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'''Wavelets for multidimensional signal analysis''' ▼
== Wavelets for multidimensional signal analysis ==▼
▲== Wavelets for multidimensional signal analysis ==
[[Wavelet
=== Multidimensional separable Discrete Wavelet Transform (DWT) ===
[[Discrete wavelet transform]] is extended to the multidimensional case is using the [[
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>]
{{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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====Implementation of multidimensional separable DWT ====
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>
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}}}}. The first chunk is passed via a low pass filter in each of these dimensions and the second one via high-pass.
====Disadvantages of M-D separable DWT====
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=== Multidimensional Complex Wavelet Transform===
Similar to 1-D complex wavelet transform
====Implementation of multidimensional (M-D) dual tree CWT ====
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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Consider an example for 2-D dual tree real oriented CWT:
Let {{math| ψ(x)}} and {{math| ψ(y)}} be complex wavelets:
{{math| ψ(x) {{=}} ψ(x)<sub>h</sub> + j ψ(x)<sub>g</sub>}} and {{math| ψ(y) {{=}} ψ(y)<sub>h</sub> + j ψ(y)<sub>g</sub>}}.
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The support of the Fourier spectrum of the wavelet above resides in first quadrant as in the diagram. When just the real part is considered, {{math|Real(ψ(x,y)) {{=}} ψ(x)<sub>h</sub>ψ(y)<sub>h</sub> - ψ(x)<sub>g</sub>ψ(x)<sub>g</sub>}} has support on opposite quadrants. Both {{math|ψ(x)<sub>h</sub>ψ(y)<sub>h</sub>}} and {{math|ψ(x)<sub>g</sub>ψ(y)<sub>g</sub>}} correspond to HH subband of two different separable 2-D DWTs. This wavelet is oriented at {{math|-45<sup>o</sup>}}.
Similarly, by considering {{math| ψ<sub>2</sub>(x,y) {{=}} ψ(x)ψ(y)<sup>*</sup>}}, wavelet oriented at {{math|45<sup>o</sup>}} is obtained. To obtain 4 more oriented real wavelets, {{math|φ(x)ψ(y)}}, {{math|ψ(x)φ(y)}}, {{math|φ(x)ψ(y)<sup>*</sup>}} and {{math|ψ(x)φ(y)<sup>*</sup>}} are considered.
For implementation of this 2 separable 2-D DWTs in parallel are needed. Then, the appropriate sum and difference of different subbands 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 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=
{{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 2-D case, this is named dual tree '''[[
The total redundancy in M-D is {{math|2<sup>m</sup>}} tight frame.
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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
===Challenges ahead===
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* [http://www.example.com www.example.com]
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