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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)


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


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.


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

Questions & Answers

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
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
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
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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