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Automatic 1-D de-noising

`XD = wden(X,TPTR,SORH,SCAL,N,`

* 'wname'*)

XD = wden(C,L,TPTR,SORH,SCAL,N,

`'wname'`

XD = wden(W,'modwtsqtwolog',SORH,'mln',N,WNAME)

[XD,CXD] = wden(...)

[XD,CXD,LXD] = wden(...)

[XD,CXD,LXD,THR] = wden(...)

[XD,CXD,THR] = wden(...)

`wden`

is a one-dimensional de-noising function.

`wden`

performs an automatic
de-noising process of a one-dimensional signal using wavelets.

`XD = wden(X,TPTR,SORH,SCAL,N,`

returns
a de-noised version * 'wname'*)

`XD`

of input signal `X`

obtained
by thresholding the wavelet coefficients.`TPTR`

character vector contains the threshold
selection rule:

`'rigrsure'`

uses the principle of Stein's Unbiased Risk.`'heursure'`

is an heuristic variant of the first option.`'sqtwolog'`

for the universal threshold $$\sqrt{2\mathrm{ln}(\xb7)}$$`'minimaxi'`

for minimax thresholding (see`thselect`

for more information)

`SORH`

(`'s'`

or `'h'`

)
is for soft or hard thresholding (see `wthresh`

for
more information).

`SCAL`

defines multiplicative threshold rescaling:

`'one'`

for no rescaling

`'sln'`

for rescaling using a single estimation
of level noise based on first-level coefficients

`'mln'`

for rescaling done using level-dependent
estimation of level noise

Wavelet decomposition is performed at level `N`

and `'`

`wname`

`'`

is
a character vector containing the name of the desired orthogonal wavelet
(see `wmaxlev`

and `wfilters`

for more information).

`XD = wden(C,L,TPTR,SORH,SCAL,N,`

returns
the same output arguments, using the same options as above, but obtained
directly from the input wavelet decomposition structure * 'wname'*)

`[C,L]`

of
the signal to be de-noised, at level `N`

and using `'wname'`

`XD = wden(W,'modwtsqtwolog',SORH,'mln',N,WNAME)`

returns
the denoised signal obtained by operating on the MODWT transform matrix `W`

,
where `W`

is the output of MODWT. You must use the
same wavelet in both `modwt`

and `wden`

.

`[XD,CXD] = wden(...)`

returns the denoised
wavelet coefficients. For DWT denoising, `CXD`

is
a vector (see `wavedec`

). For
MODWT denoising, `CXD`

is a matrix with N+1 rows
(see `modwt`

). The number of
columns is equal to the length of the input signal `X`

.

`[XD,CXD,LXD] = wden(...)`

returns the number
of coefficients by level for DWT denoising. See `wavedec`

for details. The `LXD`

output
is not supported for MODWT denoising. The additional output arguments `[CXD,LXD]`

are
the wavelet decomposition structure (see `wavedec`

for
more information) of the de-noised signal `XD`

.

`[XD,CXD,LXD,THR] = wden(...)`

returns the denoising thresholds by level
for DWT denoising.

`[XD,CXD,THR] = wden(...)`

returns the denoising thresholds by level for
MODWT denoising when you specify `'modwtsqtwolog'`

.

The underlying model for the noisy signal is basically of the following form:

$$s(n)=f(n)+\sigma e(n)$$

where time *n* is equally spaced.

In the simplest model, suppose that *e*(*n*)
is a Gaussian white noise *N*(0,1) and the noise
level σ a is supposed to be equal to 1.

The de-noising objective is to suppress the noise part of the
signal *s* and to recover *f*.

The de-noising procedure proceeds in three steps:

Decomposition. Choose a wavelet, and choose a level

`N`

. Compute the wavelet decomposition of the signal s at level`N`

.Detail coefficients thresholding. For each level from 1 to

`N`

, select a threshold and apply soft thresholding to the detail coefficients.Reconstruction. Compute wavelet reconstruction based on the original approximation coefficients of level

`N`

and the modified detail coefficients of levels from 1 to`N`

.

More details about threshold selection rules are in Wavelet Denoising and Nonparametric Function Estimation, in the
User's Guide, and in the help of the `thselect`

function.
Let us point out that

The detail coefficients vector is the superposition of the coefficients of

*f*and the coefficients of*e*, and that the decomposition of*e*leads to detail coefficients that are standard Gaussian white noises.Minimax and SURE threshold selection rules are more conservative and are more convenient when small details of function

*f*lie in the noise range. The two other rules remove the noise more efficiently. The option`'heursure'`

is a compromise.

In practice, the basic model cannot be used directly. This section
examines the options available, to deal with model deviations. The
remaining parameter `scal`

has to be specified. It
corresponds to threshold rescaling methods.

Option

`scal`

=`'one'`

corresponds to the basic model.In general, you can ignore the noise level that must be estimated. The detail coefficients

*CD*(the finest scale) are essentially noise coefficients with standard deviation equal to σ. The median absolute deviation of the coefficients is a robust estimate of σ. The use of a robust estimate is crucial because if level 1 coefficients contain_{1}*f*details, these details are concentrated in few coefficients to avoid signal end effects, which are pure artifacts due to computations on the edges.The option

`scal = 'sln'`

handles threshold rescaling using a single estimation of level noise based on the first-level coefficients.When you suspect a nonwhite noise

*e*, thresholds must be rescaled by a level-dependent estimation of the level noise. The same kind of strategy is used by estimating σlevel by level. This estimation is implemented in the file_{lev}`wnoisest`

, which handles the wavelet decomposition structure of the original signal*s*directly.The option

`scal`

=`'mln'`

handles threshold rescaling using a level-dependent estimation of the level noise.

Antoniadis, A.; G. Oppenheim, Eds. (1995), *Wavelets
and statistics*, 103, Lecture Notes in Statistics, Springer
Verlag.

Donoho, D.L. (1993), “Progress in wavelet analysis and
WVD: a ten minute tour,” in *Progress in wavelet analysis
and applications*, Y. Meyer, S. Roques, pp. 109–128.
Frontières Ed.

Donoho, D.L.; I.M. Johnstone (1994), “Ideal spatial adaptation
by wavelet shrinkage,” *Biometrika*, Vol.
81, pp. 425–455.

Donoho, D.L. (1995), “De-noising by soft-thresholding,” *IEEE
Trans. on Inf. Theory*, 42 3, pp. 613– 627.

Donoho, D.L.; I.M. Johnstone, G. Kerkyacharian, D. Picard (1995),
“Wavelet shrinkage: asymptotia,” *Jour. Roy.
Stat. Soc.*, *series B*, Vol. 57, No.
2, pp. 301–369.

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