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Best tree wavelet packet analysis
Syntax
Description
besttree is a one- or two-dimensional wavelet packet analysis function that computes the optimal subtree of an initial tree with respect to an entropy type criterion. The resulting tree may be much smaller than the initial one.
Following the organization of the wavelet packets library, it is natural to count the decompositions issued from a given orthogonal wavelet.
A signal of length N = 2L can be expanded in 
different ways, where is the number of binary subtrees of a complete binary tree of depth L.
As a result, we can conclude that 
(for more information, see the Mallat's book given in References at page 323).
This number may be very large, and since explicit enumeration is generally intractable, it is interesting to find an optimal decomposition with respect to a convenient criterion, computable by an efficient algorithm. We are looking for a minimum of the criterion.
T = besttree(T) computes the best tree T corresponding to the best entropy value.
[T,E] = besttree(T) computes the best tree T and, in addition, the best entropy value E.
The optimal entropy of the node, whose index is j-1, is E(j).
[T,E,N] = besttree(T) computes the best tree T, the best entropy value E and, in addition, the vector N containing the indices of the merged nodes.
Examples
% The current extension mode is zero-padding (see dwtmode).
% Load signal.
load noisdopp; x = noisdopp;
% Decompose x at depth 3 with db1 wavelet, using default
% entropy (shannon).
wpt = wpdec(x,3,'db1');
% Decompose the packet [3 0].
wpt = wpsplt(wpt,[3 0]);
% Plot wavelet packet tree wpt.
plot(wpt)
% Compute best tree.
bt = besttree(wpt);
% Plot best tree bt.
plot(bt)
Algorithm
Consider the one-dimensional case. Starting with the root node, the best tree is calculated using the following scheme. A node N is split into two nodes N1 and N2 if and only if the sum of the entropy of N1 and N2 is lower than the entropy of N. This is a local criterion based only on the information available at the node N.
Several entropy type criteria can be used (see wenergy for more information). If the entropy function is an additive function along the wavelet packet coefficients, this algorithm leads to the best tree.
Starting from an initial tree T and using the merging side of this algorithm, we obtain the best tree among all the binary subtrees of T.
See Also
bestlevt, wenergy, wpdec, wpdec2
References
Coifman, R.R.; M.V. Wickerhauser (1992), "Entropy-based algorithms for best basis selection," IEEE Trans. on Inf. Theory, vol. 38, 2, pp. 713-718.
Mallat, S. (1998), A wavelet tour of signal processing, Academic Press.
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