# 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.

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!