About Wavelet Packet Analysis

Wavelet Toolbox™ software contains graphical tools and command line functions that let you

• Examine and explore characteristics of individual wavelet packets

• Perform wavelet packet analysis of 1-D and 2-D data

• Use wavelet packets to compress and remove noise from signals and images

This chapter takes you step-by-step through examples that teach you how to use the Wavelet Packet 1-D and Wavelet Packet 2-D graphical tools. The last section discusses how to transfer information from the graphical tools into your disk, and back again.

Note

All the graphical user interface tools described in this chapter let you import information from and export information to either disk or workspace.

Because of the inherent complexity of packing and unpacking complete wavelet packet decomposition tree structures, we recommend using the Wavelet Packet 1-D and Wavelet Packet 2-D graphical tools for performing exploratory analyses.

The command line functions are also available and provide the same capabilities. However, it is most efficient to use the command line only for performing batch processing.

Note

For more background on the wavelet packets, you can see the section Wavelet Packets.

Some object-oriented programming features are used for wavelet packet tree structures. For more detail, refer to Introduction to Object-Oriented Features.

This chapter takes you through the features of 1-D and 2-D wavelet packet analysis using the Wavelet Toolbox software. 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 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 app functions can be found in the section Examples Using Wavelet Packet Tree Objects.

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
wenergy	Entropy
wp2wtree	Extract wavelet tree from wavelet packet tree
wpcutree	Cut wavelet packet tree

Denoising and Compression

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

In the wavelet packet framework, compression and denoising 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.

Denoising and compression are interesting applications of wavelet packet analysis. The wavelet packet denoising 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 1-D wavelet packet analysis to compress and to denoise a signal.