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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
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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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