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Tian and Wells [link] , [link] have constructed biorthogonal wavelet systems with both zero scaling function and wavelet moments. Closed form solutionsfor these biorthogonal coiflets have been found. They have approximation properties similar to the coiflets, and the filter coefficients are dyadicrationals as are the splines. The filter coefficients for these biorthogonal Coiflets are listed in [link] . Some members of this family are also in the spline family described earlier.

Lifting construction of biorthogonal systems

We have introduced several families of biorthogonal systems and their design methods. There is another method called a lifting scheme , which is very simple and general. It has a long history

Plots of Scaling Function and Wavelet and their Duals for one of the Cohen-Daubechies-Feauveau Family of Biorthogonal Wavelets that is Used in the FBI Fingerprint Compression Standard
Plots of Scaling Function and Wavelet and their Duals for one of the Cohen-Daubechies-Feauveau Family of Biorthogonal Wavelets that is Used in the FBI Fingerprint Compression Standard

[link] , [link] , [link] , [link] , [link] , [link] , and has been systematically developed recently [link] , [link] . The key idea is tobuild complicated biorthogonal systems using simple and invertible stages. The first stage does nothing but to separate even and oddsamples, and it is easily invertible. The structure is shown in [link] , and is called the lazy wavelet transform in [link] .

Coefficients for some Members of the Biorthogonal Coiflets. For Longer Filters, We only List Halfof the Coefficients.
2 h 2 h ˜
1 , 1 1 , 1
1 / 2 , 1 , 1 / 2 - 1 / 4 , 1 / 2 , 3 / 2 , 1 / 2 , - 1 / 4
3 / 8 , 1 , 3 / 4 , 0 , - 1 / 8 3 / 64 , 0 , - 3 / 16 , 3 / 8 , 41 / 32 , 3 / 4 , - 3 / 16 , - 1 / 8 , 3 / 64
- 1 / 16 , 0 , 9 / 16 , 1 , 9 / 16 , 0 , - 1 / 16 - 1 / 256 , 0 , 9 / 128 , - 1 / 16 , - 63 / 256 , 9 / 16 , 87 / 64 ,
The Lazy Wavelet Transform
The Lazy Wavelet Transform

After splitting the data into two parts, we can predict one part from the other, and keep only the prediction error, as in [link] . We can reconstruct the data by recomputing the prediction and then add back theprediction. In [link] , s and t are prediction filters.

By concatenating simple stages, we can implement the forward and inverse wavelet transforms as in [link] . It is also called the ladder structure , and the reason for the name is clear from the figure. Clearly, the system is invertible, and thusbiorthogonal. Moreover, it has been shown the orthogonal wavelet systems can also be implemented using lifting [link] . The advantages of lifting are numerous:

  • Lifting steps can be calculated inplace. As seen in [link] , the prediction outputs based on one channel of the data can be added to or subtracted from the data in other channels,and the results can be saved in the same place in the second channel. No auxiliary memory is needed.
  • The predictors s and t do not have to be linear. Nonlinear operations like the medium filter or rounding can be used, and the systemremains invertible. This allows a very simple generalization to nonlinear wavelet transform or nonlinear multiresolution analysis.
  • The design of biorthogonal systems boils down to the design of the predictors. This may lead to simple approaches that do not relay on theFourier transform [link] , and can be generalized to irregular samples or manifolds.
  • For biorthogonal systems, the lifting implementations require less numerical operations than direct implementations [link] . For orthogonal cases,the lifting schemes have the computational complexity similar to the lattice factorizations, which is almost half of the directimplementation.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
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Kyle
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Adin
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Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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.
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Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
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What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
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in general
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Graphene has a hexagonal structure
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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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