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Wavelet for multidimensional signals analysis: Difference between revisions

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== 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=1552–1556|doi=10.1109/72.471353|pmid=18263450|hdl=1903/5619|hdl-access=free}}</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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== Multidimensional Complex Wavelet Transform==
Similar to the 1-D complex wavelet transform,<ref name=kingsbury>{{cite journal|last1=Kingsbury|first1=Nick|title=Complex Wavelets for Shift Invariant Analysis and Filtering of Signals|journal= Applied and Computational Harmonic Analysis|date=2001|volume=10|issue=3|pages=234–253|doi=10.1006/acha.2000.0343|url=http://www.idealibrary.com}}</ref> tensor products of complex wavelets are considered to produce complex wavelets for multidimensional signal analysis. With further analysis it is seen that these complex wavelets are oriented.<ref name=IEEEmag>{{cite journal|last1=Selesnick|first1=Ivan|last2=Baraniuk|first2=Richard|last3=Kingsbury|first3=Nick|title=The Dual-Tree Complex Wavelet Transform|journal=IEEE Signal Processing Magazine|volume=22|issue=6|date=2005|pages=123–151|doi=10.1109/MSP.2005.1550194|bibcode=2005ISPM...22..123S|hdl=1911/20355|hdl-access=free}}</ref> This sort of orientation helps to resolve the directional ambiguity of the signal.
 
===Implementation of multidimensional (M-D) dual tree CWT ===
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