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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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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| (
This leads to the concept of multidimensional separable DWT similar in principle to the multidimensional DFT.
{{math|
{{math|
{{math|
{{math|
give detail coefficients.
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Wavelet coefficients can be computed by passing the signal to be decomposed though a series of filters. In the case of 1-D, there are two filters at every level-one low pass for approximation and one high pass for the details. In the multidimensional case, the number of filters at each level depends on the number of tensor product vector spaces. For M-D, {{math|2<sup>M</sup>}} filters are necessary at every level. Each of these is called a subband. The subband with all low pass (LLL...) gives the approximation coefficients and all the rest give the detail coefficients at that level.
For example, for {{math|M{{=}}3}} and a signal of size {{math| N1
[[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}}
[[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|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:
Let {{math|
{{math|
{{math|
{{=}}
The support of the Fourier spectrum of the wavelet above resides in the first quadrant. When just the real part is considered, {{math|Real(
Similarly, by considering {{math|
The implementation of complex oriented dual tree structure is done as follows: Two separable 2-D DWTs are implemented in parallel using the filterbank structure as in the previous section. Then, the appropriate sum and difference of different subbands (LL, LH, HL, HH) give oriented wavelets, a total of 6 in all.
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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> {
{{math| H<sub>y</sub> {
{{math| H<sub>x</sub> H<sub>y</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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