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  • Root Mean Squared Error : The mean squared error defined as (Equation )
    1 N i = 1 N f ( x i ) - f d n ( x i 2
    is computed for each realization and averaged over the 100 samples. Then, its square root is taken.
  • Maximum Deviation : The average over the 100 samples of max 1 < i < N f ( x i ) - f d n ( x i )

Computational efficiency has not been chosen as one of the criteria, since it is greatly depended on the individual programming skills of the individual. Therefore, in order to avoid a non-uniform programming approach which couldpossibly result in misleading conclusions, time efficiency has not been considered.

The test functions f ( x ) and the sample sizes N have been chosen as the factors of the comparison studies. To this aim, two samples, one of moderate moderate size ( N = 128 ) and another of larger size ( N = 1024 ) have been considered.

As far as the test functions are concerned, two smooth signals (Figures and ) and two discontinuous ones (Figures and ) were taken into account. In , the function consists of the sum of two sinusoids, whereas in , a time shifted sine is illustrated. Since the signals are smooth, linear methods are expected to be comparable to the nonlinear ones. On the other hand, nonlinear wavelet estimators areexpected to perform better for the functions in ( , ). These highly discontinuous signals have been used as examples in donoho1993

Original function with added Gaussian White noise (Wave function)
Original function with added Gaussian White noise (Time shifted sine function)
Original function with added Gaussian White noise (Blocks function)
Original function with added Gaussian White noise (Bumps function)

Results

The following plots, (Figures - ), illustrate the denoising performance for the 10 methods used. Each integer corresponds to a particular method as follows

  • VisuShrink-Hard: Universal threshold with hard thresholding rule
  • VisuShrink-Soft: Universal threshold with soft thresholding rule
  • SureShrink: SureShrink threshold
  • Translation-Invariant-Hard: Translation invariant threshold with hard thresholding rule
  • Translation-Invariant-Soft: Translation invariant threshold with soft thresholding rule
  • Minimax-Hard: Minimax threshold with hard thresholding rule
  • Minimax-Soft: Minimax threshold with soft thresholding rule
  • NeighBlock: Overlapping block thresholding (with L 0 = [ log n / 2 ] , λ = 4 . 50524 )
  • Linear Penalization: Term-by-term thresholding using linear shrinking
  • Deterministic/Stochastic: Bayesian thresholding method for shrinkage estimates
Comparison Study using Wave function. N=128
Comparison Study using Wave function. N=1024
Comparison Study using Time-shifted sine function. N=128
Comparison Study using Time-shifted sine function. N=1024
Comparison Study using Blocks function. N=128
Comparison Study using Blocks function. N=1024
Comparison Study using Bumps function. N=128
Comparison Study using Bumps function. N=1024

Conclusions

A general comment can be made related to the Root Mean Squared Error (RMSE). As expected, the bigger the sample size the lower the value of the RMSE. It is readily seen that this is true for the same test function and denoising procedure.

Focusing on the smooth Wave function, the bayesian method performs well. However, the linear penalization method and the Translation-Invariant-Hard method are very competitive. The performance of the penalization method should not besurprising since the linear estimators are expected to achieve good results in smooth functions such as the Wave signal. Similar remarks can be made about the Time-Shifted Sine signal, a function that shares with the Wave signal the smoothnesfeature.

As far as the Bumps function and the Blocks function are concerned, the Bayesian method outperform the classical ones in terms of RMSE. This leads to the conclusion that using Bayesian methods for such type of functions is preferable ifcomputational efficiency is not an issue. In fact, it is well established that non-Bayesian methods uniformly outperform Bayesian methods in terms of CPU time.

Finally, as a general remark, larger values of MaxDeviation occur for functions with many spikes and discontinuities.

Acknowledgments

The authors wish to thank Professor C. Sidney Burrus for his help and guidance through the development of this work.

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
or in general
Ebrahim
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s.
Graphene has a hexagonal structure
tahir
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Cied
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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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Cesar
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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
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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
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Damian
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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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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Smarajit Reply
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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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