# Signal denoising using wavelet-based methods  (Page 8/9)

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${\stackrel{^}{c}}_{{j}_{0}k}/\left({c}_{{j}_{0}k},,,{\sigma }^{2}\right)\sim N\left({c}_{{j}_{0}k},{\sigma }^{2}\right)$
${\stackrel{^}{d}}_{jk}/\left({c}_{jk},,,{\sigma }^{2}\right)\sim N\left({d}_{jk},{\sigma }^{2}\right)$

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 ${\stackrel{^}{d}}_{jk}$ 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 ):

${\stackrel{^}{g}}_{BR}\left(t\right)=\sum _{k=0}^{{2}^{{j}_{0}}-1}\frac{{\stackrel{^}{c}}_{{j}_{0}k}}{\sqrt{n}}{\phi }_{{j}_{0}k}\left(t\right)+\sum _{j={j}_{0}}^{J-1}\sum _{k=0}^{{2}^{j}-1}\frac{BR\left({d}_{jk}|\left({d}_{jk},{\sigma }^{2}\right)\right)}{\sqrt{n}}{\psi }_{jk}\left(t\right)$

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 ${\stackrel{^}{\mathbf{c}}}_{{j}_{0}}$ is the vector of empirical scaling coefficients, ${\stackrel{^}{\mathbf{d}}}_{j}$ the vector of empirical wavelet coefficients, ${\mathbf{c}}_{{j}_{0}}$ the vector of underlying scaling coefficients, and ${\mathbf{d}}_{j}$ the vector of underlying wavelet coefficients, then the Bayesian model (Equation ):

$\omega /\left(\beta ,{\sigma }^{2}\right)\sim N\left(\beta ,{\sigma }^{2}I\right)$

with $\omega ={\left({\stackrel{^}{\mathbf{c}}}_{{j}_{0}},{\stackrel{^}{\mathbf{d}}}_{{j}_{0}},...,{\stackrel{^}{\mathbf{d}}}_{J-1}^{\text{'}}\right)}^{\text{'}}$ and the underlying signal $\beta ={\left({\mathbf{c}}_{{j}_{0}}^{\text{'}},{\mathbf{d}}_{{j}_{0}}^{\text{'}},...,{\mathbf{d}}_{J-1}^{\text{'}}\right)}^{\text{'}}$ is assumed to follow an apriori distribution (Equation )

$\beta /\left(\mu ,\theta \right)\sim N\left(\mu ,\Sigma \left(\theta \right)\right)$

where $\mu$ is the deterministic mean structure and $\Sigma \left(\theta \right)$ accounts for the uncertainty and value correlation in the underlying signal. Notice that if $\eta$ following a distribution $N\left(0,\Sigma \left(\theta \right)\right)$ is defined as the stochastic component representing small variation (high frequency) in the signal, then $\mu$ can be interpretated as the stochastic component accounting for the large-scale variation in $\beta$ . So, it is possible to rewrite $\beta$ as (Equation ),

$\beta =\mu +\eta$

Using this model, a shrinkage rule can be established by calculating the mean of $\beta$ conditional on ${\sigma }^{2}$ which is expressed as (Equation ),

$E\left(\beta ,/,\left(,\omega ,,,{\sigma }^{2},\right)\right)=\mu +\frac{\Sigma \left(\theta \right)}{\left(\Sigma \left(\theta \right)+{\sigma }^{2}I\right)}\left(\omega -\mu \right)$

## 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

${y}_{i}=f\left({x}_{i}\right)+{ϵ}_{i},\left\{{ϵ}_{i}\right\}\phantom{\rule{3.33333pt}{0ex}}N\left(0,{\sigma }^{2}\right)$

where the data $\left\{{x}_{i}\right\}$ are considered equally spaced in the interval $\left[0,1\right]$ . 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 $\left\{{ϵ}_{i}\right\}$ , 100 realizations of the function $\left\{{y}_{i}\right\}$ 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:

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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