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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 somebody tell me a best nano engineering book for beginners?
s. Reply
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Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
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.
Harper
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SUYASH Reply
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SUYASH
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s. Reply
of graphene you mean?
Ebrahim
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Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
what is the function of carbon nanotubes?
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I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
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Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Elec 301 projects fall 2008. OpenStax CNX. Jan 22, 2009 Download for free at http://cnx.org/content/col10633/1.1
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