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c ^ j 0 k / c j 0 k , σ 2 N ( c j 0 k , σ 2 )
d ^ j k / c j k , σ 2 N ( d j k , σ 2 )

The Bayesian approach imposes an apriori model for the wavelets coefficients designed to capture the sparseness of the wavelet expansion common to most applications. An usual prior model for each wavelet coefficient d ^ j k is a mixture of two distributions, one of them associated to negligable coefficients and the other to significant coefficients. Two types of mixtures have been widely used. One of them employs two normal distributions while theother uses one normal distribution and one point mass at zero.

After mathematical manipulation, it can be shown that an estimator for the underlying signal can be written as (Equation ):

g ^ B R ( t ) = k = 0 2 j 0 - 1 c ^ j 0 k n φ j 0 k ( t ) + j = j 0 J - 1 k = 0 2 j - 1 B R ( d j k | ( d j k , σ 2 ) ) n ψ j k ( t )

i.e. the scaling coefficients are estimated by the empirical scaling coefficients while the wavelet coefficients are estimated by a Bayesian rule (BR), taking into account the obtained empirical wavelet coefficient and the noise level.

Shrinkage estimates based on deterministic/stochastic decompositions

huang2000 proposed a method that takes into account the value of the prior mean for each wavelet coefficient, by introducing a estimator for the parameter into the general wavelet shrinkage model. These authorsassumed thatthe undelying signal is composed of a piecewise deterministic portion with an added zero mean stochastic part.

If c ^ j 0 is the vector of empirical scaling coefficients, d ^ j the vector of empirical wavelet coefficients, c j 0 the vector of underlying scaling coefficients, and d j the vector of underlying wavelet coefficients, then the Bayesian model (Equation ):

ω / ( β , σ 2 ) N ( β , σ 2 I )

with ω = ( c ^ j 0 , d ^ j 0 , ... , d ^ J - 1 ' ) ' and the underlying signal β = ( c j 0 ' , d j 0 ' , ... , d J - 1 ' ) ' is assumed to follow an apriori distribution (Equation )

β / ( μ , θ ) N ( μ , Σ ( θ ) )

where μ is the deterministic mean structure and Σ ( θ ) accounts for the uncertainty and value correlation in the underlying signal. Notice that if η following a distribution N ( 0 , Σ ( θ ) ) is defined as the stochastic component representing small variation (high frequency) in the signal, then μ can be interpretated as the stochastic component accounting for the large-scale variation in β . So, it is possible to rewrite β as (Equation ),

β = μ + η

Using this model, a shrinkage rule can be established by calculating the mean of β conditional on σ 2 which is expressed as (Equation ),

E β / ( ω , σ 2 ) = μ + Σ ( θ ) ( Σ ( θ ) + σ 2 I ) ( ω - μ )

Numerical simulations

Description of the scheme

In order to assess the efficiency and accuracy of the proposed methods, a number of simulations have been conducted. To this aim, data have been generated according to the following scheme

y i = f ( x i ) + ϵ i , { ϵ i } N ( 0 , σ 2 )

where the data { x i } are considered equally spaced in the interval [ 0 , 1 ] . The signal-to-noise ratio has been taken equal to 3. In these simulations the Symmlet 8 wavelet basis has been used. Given the random nature of { ϵ i } , 100 realizations of the function { y i } have been produced. This has been done in order to apply the comparison criteria to the ensemble average of the realizations. Since the primary goal of the simulations is the comparison ofthe different denoising methods, the following criteria are introduced:

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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?
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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