Originally known as optimal subband tree structuring (SB-TS), also called wavelet packet decomposition (WPD)
(sometimes known as just wavelet packets or subband tree), is a
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
transform where the discrete-time (sampled) signal is passed through more filters than the
discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal ...
(DWT).
Introduction
In the DWT, each level is calculated by passing only the previous wavelet approximation coefficients (''cA
j'') through discrete-time low- and high-pass
quadrature mirror filter In digital signal processing, a quadrature mirror filter is a filter whose magnitude response is the mirror image around \pi/2 of that of another filter. Together these filters, first introduced by Croisier et al., are known as the quadrature mirror ...
s.
[ However, in the WPD, both the detail (''cDj'' (in the 1-D case), ''cHj'', ''cVj'', ''cDj'' (in the 2-D case)) and approximation coefficients are decomposed to create the full binary tree.][Daubechies, I. (1992), Ten lectures on wavelets, SIAM.][
For ''n'' levels of decomposition the WPD produces 2''n'' different sets of coefficients (or nodes) as opposed to sets for the DWT. However, due to the ]downsampling In digital signal processing, downsampling, compression, and decimation are terms associated with the process of ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be synonymous with ''comp ...
process the overall number of coefficients is still the same and there is no redundancy.
From the point of view of compression, the standard wavelet transform may not produce the best result, since it is limited to wavelet bases that increase by a power of two towards the low frequencies. It could be that another combination of bases produce a more desirable representation for a particular signal. [A. N. Akansu and R. A. Haddad]
Multiresolution Signal Decomposition: Transforms, Subbands, and Wavelets
Boston, MA: Academic Press, , 1992. There are several algorithms for subband tree structuring that find a set of optimal bases that provide the most desirable representation of the data relative to a particular cost function (entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, energy compaction, etc.).[Coifman R. R. & Wickerhauser M. V., 1992]
Entropy-Based Algorithms for Best Basis Selection
IEEE Transactions on Information Theory, 38(2).
[A. N. Akansu and Y. Liu]
On Signal Decomposition Techniques
(Invited Paper), Optical Engineering Journal, special issue Visual Communications and Image Processing, vol. 30, pp. 912–920, July 1991.
There were relevant studies in signal processing and communications fields to address the selection of subband trees (orthogonal basis) of various kinds, e.g. regular, dyadic, irregular, with respect to performance metrics of interest including energy compaction (entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
), subband correlations and others.
[H. Caglar, Y. Liu and A. N. Akansu]
Statistically Optimized PR-QMF Design
Proc. SPIE Visual Communications and Image Processing, vol. 1605, pp. 86–94, 1991.
[A. Benyassine and A. N. Akansu]
Performance Analysis and Optimal Structuring of Subchannels for Discrete Multitone Transceivers
Proc. IEEE International Symposium on Circuits and Systems (ISCAS), pp. 1456–1459, April 1995.
[M. V. Tazebay and A. N. Akansu]
Adaptive Subband Transforms in Time-frequency Excisers for DSSS Communications Systems
IEEE Trans. Signal Process., vol. 43, pp. 2776–2782, Nov. 1995.
Discrete wavelet transform theory (continuous in the time variable) offers an approximation to transform discrete (sampled) signals. In contrast, the discrete-time subband transform theory enables a perfect representation of already sampled signals.[
]
Gallery
Image:Daubechies12-packet-functions.png, Daubechies D12 packet functions
Image:Daubechies12-packet-spectrum.png, Their Fourier spectra
Applications
Wavelet packets were successfully applied in preclinical diagnosis.
References
External links
* An implementation of wavelet packet decomposition can be found in MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
br>wavelet toolbox
* An implementation for R can be found in th
wavethresh package
* An illustration and implementation of wavelet packets along with its code in C++ can be found at: {{cite web , url=http://www.bearcave.com/misl/misl_tech/wavelets/packet/index.html , title=The Wavelet Packet Transform , author=Ian Kaplan , website=Bearcave , date=March 2002
JWave
An implementation in Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
for 1-D and 2-D wavelet packets using Haar, Daubechies, Coiflet
Coiflets are discrete wavelets designed by Ingrid Daubechies, at the request of Ronald Coifman, to have scaling functions with vanishing moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
...
, and Legendre wavelets.
Wavelets
Signal processing