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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.|
▲[[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 discrete wavelet transform (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|doi-access=free}}</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|s2cid=833630 |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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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|s2cid=5994808 }}</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 book |doi = 10.1109/ICASSP.2004.1326715|chapter = Directional hypercomplex wavelets for multidimensional signal analysis and processing|title = 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing|volume = 3|pages = iii–996–9|year = 2004|last1 = Wai Lam Chan|last2 = Hyeokho Choi|last3 = Baraniuk|first3 = R.G.|isbn = 0-7803-8484-9|hdl = 1911/19796| s2cid=8287497 }}</ref> consists of a standard DWT tensor and {{math|2<sup>m -1</sup>}} wavelets obtained from combining the 1-D Hilbert transform of these wavelets along the n-coordinates. In particular a 2-D HWT consists of the standard 2-D separable DWT tensor and three additional components:
{{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|pmid=18586616|bibcode=2008ITIP...17.1069C|
The total redundancy in M-D is {{math|2<sup>m</sup>}} tight frame.
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