wpdec
Wavelet packet decomposition 1-D
Syntax
T = wpdec(X,N,wname,E,P)
T = wpdec(X,N,wname)
T = wpdec(X,N,wname,'shannon')
Description
wpdec is a one-dimensional
wavelet packet analysis function.
T = wpdec(X,N, returns a wavelet packet
tree wname,E,P)T corresponding to the wavelet packet decomposition of
the vector X at level N, using the wavelet
specified by wname (see wfilters for more information).
T = wpdec(X,N, is equivalent to
wname)T = wpdec(X,N,.wname,'shannon')
E is a character vector or string scalar containing the type of entropy and
P is an optional parameter depending on the value of
T (see wentropy for more information).
Entropy Type Name (E) | Parameter (P) | Comments |
|---|---|---|
'shannon' | P is not used. | |
'log energy' | P is not used. | |
'threshold' | 0 ≤ P | P is the threshold. |
'sure' | 0 ≤ P | P is the threshold. |
'norm' | 1 ≤ P | P is the power. |
'user' | character vector or string scalar | P is a character vector or string scalar
containing the file name of your own entropy function, with a single
input X. |
FunName | No constraints on P |
FunName contains
the file name of your own entropy function, with
X as input and P as
additional parameter to your entropy function. |
Note
The 'user' option is historical and still
kept for compatibility, but it is obsoleted by the last option described
in the table above. The FunName option do the same
as the 'user' option and in addition gives the
possibility to pass a parameter to your own entropy function.
The wavelet packet method is a generalization of wavelet decomposition that offers a richer signal analysis. Wavelet packet atoms are waveforms indexed by three naturally interpreted parameters: position and scale as in wavelet decomposition, and frequency.
For a given orthogonal wavelet function, a library of wavelet packets bases is generated. Each of these bases offers a particular way of coding signals, preserving global energy and reconstructing exact features. The wavelet packets can then be used for numerous expansions of a given signal.
Simple and efficient algorithms exist for both wavelet packets decomposition and optimal decomposition selection. Adaptive filtering algorithms with direct applications in optimal signal coding and data compression can then be produced.
In the orthogonal wavelet decomposition procedure, the generic step splits the approximation coefficients into two parts. After splitting we obtain a vector of approximation coefficients and a vector of detail coefficients, both at a coarser scale. The information lost between two successive approximations is captured in the detail coefficients. The next step consists in splitting the new approximation coefficient vector; successive details are never re-analyzed.
In the corresponding wavelet packets situation, each detail coefficient vector is also decomposed into two parts using the same approach as in approximation vector splitting. This offers the richest analysis: the complete binary tree is produced in the one-dimensional case or a quaternary tree in the two-dimensional case.
Examples
% The current extension mode is zero-padding (see dwtmode).
% Load signal.
load noisdopp; x = noisdopp;
% Decompose x at depth 3 with db1 wavelet packets
% using Shannon entropy.
wpt = wpdec(x,3,'db1','shannon');
% The result is the wavelet packet tree wpt.
% Plot wavelet packet tree (binary tree, or tree of order 2).
plot(wpt)
Algorithms
The algorithm used for the wavelet packets decomposition follows
the same line as the wavelet decomposition process (see dwt and wavedec for
more information).
References
Coifman, R.R., and M.V. Wickerhauser. “Entropy-Based Algorithms for Best Basis Selection.” IEEE Transactions on Information Theory 38, no. 2 (March 1992): 713–18. https://doi.org/10.1109/18.119732.
Meyer, Yves. Les ondelettes. Algorithmes et applications, Colin Ed., Paris, 2nd edition, 1994. (English translation: Wavelets: Algorithms and Applications, SIAM).
Wickerhauser, M.V. (1991), “INRIA lectures on wavelet packet algorithms,” Proceedings ondelettes et paquets d'ondes, 17–21 June, Rocquencourt, France, pp. 31–99.
Wickerhauser, Mladen Victor. Adapted Wavelet Analysis from Theory to Software. Wellesley, MA: A.K. Peters, 1994.