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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 one- and two-dimensional 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 one- and two-dimensional 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 GUI functions can be found in the section Examples Using 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

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.

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