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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 such as using wavelets.<ref>{{cite web|last1=
▲[[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 such as using wavelets.<ref>{{cite web|last1=DeVORE|first1=RONALD|last2=JAWERTH|first2=BJORN|last3=LUCIER|first3=BRADLEY|title=DATA COMPRESSION USING WAVELETS: ERROR, SMOOTHNESS, AND QUANTIZATION|url=https://www.math.purdue.edu/~lucier/692/data-compression.pdf}}</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) ===
[[Discrete wavelet transform]] is extended to the multidimensional case is 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 as [http://www.uio.no/studier/emner/matnat/math/MAT-INF2360/v12/tensorwavelet.pdf<ref>{{cite web|title=Tensor products in a wavelet setting|url=http://www.uio.no/studier/emner/matnat/math/MAT-INF2360/v12/tensorwavelet.pdf|website=University of Oslo}}</ref>]
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give detail coefficients
Wavelet coefficients can be computed by passing the signal to be decomposed though a series of filters.
For example, for {{math|M{{=}}3}} and a signal of size {{math| N1 × N2 × N3}} , a separable DWT can be implemented as follows:<ref name=WavPoly>{{cite web|last1=Cai|first1=Shihua|last2=Li|first2=Keyong|title=Matlab implementation of wavelet transforms|url=http://eeweb.poly.edu/iselesni/WaveletSoftware/index.html}}</ref>
[[File:Wiki figures1.pdf|thumbnail|center|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]]
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[[File:Filterbank.pdf|thumbnail|The figure shows the 3-D analysis filterbank for 3-D separable DWT]]
The wavelets generated by the separable DWT procedure are highly shift variant. A small shift in the input signal changes the wavelet coefficients to a large extent. Also, these wavelets are almost equal in their magnitude in all directions and thus do not reflect the orientation or directivity that could be present in the multidimensional signal. For example, there could be an edge discontinuity in an image or an object moving smoothly along a straight line in the space-time 4D dimension. A separable DWT does not fully capture the same.
In order to overcome these difficulties, a method of wavelet transform called [[Complex wavelet transform]] (CWT) was developed.
Similar to 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|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 directional ambiguity of the signal.
Dual tree CWT in 1-D uses 2 real DWTs, where the first one gives the real part of CWT and the second DWT gives the imaginary part of the CWT. M-D dual tree CWT is analyzed in terms of tensor products. However, it is possible to implement M-D CWTs efficiently using separable M-D DWTs and considering sum and difference of subbands obtained. Additionally, these wavelets tend to be oriented in specific directions.
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Similarly, in 3-D, 4 separable 3-D DWTs in parallel are needed and a total of 28 oriented wavelets are obtained.
Although M-D CWT provides one with oriented wavelets, these orientations are only appropriate to represent the orientation along {{math|m-1}} 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, we need oriented wavelets that are smooth in the direction of the manifold and change rapidly in the direction normal to it. A new transform, Hypercomplex Wavelet transform was developed in order to address this issue.
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=ICASSP|date=2004|volume=3|pages=996–999|url=http://citeseerx.ist.psu.edu/viewdoc/download?}}</ref> consists of standard DWT tensor and {{math|2<sup>m -1</sup>}} wavelets obtained from combining 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 as described and three additional components:
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The total redundancy in M-D is {{math|2<sup>m</sup>}} tight frame.
The hypercomplex transform described above serves as a building block to construct the '''Directional Hypercomplex Wavelet Transform (DHWT)'''. A linear combination of the wavelets obtained using the hypercomplex transform give a wavelet oriented in a particular direction. For the 2-D DHWT, it is seen that these linear combinations correspond to the exact 2-D dual tree CWT case.
For 3-D, 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.
The wavelet transforms for multidimensional signals are often computationally challenging which is the case with most multidimensional signals. Also, the methods of CWT and DHWT are redundant even though they offer directivity and shift invariance.
== References ==
{{Reflist}}
[[Category:Articles created via the Article Wizard]]
[[Category:Multidimensional signal processing]]
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