# Signal denoising using wavelet-based methods  (Page 5/9)

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In the linear penalization method every wavelet coefficient is affected by a linear shrinkage particular associated to the resolution level of the coefficient. A mathematical expression for this type of approach using linear shrinkage is shown in Equation .

${\stackrel{˜}{d}}_{jk}=\frac{{\stackrel{^}{d}}_{jk}}{1+\lambda {2}^{2js}}$

In Equation , parameter $s$ is the known smoothness index of the underlying signal $g$ , while parameter $\lambda$ is a smooting factor whose determination is critical for this type of analysis.

It must be said that linear thresholding is adequate only for spatially homogenous signal with important levels of regularity. When homegeneity and regularity conditions are not met nonlinear wavelet thresholding or shrinkage methods are usuallymore suitable.

donoho1995 and donoho1995b proposed a nonlinear strategy for thresholding. Under their approach, the thresholding can be done by implementing either a hard or a soft thresholding rule. Their mathematicalexpressions are shown in Equation and Equation respectively.

In both methods, the role of the parameter $\lambda$ as a threshold value is critical as the estimator leading to destruction,reduction, or increase in the value of a wavelet coefficient.

${\delta }_{\lambda }^{H}\left({\stackrel{^}{d}}_{jk}\right)=\left\{\begin{array}{cc}0\hfill & \phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}|{\stackrel{^}{d}}_{jk}|\le \lambda \hfill \\ {\stackrel{^}{d}}_{jk}\hfill & \phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}|{\stackrel{^}{d}}_{jk}|>\lambda \hfill \end{array}\right)$
${\delta }_{\lambda }^{S}\left({\stackrel{^}{d}}_{jk}\right)=\left\{\begin{array}{cc}0\hfill & \phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}|{\stackrel{^}{d}}_{jk}|\le \lambda \hfill \\ {\stackrel{^}{d}}_{jk}-\lambda \hfill & \phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}{\stackrel{^}{d}}_{jk}>\lambda \hfill \\ {\stackrel{^}{d}}_{jk}+\lambda \hfill & \phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}{\stackrel{^}{d}}_{jk}<-\lambda \hfill \end{array}\right)$

Several authors have discussed the properties and limitations of these two strategies; hard thresholding, due to its induced discontinuity, can be unstable and sensitive even to small changes in the data. On the other hand, soft thresholdingcan create unnecessary bias when the true coefficients are large. Although more sophisticated methods has been introduced to account for the drawbacks of the described nonlinear strategies, the discussion in this report is limited to the hardand soft approaches.

## Term-by-term thresholding

One apparent problem in applying wavelet thresholding methods is the way of selecting an appropriate value for the threshold, $\lambda$ . There are indeed several approaches for specifying the value of the parameter in question. In a general sense, these strategies can be classified in two groups: global thresholds and level-dependent thresholds. Global threshold implies the selection of one $\lambda$ value, applied to all the wavelet coefficients. Level-dependent thresholds implies that a (possibly) different threshold value $lambd{a}_{j}$ is applied for each resolution level. All the alternatives require an estimate of the noise level $\sigma$ . The standard deviation of the data values is clearly not a good estimator, unless the underlying response function $g$ is reasonably flat. donoho1995 considered estimating $\sigma$ in the wavelet domain by using the expression in Equation .

$\stackrel{^}{\sigma }=\frac{median\left(|{\stackrel{^}{d}}_{J-1,k}|\right)}{0.6745},\phantom{\rule{1.em}{0ex}}k=0,1,...,{2}^{J-1}-1$

## The minimax threshold

donoho1995 obtained an optimal threshold value ${\lambda }^{M}$ by minimizing the risk involved in estimating a function. The porposed minimax threshold depends of the available data and also takes into account the noise level contaminating the signal (Equation ).

${\lambda }^{M}=\stackrel{^}{\sigma }{\lambda }_{n}^{*}$

Where, ${\lambda }_{n}^{*}$ is equal to the value of $\lambda$ satisfying Equation

${\lambda }_{n}^{*}=\underset{\lambda }{inf}\underset{d}{sup}\left\{\frac{{R}_{\lambda }\left(d\right)}{{n}^{-1}+{R}_{oracle}\left(d\right)}\right\}$

In Equation , ${R}_{\lambda }\left(d\right)$ is calculated following Equation .

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I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
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can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
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I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
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oops. ignore that.
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hmm
Abhi
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Abhi
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Abhi
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salma
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salma
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Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
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Cied
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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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Yasmin
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Cesar
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Uday
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Stotaw
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Azam
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Prasenjit
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Azam
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Prasenjit
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Damian
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Azam
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Prasenjit
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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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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