Wavelet for multidimensional signals analysis: Difference between revisions

Revision as of 14:49, 23 June 2020

Revision as of 14:50, 23 June 2020

Line 4:

== Multidimensional separable discrete wavelet transform (DWT) ==

The [[Discretediscrete wavelet transform]] is extended to the multidimensional case 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<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;(x) ⨁ &psi;(x) ) ⊗ ( &phi;(y) ⨁ &psi;(y) ) {{=}} { &phi;(x)&phi;(y), &phi;(x)&psi;(y), &psi;(x)&phi;(y), &psi;(x)&psi;(y) }}}

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

Line 75:

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

For the 2-D case, this is named dual tree '''[[quaternion]] Waveletwavelet Transformtransform (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>

The total redundancy in M-D is {{math|2^m}} tight frame.