Wavelet for multidimensional signals analysis: Difference between revisions

Revision as of 05:54, 11 December 2015

Revision as of 17:26, 18 May 2016

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{{Orphan|date=November 2015}}

[[Wavelet]]s are often used to analyse piece-wise smooth signals.<ref>{{cite book|last1=Mallat|first1=Stéphane|title=A Wavelet Tour of Signal Processing|date=2008|publisher=Academic Press}}</ref> Wavelet coefficients can efficiently represent a signal which has led to data compression algorithms using wavelets. <ref>{{cite journal|last1=Devore|first1=Ronald|last2=Jawerth|first2=Bjorn|last3=Lucier|first3=Bradley|title=Data compression using wavelets: error, smoothness and quantization|journal=Data Compression Conference,IEEE|date=8 April 1991|pagepages=186 - 195186–195|doi=10.1109/DCC.1991.213386}}</ref> Wavelet analysis is extended for [[multidimensional signal processing]] as well. This article introduces a few methods for wavelet synthesis and analysis for multidimensional signals. There also occur challenges such as directivity in multidimensional case.

== Multidimensional separable Discrete Wavelet Transform (DWT) ==

The [[Discrete wavelet transform]] is extended to the multidimensional case using the [[tensor product]] of well known 1-D wavelets.

In 2-D for example, the tensor product space for 2-D is decomposed into four tensor product vector spaces<ref name=Tensor_products>{{cite journal|last1=Kugarajah|first1=Tharmarajah|last2=Zhang|first2=Qinghua|title=Multidimensional wavelet frames|journal=IEEE Transactions on Neural Networks|date=November 1995|volume=6|issue=6|pages=1552 - 15561552–1556|doi=10.1109/72.471353|url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=471353&tag=1}}</ref> as

{{math| ( &phi;(x) ⨁ &psi;(x) ) ⊗ ( &phi;(y) ⨁ &psi;(y) ) {{=}} { &phi;(x)&phi;(y), &phi;(x)&psi;(y), &psi;(x)&phi;(y), &psi;(x)&psi;(y) }}}

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[[Image:Wiki figures mod.001.png|framed|none|The figure depicts 3-D separable DWT procedure by applying 1-D DWT for each dimension and splitting the data into chunks to obtain wavelets for different subbands]]

Applying the 1-D DWT analysis filterbank in dimension {{math|N1}}, it is now split into two chunks of size {{math| {{frac|N1|2}} &times; N2 &times; N3}}. Applying 1-D DWT in {{math|N2}} dimension, each of these chunks is split into two more chunks of {{math|{{frac|N1|2}} &times; {{frac|N2|2}} &times; N3}}. This repeated in 3-D gives a total of 8 chunks of size {{math| {{frac|N1|2}} &times; {{frac|N2|2}} &times; {{frac|N3|2}}}}. <ref>{{cite journal|last1=Cheng-Wu|first1=Po|last2=Gee-Chen|first2=Liang|title=An efficient architecture for two-dimensional discrete wavelet transform|journal=IEEE Transactions on Circuits and Systems for Video technology|date=7 August 2002|volume=11|issue=4|pages=536-545536–545|doi=10.1109/76.915359|url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=915359&tag=1}}</ref>

[[Image:Filterbank mod try 2.001.png|framed|none|The figure shows the 3-D analysis filterbank for 3-D separable DWT]]

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If both real and imaginary parts of the tensor products of complex wavelets are considered, complex oriented dual tree CWT which is 2 times more expansive than real oriented dual tree CWT is obtained. So there are two wavelets oriented in each of the directions.

Although implementing complex oriented dual tree structure takes more resources, it is used in order to ensure an approximate shift invariance property that a complex analytical wavelet can provide in 1-D. In the 1-D case, it is required that the real part of the wavelet and the imaginary part are [[Hilbert transform]] pairs for the wavelet to be analytical and to exhibit shift invariance. Similarly in the M-D case, the real and imaginary parts of tensor products are made to be approximate Hilbert transform pairs in order to be analytic and shift invariant.<ref name=IEEEmag /><ref>{{cite journal|last1=Selesnick|first1=I.W.|title=Hilbert transform pairs of wavelet bases|journal=IEEE Signal Processing Letters|date=June 2001|volume=8|issue=6|pagepages=170 - 173|page=170-173170–173|doi=10.1109/97.923042}}</ref>

Consider an example for 2-D dual tree real oriented CWT:

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{{Reflist}}

== External Linkslinks ==

*[http://www.uio.no/studier/emner/matnat/math/MAT-INF2360/v12/tensorwavelet.pdf Tensor products in wavelet settings]

*[http://eeweb.poly.edu/iselesni/WaveletSoftware/index.html Matlab implementation of wavelet transforms]

[[Category:Articles created via the Article Wizard]]

[[Category:Multidimensional signal processing]]