# 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.

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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