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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
== 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|
{{math| ( φ(x) ⨁ ψ(x) ) ⊗ ( φ(y) ⨁ ψ(y) ) {{=}} { φ(x)φ(y), φ(x)ψ(y), ψ(x)φ(y), ψ(x)ψ(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}} × N2 × N3}}. Applying 1-D DWT in {{math|N2}} dimension, each of these chunks is split into two more chunks of {{math|{{frac|N1|2}} × {{frac|N2|2}} × N3}}. This repeated in 3-D gives a total of 8 chunks of size {{math| {{frac|N1|2}} × {{frac|N2|2}} × {{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
[[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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== 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|date=2005|pages=123–151|url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1550194&tag=1}}</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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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|pages=170–173|doi=10.1109/97.923042|citeseerx=10.1.1.139.5369}}</ref>
Consider an example for 2-D dual tree real oriented CWT:
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==Hypercomplex Wavelet Transform==
The dual tree '''Hypercomplex Wavelet Transform (HWT)''' developed in <ref name=DHWT>{{cite journal|last1=Lam Chan|first1=Wai|last2=Choi|first2=Hyeokho|last3=Baraniuk|first3=Richard|title=DIRECTIONAL HYPERCOMPLEX WAVELETS FOR MULTIDIMENSIONAL SIGNAL ANALYSIS AND PROCESSING|journal=IEEE
{{math| H<sub>x</sub> {ψ(x)<sub>h</sub>ψ(y)<sub>h</sub>} {{=}} ψ(x)<sub>g</sub>ψ(y)<sub>h</sub> }}
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{{math| H<sub>x</sub> H<sub>y</sub> {ψ(x)<sub>h</sub>ψ(y)<sub>h</sub>} {{=}} ψ(x)<sub>g</sub>ψ(y)<sub>g</sub> }}
For the 2-D case, this is named dual tree '''[[quaternion]] Wavelet Transform (QWT)'''.<ref>{{cite journal|last1=Lam Chan|first1=Wai|last2=Choi|first2=Hyeokho|last3=Baraniuk|first3=Richard|title=Coherent Multiscale Image Processing Using Dual-Tree Quaternion Wavelets|journal=IEEE Transactions on Image Processing|volume=17|issue=7|pages=1069–1082|date=2008|doi=10.1109/TIP.2008.924282|
The total redundancy in M-D is {{math|2<sup>m</sup>}} tight frame.
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