different entropy methods as our criteria because the describe information-related properties for an accurate representation of a given signal. The entropy methods currently available within our package are, ShannonEntropy, LogEnergyEntropy, and NormEntropy. For a more detailed description of each method, refer to the AC Wavelet Utils documentation.

A simple visualization of the best basis decomposition can be obtained using the selectednodes_plot function, which highlights the selected nodes on a predefined grid. The autocorrelation wavelet decomposition is redundant, meaning that each node of the decomposition tree will be the same length as the original signal. Therefore the size of each grid cell does not accurately represent the length of the coefficient vector in each node. However, it is sufficient to understand which nodes are selected in the best basis tree.

selectednodes_plot(x::BitArray, nodecolor::Symbol = :red)