# 0.7 Generalizations of the basic multiresolution wavelet system  (Page 20/28)

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Recent results indicate this nondecimated DWT, together with thresholding, may be the best denoising strategy [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] . The nondecimated DWT is shift invariant, is less affected by noise,quantization, and error, and has order $Nlog\left(N\right)$ storage and arithmetic complexity. It combines with thresholding to give denoising andcompression superior to the classical Donoho method for many examples. Further discussion of use of the RDWT can be found in Section: Nonlinear Filtering or Denoising with the DWT .

## Adaptive construction of frames and bases

In the case of the redundant discrete wavelet transform just described, an overcomplete expansion system was constructed in such a way as to be atight frame. This allowed a single linear shift-invariant system to describe a very wide set of signals, however, the description was adaptedto the characteristics of the signal. Recent research has been quite successful in constructing expansion systems adaptively soas to give high sparsity and superresolution but at a cost of added computation and being nonlinear. This section will look at some of therecent results in this area [link] , [link] , [link] , [link] .

While use of an adaptive paradigm results in a shift-invariant orthogonal transform, it is nonlinear. It has the property of $DWT\left\{a\phantom{\rule{0.166667em}{0ex}}f\left(x\right)\right\}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}a\phantom{\rule{0.166667em}{0ex}}DWT\left\{f\left(x\right)\right\}$ , but it does not satisfy superposition, i.e. $DWT\left\{\phantom{\rule{0.166667em}{0ex}}f\left(x\right)+g\left(x\right)\right\}\phantom{\rule{0.166667em}{0ex}}\ne \phantom{\rule{0.166667em}{0ex}}DWT\left\{f\left(x\right)\right\}+DWT\left\{g\left(x\right)\right\}$ . That can sometimes be a problem.

Since these finite dimensional overcomplete systems are a frame, a subset of the expansion vectors can be chosen to be a basis while keepingmost of the desirable properties of the frame. This is described well by Chen and Donoho in [link] , [link] . Several of these methods are outlined as follows:

• The method of frames (MOF) was first described by Daubechies [link] , [link] , [link] and uses the rather straightforward idea of solving the overcomplete frame (underdetermined set of equations) in [link] by minimizing the ${L}^{2}$ norm of $\alpha$ . Indeed, this is one of the classical definitions of solving the normal equations or use of apseudo-inverse. That can easily be done in Matlab by a = pinv(X)*y . This gives a frame solution, but it is usually not sparse.
• The best orthogonal basis method (BOB) was proposed by Coifman and Wickerhauser [link] , [link] to adaptively choose a best basis from a large collection. The method is fast (order $NlogN$ ) but not necessarily sparse.
• Mallat and Zhang [link] proposed a sequential selection scheme called matching pursuit (MP) which builds a basis, vector by vector. Theefficiency of the algorithm depends on the order in which vectors are added. If poor choices are made early, it takes many terms to correct them. Typicallythis method also does not give sparse representations.
• A method called basis pursuit (BP) was proposed by Chen and Donoho [link] , [link] which solves [link] while minimizing the ${L}^{1}$ norm of $\alpha$ . This is done by linear programming and results in a globally optimal solution. It is similar in philosophy to the MOFs butuses an ${L}^{1}$ norm rather than an ${L}^{2}$ norm and uses linear programming to obtain the optimization. Using interior point methods, it is reasonablyefficient and usually gives a fairly sparse solution.
• Krim et al. describe a best basis method in [link] . Tewfik et al. propose a method called optimal subset selection in [link] and others are [link] , [link] .

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
hii
Uday
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
I'm interested in nanotube
Uday
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
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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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