<< Chapter < Page Chapter >> Page >
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

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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
How can I make nanorobot?
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
how can I make nanorobot?
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Elec 301 projects fall 2008. OpenStax CNX. Jan 22, 2009 Download for free at http://cnx.org/content/col10633/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elec 301 projects fall 2008' conversation and receive update notifications?