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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

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
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
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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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