# 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.

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
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