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Lattices and lifting

An alternative to using the basic two-band tree-structured filter bank is a lattice-structured filter bank. Because of the relationship betweenthe scaling filter h ( n ) and the wavelet filter h 1 ( n ) given in [link] , some of the calculation can be done together with a significant savings in arithmetic. This is developed in Chapter: Calculation of the Discrete Wavelet Transform [link] .

Still another approach to the calculation of discrete wavelet transforms and to the calculations of the scaling functions and wavelets themselvesis called “lifting." [link] , [link] Although it is related to several other schemes [link] , [link] , [link] , [link] , this idea was first explained by Wim Sweldens as a time-domainconstruction based on interpolation [link] . Lifting does not use Fourier methods and can be applied to more general problems(e.g., nonuniform sampling) than the approach in this chapter. It was first applied to the biorthogonal system [link] and then extended to orthogonal systems [link] . The application of lifting to biorthogonal is introduced in Section: Biorthogonal Wavelet Systems later in this book. Implementations based on lifting also achievethe same improvement in arithmetic efficiency as the lattice structure do.

Different points of view

Multiresolution versus time-frequency analysis

The development of wavelet decomposition and the DWT has thus far been in terms of multiresolution where the higher scale wavelet components areconsidered the “detail" on a lower scale signal or image. This is indeed a powerful point of view and an accurate model for many signals and images,but there are other cases where the components of a composite signal at different scales and/or time are independent or, at least, not details ofeach other. If you think of a musical score as a wavelet decomposition, the higher frequency notes are not details on a lower frequency note; theyare independent notes. This second point of view is more one of the time-frequency or time-scale analysis methods [link] , [link] , [link] , [link] , [link] and may be better developed with wavelet packets (see Section: Wavelet Packets ), M-band wavelets (see Section: Multiplicity-M (M-band) Scaling Functions and Wavelets ), or a redundant representation (see Section: Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases ), but would still be implemented by some sort of filter bank.

Periodic versus nonperiodic discrete wavelet transforms

Unlike the Fourier series, the DWT can be formulated as a periodic or a nonperiodic transform. Up until now, we have considered a nonperiodicseries expansion [link] over - < t < with the calculations made by the filter banks being an on-going string ofcoefficients at each of the scales. If the input to the filter bank has a certain rate, the output at the next lower scale will be twosequences, one of scaling function coefficients c j - 1 , k - 1 and one of wavelet coefficients d j - 1 , k - 1 , each, after down-sampling, being at half the rate of the input. At the next lower scale, the sameprocess is done on the scaling coefficients to give a total output of three strings, one at half rate and two at quarter rate. In otherwords, the calculation of the wavelet transform coefficients is a multirate filter bank producing sequences of coefficients at differentrates but with the average number at any stage being the same. This approach can be applied to any signal, finite or infinite in length,periodic or nonperiodic. Note that while the average output rate is the same as the average input rate, the number of output coefficientsis greater than the number of input coefficients because the length of the output of convolution is greater than the length of the input.

Questions & Answers

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
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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