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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.
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 |
---|---|
Wavelet packet coefficients | |
Full decomposition | |
Decompose packet |
Synthesis-Reconstruction Functions
Function Name | Purpose |
---|---|
Reconstruct coefficients | |
Full reconstruction | |
Recompose packet |
Decomposition Structure Utilities
Function Name | Purpose |
---|---|
Update wavelet packets entropy | |
Get WPTREE object fields contents | |
Read values in WPTREE object fields | |
Entropy | |
Extract wavelet tree from wavelet packet tree | |
Cut wavelet packet tree |
De-Noising and Compression
Function Name | Purpose |
---|---|
Penalized threshold for wavelet packet de-noising | |
De-noising and compression using wavelet packets | |
Wavelet packets coefficients thresholding | |
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:
For a given wavelet, compute the wavelet packet decomposition of signal x at level N.
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.
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.
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.