Main Content

Automatic 1-D denoising

`wden`

is no longer recommended. Use `wdenoise`

instead.

returns a denoised version `XD`

= wden(`X`

,`TPTR`

,`SORH`

,`SCAL`

,`N`

,`wname`

)`XD`

of the signal `X`

. The
function uses an `N`

-level wavelet decomposition of
`X`

using the specified orthogonal or biorthogonal wavelet
`wname`

to obtain the wavelet coefficients. The thresholding selection
rule `TPTR`

is applied to the wavelet decomposition.
`SORH`

and `SCAL`

define how the rule is
applied.

`[`

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

,`CXD`

,`LXD`

] = wden(___)`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 denoised signal
`XD`

.

The most general model for the noisy signal has 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 is equal to 1. The denoising objective is
to suppress the noise part of the signal *s* and to recover
*f*.

The denoising procedure has 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 and in the help of the `thselect`

function. Note that:

The detail coefficients vector is the superposition of the coefficients of

*f*and the coefficients of*e*. The decomposition of*e*leads to detail coefficients that are standard Gaussian white noises.Minimax and SURE threshold selection rules are more conservative and 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. To deal with model deviations, the
remaining parameter `scal`

must be specified. It corresponds to threshold
rescaling methods.

The option

`scal`

=`'one'`

corresponds to the basic model.The option

`scal = 'sln'`

handles threshold rescaling using a single estimation of level noise based on the first-level coefficients.In general, you can ignore the noise level that must be estimated. The detail coefficients

*CD*_{1}(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. If level 1 coefficients contain*f*details, these details are concentrated in a few coefficients to avoid signal end effects, which are pure artifacts due to computations on the edges.The option

`scal`

=`'mln'`

handles threshold rescaling using a level-dependent estimation of the level noise.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*σ*_{lev}level by level. This estimation is implemented in the file`wnoisest`

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

[1] Antoniadis, A., and G. Oppenheim,
eds. *Wavelets and Statistics*, 103. Lecture Notes in Statistics. New York:
Springer Verlag, 1995.

[2] Donoho, D. L. “Progress in
Wavelet Analysis and WVD: A Ten Minute Tour.” *Progress in Wavelet Analysis
and Applications* (Y. Meyer, and S. Roques, eds.). Gif-sur-Yvette: Editions
Frontières, 1993.

[3] Donoho, D. L., and Johnstone, I.
M. “Ideal Spatial Adaptation by Wavelet Shrinkage.”
*Biometrika*, Vol. 81, pp. 425–455, 1994.

[4] Donoho, D. L. “De-noising
by Soft-Thresholding.” *IEEE Transactions on Information Theory*,
Vol. 42, Number 3, pp. 613–627, 1995.

[5] Donoho, D. L., I. M. Johnstone, G.
Kerkyacharian, and D. Picard. “Wavelet Shrinkage: Asymptopia?” *Journal
of the Royal Statistical Society*, *series B*. Vol. 57, Number
2, pp. 301–369, 1995.