# 0.10 Wavelet-based signal processing and applications  (Page 7/13)

 Page 7 / 13

The new nonlinear method is entirely different. The spectra can overlap as much as they want. The idea is to have the amplitude, rather than thelocation of the spectra be as different as possible. This allows clipping, thresholding, and shrinking of the amplitude of the transform toseparate signals or remove noise. It is the localizing or concentrating properties of the wavelet transform that makes it particularly effectivewhen used with these nonlinear methods. Usually the same properties that make a system good for denoising or separation by nonlinear methods, makesit good for compression, which is also a nonlinear process.

## Denoising by thresholding

We develop the basic ideas of thresholding the wavelet transform using Donoho's formulations [link] , [link] , [link] . Assume a finite length signal with additive noise of the form

${y}_{i}={x}_{i}+ϵ{n}_{i},\phantom{\rule{1.em}{0ex}}i=1,...,N$

as a finite length signal of observations of the signal ${x}_{i}$ that is corrupted by i.i.d.  zero mean, white Gaussian noise ${n}_{i}$ with standard deviation $ϵ$ , i.e., ${n}_{i}\stackrel{iid}{\sim }\mathcal{N}\left(0,1\right)$ . The goal is to recover the signal $x$ from the noisy observations $y$ . Here and in the following, $v$ denotes a vector with the ordered elements ${v}_{i}$ if the index $i$ is omitted. Let $W$ be a left invertible wavelet transformation matrix of the discrete wavelet transform(DWT). Then Eq. [link] can be written in the transformation domain

$Y=X+N,\phantom{\rule{1.em}{0ex}}\text{or},\phantom{\rule{1.em}{0ex}}{Y}_{i}={X}_{i}+{N}_{i},$

where capital letters denote variables in the transform domain, i.e., $Y=Wy$ . Then the inverse transform matrix ${W}^{-1}$ exists, and we have

${W}^{-1}W=I.$

The following presentation follows Donoho's approach [link] , [link] , [link] , [link] , [link] that assumes an orthogonal wavelet transform with a square $W$ ; i.e., ${W}^{-1}={W}^{T}$ . We will use the same assumption throughout this section.

Let $\stackrel{^}{X}$ denote an estimate of $X$ , based on the observations $Y$ . We consider diagonal linear projections

$\text{Δ}=\text{diag}\left({\delta }_{1},...,{\delta }_{N}\right),\phantom{\rule{1.em}{0ex}}{\delta }_{i}\in \left\{0,1\right\},\phantom{\rule{1.em}{0ex}}i=1,...,N,$

which give rise to the estimate

$\stackrel{^}{x}={W}^{-1}\stackrel{^}{X}={W}^{-1}\text{Δ}Y={W}^{-1}\text{Δ}Wy.$

The estimate $\stackrel{^}{X}$ is obtained by simply keeping or zeroing the individual wavelet coefficients. Since we are interested inthe ${l}_{2}$ error we define the risk measure

$\mathcal{R}\left(\stackrel{^}{X},X\right)=E\left[\parallel ,\stackrel{^}{x},{-x\parallel }_{2}^{2}\right]=E\left[\parallel ,{W}^{-1},\left(\stackrel{^}{X}-X\right),{\parallel }_{2}^{2}\right]=E\left[\parallel ,\stackrel{^}{X},{-X\parallel }_{2}^{2}\right].$

Notice that the last equality in Eq. [link] is a consequence of the orthogonality of $W$ . The optimal coefficients in the diagonal projection scheme are ${\delta }_{i}={1}_{{X}_{i}>ϵ}$ ; It is interesting to note that allowing arbitrary ${\delta }_{i}\in \mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{R}$ improves the ideal risk by at most a factor of 2 [link] i.e., only those values of $Y$ where the corresponding elements of $X$ are larger than $ϵ$ are kept, all others are set to zero. This leads to the ideal risk

${\mathcal{R}}_{id}\left(\stackrel{^}{X},X\right)=\sum _{n=1}^{N}min\left({X}^{2},{ϵ}^{2}\right).$

The ideal risk cannot be attained in practice, since it requires knowledge of $X$ , the wavelet transform of the unknown vector $x$ . However, it does give us a lower limit for the ${l}_{2}$ error.

Donoho proposes the following scheme for denoising:

1. compute the DWT $Y=Wy$
2. perform thresholding in the wavelet domain, according to so-called hard thresholding
$\stackrel{^}{X}={T}_{h}\left(Y,t\right)=\left\{\begin{array}{cc}Y,& |Y|\ge t\hfill \\ 0,& |Y|
or according to so-called soft thresholding
$\stackrel{^}{X}={T}_{S}\left(Y,t\right)=\left\{\begin{array}{cc}\text{sgn}\left(Y\right)\left(|Y|-t\right),& |Y|\ge t\hfill \\ 0,& |Y|
3. compute the inverse DWT $\stackrel{^}{x}={W}^{-1}\stackrel{^}{X}$

This simple scheme has several interesting properties. It's risk is within a logarithmic factor ( $logN$ ) of the ideal risk for both thresholding schemes and properly chosen thresholds $t\left(N,ϵ\right)$ . If one employs soft thresholding, then the estimate is with high probability at least assmooth as the original function. The proof of this proposition relies on the fact that wavelets are unconditional bases for a variety of smoothnessclasses and that soft thresholding guarantees (with high probability) that the shrinkage condition $|{\stackrel{^}{X}}_{i}|<|{X}_{i}|$ holds. The shrinkage condition guarantees that $\stackrel{^}{x}$ is in the same smoothness class as is $x$ . Moreover, the soft threshold estimate is the optimal estimate that satisfies the shrinkage condition. The smoothness property guarantees anestimate free from spurious oscillations which may result from hard thresholding or Fourier methods. Also, it can be shown that it is notpossible to come closer to the ideal risk than within a factor $logN$ . Not only does Donoho's method have nice theoretical properties, but italso works very well in practice.

what is variations in raman spectra for nanomaterials
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
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
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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!    By By Mistry Bhavesh By Sam Luong By  By By