<< Chapter < Page Chapter >> Page >

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 , 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 i i d N ( 0 , 1 ) . 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 , or , Y i = X i + N i ,

where capital letters denote variables in the transform domain, i.e., Y = W y . 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 X ^ denote an estimate of X , based on the observations Y . We consider diagonal linear projections

Δ = diag ( δ 1 , ... , δ N ) , δ i { 0 , 1 } , i = 1 , ... , N ,

which give rise to the estimate

x ^ = W - 1 X ^ = W - 1 Δ Y = W - 1 Δ W y .

The estimate 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

R ( X ^ , X ) = E x ^ - x 2 2 = E W - 1 ( X ^ - X ) 2 2 = E X ^ - X 2 2 .

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 δ i = 1 X i > ϵ ; It is interesting to note that allowing arbitrary δ i I 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

R i d ( X ^ , X ) = n = 1 N min ( X 2 , ϵ 2 ) .

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 = W y
  2. perform thresholding in the wavelet domain, according to so-called hard thresholding
    X ^ = T h ( Y , t ) = Y , | Y | t 0 , | Y | < t
    or according to so-called soft thresholding
    X ^ = T S ( Y , t ) = sgn ( Y ) ( | Y | - t ) , | Y | t 0 , | Y | < t
  3. compute the inverse DWT x ^ = W - 1 X ^

This simple scheme has several interesting properties. It's risk is within a logarithmic factor ( log N ) of the ideal risk for both thresholding schemes and properly chosen thresholds t ( N , ϵ ) . 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 | X ^ i | < | X i | holds. The shrinkage condition guarantees that 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 log N . Not only does Donoho's method have nice theoretical properties, but italso works very well in practice.

Questions & Answers

so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
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
Hello
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Wavelets and wavelet transforms' conversation and receive update notifications?

Ask