Wavelet Toolbox

About Wavelet Packet Analysis

This chapter takes you through the features of one- and two-dimensional wavelet packet analysis using the MATLAB Wavelet Toolbox. You'll learn how to:

• Load a signal or image
• Perform a wavelet packet analysis of a signal or image
• Compress a signal
• Remove noise from a signal
• Compress an image
• Show statistics and histograms

The Wavelet Toolbox provides these functions for wavelet packet analysis. For more information, see the reference pages. The reference entries for these functions include examples showing how to perform wavelet packet analysis via the command line.

Some more advanced examples mixing command line and GUI functions can be found in the section Simple Use of Objects Through Four Examples.

Analysis-Decomposition Functions.   

Function Name	Purpose
wpcoef	Wavelet packet coefficients
wpdec and wpdec2	Full decomposition
wpsplt	Decompose packet

Synthesis-Reconstruction Functions.   

Function Name	Purpose
wprcoef	Reconstruct coefficients
wprec and wprec2	Full reconstruction
wpjoin	Recompose packet

Decomposition Structure Utilities.   

Function Name	Purpose
besttree	Find best tree
bestlevt	Find best level tree
entrupd	Update wavelet packets entropy
get	Get WPTREE object fields contents
read	Read values in WPTREE object fields
wentropy	Entropy
wp2wtree	Extract wavelet tree from wavelet packet tree
wpcutree	Cut wavelet packet tree

De-noising and Compression.   

Function Name	Purpose
ddencmp	Default values for de-noising and compression
wpbmpen	Penalized threshold for wavelet packet de-noising
wpdencmp	De-noising and compression using wavelet packets
wpthcoef	Wavelet packets coefficients thresholding
wthrmngr	Threshold settings manager

In the wavelet packet framework, compression and de-noising ideas are exactly the same as those developed in the wavelet framework. The only difference is that wavelet packets offer a more complex and flexible analysis, because in wavelet packet analysis, the details as well as the approximations are split.

A single wavelet packet decomposition gives a lot of bases from which you can look for the best representation with respect to a design objective. This can be done by finding the "best tree" based on an entropy criterion.

De-noising and compression are interesting applications of wavelet packet analysis. The wavelet packet de-noising or compression procedure involves four steps:

1. Decomposition

For a given wavelet, compute the wavelet packet decomposition of signal x at level N.

2. Computation of the best tree

For a given entropy, compute the optimal wavelet packet tree. Of course, this step is optional. The graphical tools provide a Best Tree button for making this computation quick and easy.

3. Thresholding of wavelet packet coefficients

For each packet (except for the approximation), select a threshold and apply thresholding to coefficients.

The graphical tools automatically provide an initial threshold based on balancing the amount of compression and retained energy. This threshold is a reasonable first approximation for most cases. However, in general you will have to refine your threshold by trial and error so as to optimize the results to fit your particular analysis and design criteria.

The tools facilitate experimentation with different thresholds, and make it easy to alter the tradeoff between amount of compression and retained signal energy.

4. Reconstruction

Compute wavelet packet reconstruction based on the original approximation coefficients at level N and the modified coefficients.

In this example, we'll show how you can use one-dimensional wavelet packet analysis to compress and to de-noise a signal.

Using Wavelet Packets

One-Dimensional Wavelet Packet Analysis