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Vector Space Decomposition
Vector Space Decomposition for a M = 2 Full Wavelet Packet System
Frequency Responses for the Two-Band Wavelet Packet Filter Bank
Frequency Responses for the Two-Band Wavelet Packet Filter Bank

Another is the full packet decomposition shown in [link] . Any pruning of this full tree would generate a valid packet basis systemand would allow a very flexible tiling of the time-scale plane.

We can choose a set of basic vectors and form an orthonormal basis, such that some cost measure on the transformed coefficients is minimized.Moreover, when the cost is additive, the

Wavelet Packets Generated by PHI_D
Wavelet Packets Generated by φ D 8 '
The Haar Wavelet Packet
The Haar Wavelet Packet

best orthonormal wavelet packet transform can be found using a binary searching algorithm [link] in O ( N log N ) time.

Some examples of the resulting time-frequency tilings are shown in [link] . These plots demonstrate the frequency adaptation power of the wavelet packet transform.

Examples of Time-Frequency Tilings of Different Three-Scale Orthonormal Wavelet Packet Transforms.
Examples of Time-Frequency Tilings of Different Three-Scale OrthonormalWavelet Packet Transforms.

There are two approaches to using adaptive wavelet packets. One is to choose a particular decomposition (filter bank pruning) based on thecharacteristics of the class of signals to be processed, then to use the transform nonadaptively on the individual signals. The other is to adaptthe decomposition for each individual signal. The first is a linear process over the class of signals. The second is not and will not obeysuperposition.

Let P ( J ) denote the number of different J -scale orthonormal wavelet packet transforms. We can easily see that

P ( J ) = P ( J - 1 ) 2 + 1 , P ( 1 ) = 1 .

So the number of possible choices grows dramatically as the scale increases. This is another reason for the wavelet packets to be a verypowerful tool in practice. For example, the FBI standard for fingerprint image compression [link] , [link] is based on wavelet packet transforms. The wavelet packets are successfully used for acoustic signalcompression [link] . In [link] , a rate-distortion measure is used with the wavelet packet transform toimprove image compression performance.

M -band DWTs give a flexible tiling of the time-frequency plane. They are associated with a particular tree-structured filter bank, where the lowpass channelat any depth is split into M bands. Combining the M -band and wavelet packet structure gives a rather arbitrarytree-structured filter bank, where all channels are split into sub-channels (using filter banks with a potentially different number of bands), andwould give a very flexible signal decomposition. The wavelet analog of this is known as the wavelet packet decomposition [link] . For a given signal or class of signals, one can, for a fixed set offilters, obtain the best (in some sense) filter bank tree-topology. For a binary tree an efficient scheme using entropy as the criterion has beendeveloped—the best wavelet packet basis algorithm [link] , [link] .

Biorthogonal wavelet systems

Requiring the wavelet expansion system to be orthogonal across both translations and scale gives a clean, robust, and symmetric formulationwith a Parseval's theorem. It also places strong limitations on the possibilities of the system. Requiring orthogonality uses upa large number of the degrees of freedom, results in complicated design equations, prevents linear phase analysis and synthesis filter banks, andprevents asymmetric analysis and synthesis systems. This section will develop the biorthogonal wavelet system using a nonorthogonal basis anddual basis to allow greater flexibility in achieving other goals at the expense of the energy partitioning property that Parseval's theorem states [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] . Some researchers have considered “almost orthogonal" systems where thereis some relaxation of the orthogonal constraints in order to improve other characteristics [link] . Indeed, many image compression schemes (including the fingerprint compression used by the FBI [link] , [link] ) use biorthogonal systems.

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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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