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[[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 book |doi=10.1109/DCC.1991.213386|isbn=978-0-8186-9202-4|chapter=Data compression using wavelets: Error, smoothness and quantization|title=[1991] Proceedings. Data Compression Conference|pages=186–195|year=1991|last1=Devore|first1=R.A.|last2=Jawerth|first2=B.|last3=Lucier|first3=B.J.|chapter-url=https://www.semanticscholar.org/paper/8b93bc5d02cc102b82c17d3db9c98909275d8132}}</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
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
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In order to overcome these difficulties, a method of wavelet transform called [[Complex wavelet transform]] (CWT) was developed.
== Multidimensional
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.
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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|bibcode=2001ISPL....8..170S}}</ref>
Consider an example for 2-D dual tree real oriented CWT:
Let {{math| ψ(x)}} and {{math| ψ(y)}} be complex wavelets:
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Although the M-D CWT provides one with oriented wavelets, these orientations are only appropriate to represent the orientation along the (m-1)<sup>th</sup> dimension of a signal with {{math|m}} dimensions. When singularities in [[Manifolds|manifold]]<ref>{{cite book|last1=Boothby|first1=W|title=An Introduction to Differentiable Manifolds and Riemannian Geometry|date=2003|publisher=Academic|___location=San Diego}}</ref> of lower dimensions are considered, such as a bee moving in a straight line in the 4-D space-time, oriented wavelets that are smooth in the direction of the manifold and change rapidly in the direction normal to it are needed. A new transform, Hypercomplex Wavelet transform was developed in order to address this issue.
==Hypercomplex
The dual tree '''
{{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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The total redundancy in M-D is {{math|2<sup>m</sup>}} tight frame.
==Directional
The hypercomplex transform described above serves as a building block to construct the '''
For 3-D, the DHWT can be considered in two dimensions, one DHWT for {{math|n {{=}} 1}} and another for {{math|n {{=}} 2}}. For {{math|n {{=}} 2}}, {{math|n {{=}} m-1}}, so, as in the 2-D case, this corresponds to 3-D dual tree CWT. But the case of {{math|n {{=}} 1}} gives rise to a new DHWT transform. The combination of 3-D HWT wavelets is done in a manner to ensure that the resultant wavelet is lowpass along 1-D and bandpass along 2-D.
In,<ref name=DHWT /> this was used to detect line singularities in 3-D space.
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