Wavelet for multidimensional signals analysis: Difference between revisions

Revision as of 17:42, 2 July 2020

Revision as of 22:32, 22 March 2021

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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| (&phi;φ(x) ⨁ &psi;ψ(x)) ⊗ (&phi;φ(y) ⨁ &psi;ψ(y)) {{=}} { &phi;φ(x)&phi;φ(y), &phi;φ(x)&psi;ψ(y), &psi;ψ(x)&phi;φ(y), &psi;ψ(x)&psi;ψ(y) }}}

This leads to the concept of multidimensional separable DWT similar in principle to the multidimensional DFT.

{{math|&phi;φ(x)&phi;φ(y)}} gives the approximation coefficients and other subbands:

{{math|&phi;φ(x)&psi;ψ(y)}} low-high (LH) subband,

{{math|&psi;ψ(x)&phi;φ(y)}} high-low (HL) subband,

{{math|&psi;ψ(x)&psi;ψ(y)}} high-high (HH) subband,

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^M}} 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 &times;× N2 &times;× N3}} , a separable DWT can be implemented as follows:

[[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}} &times;× N2 &times;× N3}}. Applying 1-D DWT in {{math|N2}} dimension, each of these chunks is split into two more chunks of {{math|{{frac|N1|2}} &times;× {{frac|N2|2}} &times;× N3}}. This repeated in 3-D gives a total of 8 chunks of size {{math| {{frac|N1|2}} &times;× {{frac|N2|2}} &times;× {{frac|N3|2}}}}.<ref>{{cite journal|last1=Cheng-Wu|first1=Po|last2=Gee-Chen|first2=Liang|title=An efficient architecture for two-dimensional discrete wavelet transform|journal=IEEE Transactions on Circuits and Systems for Video Technology|date=7 August 2002|volume=11|issue=4|pages=536–545|doi=10.1109/76.915359}}</ref>

[[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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Consider an example for 2-D dual tree real oriented CWT:

Let {{math| &psi;ψ(x)}} and {{math| &psi;ψ(y)}} be complex wavelets:

{{math| &psi;ψ(x) {{=}} &psi;ψ(x)_h + j &psi;ψ(x)_g}} and {{math| &psi;ψ(y) {{=}} &psi;ψ(y)_h + j &psi;ψ(y)_g}}.

{{math| &psi;ψ(x,y) {{=}} [&psi;ψ(x)_h + j &psi;ψ(x)_g][ &psi;ψ(y)_h + j &psi;ψ(y)_g]

{{=}} &psi;ψ(x)_h&psi;ψ(y)_h - &psi;ψ(x)_g&psi;ψ(x)_g + j [&psi;ψ(x)_h&psi;ψ(y)_g - &psi;ψ(x)_h&psi;ψ(x)_g]}}

The support of the Fourier spectrum of the wavelet above resides in the first quadrant. When just the real part is considered, {{math|Real(&psi;ψ(x,y)) {{=}} &psi;ψ(x)_h&psi;ψ(y)_h - &psi;ψ(x)_g&psi;ψ(x)_g}} has support on opposite quadrants (see (a) in figure). Both {{math|&psi;ψ(x)_h&psi;ψ(y)_h}} and {{math|&psi;ψ(x)_g&psi;ψ(y)_g}} correspond to the HH subband of two different separable 2-D DWTs. This wavelet is oriented at {{math|-45^o}}.

Similarly, by considering {{math| &psi;ψ₂(x,y) {{=}} &psi;ψ(x)&psi;ψ(y)^*}}, a wavelet oriented at {{math|45^o}} is obtained. To obtain 4 more oriented real wavelets, {{math|&phi;φ(x)&psi;ψ(y)}}, {{math|&psi;ψ(x)&phi;φ(y)}}, {{math|&phi;φ(x)&psi;ψ(y)^*}} and {{math|&psi;ψ(x)&phi;φ(y)^*}} are considered.

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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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}}</ref> consists of a standard DWT tensor and {{math|2^{m -1}}} 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_x {&psi;ψ(x)_h&psi;ψ(y)_h} {{=}} &psi;ψ(x)_g&psi;ψ(y)_h }}

{{math| H_y {&psi;ψ(x)_h&psi;ψ(y)_h} {{=}} &psi;ψ(x)_h&psi;ψ(y)_g }}

{{math| H_x H_y {&psi;ψ(x)_h&psi;ψ(y)_h} {{=}} &psi;ψ(x)_g&psi;ψ(y)_g }}

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|url=https://www.semanticscholar.org/paper/c7fd84b91df62e895c85d8afbcae76a0f7af0908}}</ref>