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Energy concentration is one of the most important properties for low bitrate transform coding. Suppose for the sample quantization step size T , we have a second set of basis that generate less significant coefficients. The distribution of the significant map indicators is moreskewed, thus require less bits to code. Also, we need to code less number of significant values, thus it may require less bits. In the mean time, asmaller M reduces the second error term as in  [link] . Overall, it is very likely that the new basis improves the rate-distortionperformance. Wavelets have better energy concentration property than theFourier transform for signals with discontinuities. This is one of the main reasons that wavelet based compression methods usually out perform DCTbased JPEG, especially at low bitrate.

Improved wavelet based compression algorithms

The above prototype algorithm works well [link] , [link] , but can be further improved for its various building blocks [link] . As we can see from [link] , the significant map still has considerable structure, which could be exploited.Modifications and improvements use the following ideas:

  • Insignificant coefficients are often clustered together. Especially, they often cluster around the same location across severalscales. Since the distance between nearby coefficients doubles for every scale, the insignificant coefficients often form a tree shape, as we cansee from Figure: Discrete Wavelet Transform of the Houston Skyline, using ψ D 8 ' with a Gain of 2 for Each Higher Scale . These so called zero-trees can be exploited [link] , [link] to achieve excellent results.
  • The choice of basis is very important. Methods have been developed to adaptively choose the basis for the signal [link] , [link] . Although they could be computationally very intensive, substantial improvement canbe realized.
  • Special run-length codes could be used to code significant map and values [link] , [link] .
    The Significant Map for the Lenna image.
    The Significant Map for the Lenna image.
  • Advanced quantization methods could be used to replace the simple scalar quantizer [link] .
  • Method based on statistical analysis like classification, modeling, estimation,and prediction also produces impressive result [link] .
  • Instead of using one fixed quantization step size, we can successively refine the quantization by using smaller and smaller stepsizes. These embedded schemes allow both the encoder and the decoder to stop at any bit rate [link] , [link] .
  • The wavelet transform could be replaced by an integer-to-integer wavelet transform, no quantization is necessary, and the compression islossless [link] .

Other references are: [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] .

Why are wavelets so useful?

The basic wavelet in wavelet analysis can be chosen so that it is smooth , where smoothness is measured in a variety of ways [link] . To represent f ( t ) with K derivatives, one can choose a wavelet ψ ( t ) that is K (or more) times continuously differentiable; the penalty for imposing greater smoothness in this senseis that the supports of the basis functions, the filter lengths and hence the computational complexity all increase. Besides, smooth wavelet basesare also the “best bases” for representing signals with arbitrarily many singularities [link] , a remarkable property.

Questions & Answers

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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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