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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

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
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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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