Wavelet Toolbox

wpdec

Wavelet packet decomposition 1-D.

Syntax

T = wpdec(X,N,'wname',E,P)
T = wpdec(X,N,'wname')

Description

wpdec is a one-dimensional wavelet packet analysis function.

T = wpdec(X,N,'wname',E,P) returns a wavelet packet tree T corresponding to the wavelet packet decomposition of the vector X at level N, with a particular wavelet ('wname', see wfilters for more information).

E is a string containing the type of entropy (see wentropy for more information):

E = 'shannon', 'threshold', 'norm', 'log energy', 'sure', 'user'

P is an optional parameter:

'shannon' or 'log energy': P is not used

'threshold' or 'sure': P is the threshold (0 P)

'norm': P is a power (1 P)

'user': P is a string containing the name of a user-defined function.

T = wpdec(X,N,'wname') is equivalent to T = wpdec(X,N,'wname','shannon').

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)

Algorithm

The algorithm used for the wavelet packets decomposition follows the same line as the wavelet decomposition process (see dwt and wavedec for more information).

See Also
wavedec, waveinfo, wentropy, wpdec, wprec

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.

Meyer, Y. (1993), Les ondelettes. Algorithmes et applications, Colin Ed., Paris, 2nd edition. (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, M.V. (1994), Adapted wavelet analysis from theory to software algorithms, A.K. Peters.

wpcutree

wpdec2