# 0.7 Generalizations of the basic multiresolution wavelet system  (Page 14/28)

 Page 14 / 28

## Support

In general, the finite length of $H\left(k\right)$ and $G\left(k\right)$ ensure the finite support of $\text{Φ}\left(t\right)$ and $\Psi \left(t\right)$ . However, there are no straightforward relations between the support length and the number of nonzero coefficients in $H\left(k\right)$ and $G\left(k\right)$ . An explanation is the existence of nilpotent matrices [link] . A method to estimate the support is developed in [link] .

## Orthogonality

For these scaling functions and wavelets to be orthogonal to each other and orthogonal to their translations, we need [link]

$\mathbf{H}\left(\text{ω}\right){\mathbf{H}}^{†}\left(\text{ω}\right)+\mathbf{H}\left(\text{ω}+\pi \right){\mathbf{H}}^{†}\left(\text{ω}+\pi \right)={I}_{R},$
$\mathbf{G}\left(\text{ω}\right){\mathbf{G}}^{†}\left(\text{ω}\right)+\mathbf{G}\left(\text{ω}+\pi \right){\mathbf{G}}^{†}\left(\text{ω}+\pi \right)={I}_{R},$
$\mathbf{H}\left(\text{ω}\right){\mathbf{G}}^{†}\left(\text{ω}\right)+\mathbf{H}\left(\text{ω}+\pi \right){\mathbf{G}}^{†}\left(\text{ω}+\pi \right)={0}_{R},$

where $†$ denotes the complex conjugate transpose, ${I}_{R}$ and ${0}_{R}$ are the $R×R$ identity and zero matrix respectively. These are the matrix versions of [link] and [link] . In the scalar case, [link] can be easily satisfied if we choose the wavelet filter by time-reversing the scaling filter and changing the signsof every other coefficients. However, for the matrix case here, since matrices do notcommute in general, we cannot derive the $G\left(k\right)$ 's from $H\left(k\right)$ 's so straightforwardly. This presents some difficulty in finding the wavelets from the scaling functions; however, this also gives usflexibility to design different wavelets even if the scaling functions are fixed [link] .

The conditions in [link][link] are necessary but notsufficient. Generalization of Lawton's sufficient condition (Theorem  Theorem 14 in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients ) has been developed in [link] , [link] , [link] .

## Implementation of multiwavelet transform

Let the expansion coefficients of multiscaling functions and multiwavelets be

${c}_{i,j}\left(k\right)=〈f,\left(t\right),,,{\phi }_{i,j,k},\left(t\right)〉,$
${d}_{i,j}\left(k\right)=〈f,\left(t\right),,,{\psi }_{i,j,k},\left(t\right)〉.$

We create vectors by

${C}_{j}\left(k\right)={\left[{c}_{1,j}\left(k\right),...,{c}_{R,j}\left(k\right)\right]}^{T},$
${D}_{j}\left(k\right)={\left[{d}_{1,j}\left(k\right),...,{d}_{R,j}\left(k\right)\right]}^{T}.$

For $f\left(t\right)$ in ${V}_{0}$ , it can be written as linear combinations of scaling functions and wavelets,

$f\left(t\right)=\sum _{k}{C}_{{j}_{0}}{\left(k\right)}^{T}{\text{Φ}}_{{J}_{0},k}\left(t\right)+\sum _{j={j}_{0}}^{\infty }\sum _{k}{D}_{j}{\left(k\right)}^{T}{\Psi }_{j,k}\left(t\right).$

${C}_{j-1}\left(k\right)=\sqrt{2}\sum _{n}H\left(n\right){C}_{j}\left(2k+n\right)$

and

${D}_{j-1}\left(k\right)=\sqrt{2}\sum _{n}G\left(n\right){C}_{j}\left(2k+n\right).$

Moreover,

${C}_{j}\left(k\right)=\sqrt{2}\sum _{k}\left(H,{\left(k\right)}^{†},{C}_{j-1},\left(2k+n\right),+,G,{\left(k\right)}^{†},{D}_{j-1},\left(2k+n\right)\right).$

These are the vector forms of [link] , [link] , and [link] . Thus the synthesis and analysis filter banks for multiwavelet transforms have similar structures as the scalar case. Thedifference is that the filter banks operate on blocks of $R$ inputs and the filtering and rate-changing are all done in terms of blocks of inputs.

To start the multiwavelet transform, we need to get the scaling coefficients at high resolution. Recall that in the scalar case, thescaling functions are close to delta functions at very high resolution, so the samples of the function are used as the scaling coefficients. However, for multiwavelets we need the expansion coefficients for $R$ scaling functions. Simply using nearby samples as the scaling coefficients is abad choice. Data samples need to be preprocessed ( prefiltered ) to produce reasonable values of the expansion coefficients for multi-scalingfunction at the highest scale. Prefilters have been designed based on interpolation [link] , approximation [link] , and orthogonal projection [link] .

## Examples

Because of the larger degree of freedom, many methods for constructing multiwavelets have been developed.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!