# 0.7 Generalizations of the basic multiresolution wavelet system  (Page 15/28)

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## Geronimo-hardin-massopust multiwavelets

A set of multiscaling filters based on fractal interpolation functions were developed in [link] , and the corresponding multiwavelets were constructed in [link] . As shown in [link] , they

are both symmetrical and orthogonal—a combination which is impossible for two-band orthogonal scalarwavelets. They also have short support, and can exactly reproduce the hat function. These interesting properties make multiwavelet a promisingexpansion system.

## Spline multiwavelets

Spline bases have a maximal approximation order with respect to their length, however spline uniwavelets are only semiorthogonal [link] . A family of spline multiwavelets that are symmetric and orthogonal is developed in [link] .

## Other constructions

Other types of multiwavelets are constructed using Hermite interpolating conditions [link] , matrix spectral factorization [link] , finite elements [link] , and oblique projections [link] . Similar to multiwavelets, vector-valued wavelets and vector filter banks are also developed [link] .

## Applications

Multiwavelets have been used in data compression [link] , [link] , [link] , noise reduction [link] , [link] , and solution of integral equations [link] . Because multiwavelets are able to offer a combination of orthogonality,symmetry, higher order of approximation and short support, methods using multiwavelets frequently outperform those using the comparable scalewavelets. However, it is found that prefiltering is very important, and should be chosen carefully for the applications [link] , [link] , [link] . Also, since discrete multiwavelettransforms operate on size- $R$ blocks of data and generate blocks of wavelet coefficients, the correlation within each block of coefficients needs to beexploited. For image compression, predictions rules are proposed to exploit the correlation in order to reduce the bit rate [link] . For noise reduction, joint thresholding coefficients within each block improve the performance [link] .

## Overcomplete representations, frames, redundant transforms, and adaptive bases

In this chapter, we apply the ideas of frames and tight frames introduced in Chapter: Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Right Frames, and unconditional Bases as well as bases to obtain a more efficient representation of many interesting signal classes. It might be helpful to review the material onbases and frames in that chapter while reading this section.

Traditional basis systems such as Fourier, Gabor, wavelet, and wave packets are efficient representations for certain classes of signals, butthere are many cases where a single system is not effective. For example, the Fourier basis is an efficient system for sinusoidal or smooth periodicsignals, but poor for transient or chirp-like signals. Each system seems to be best for a rather well-defined but narrow class of signals.Recent research indicates that significant improvements in efficiency can be achieved by combining several basis systems. One can intuitivelyimagine removing Fourier components until the expansion coefficients quit dropping off rapidly, then switching to a different basis system to expandthe residual and, after that expansion quits dropping off rapidly, switching to still another. Clearly, this is not a unique expansionbecause the order of expansion system used would give different results. This is because the total expansion system is a linear combination of theindividual basis systems and is, therefore, not a basis itself but a frame. It is an overcomplete expansion system and a variety of criteriahave been developed to use the freedom of the nonuniqueness of the expansion to advantage. The collection of basis systems from which asubset of expansion vectors is chosen is sometimes called a dictionary.

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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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