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Denoise data using an empirical Bayesian method with a Cauchy prior

`XDEN = wdenoise(X)`

`XDEN = wdenoise(X,LEVEL)`

`XDEN = wdenoise(___,Name,Value)`

`[XDEN,DENOISEDCFS] = wdenoise(___)`

`[XDEN,DENOISEDCFS,ORIGCFS] = wdenoise(___)`

denoises the data in `XDEN`

= wdenoise(`X`

)`X`

using an empirical Bayesian method
with a Cauchy prior. By default, the `sym4`

wavelet is used
with a posterior median threshold rule. Denoising is down to the minimum of
`floor(log`

and _{2}*N*)`wmaxlev(N,'sym4')`

where *N* is the
number of samples in the data. (For more information, see `wmaxlev`

.)
`X`

is a real-valued vector, matrix, or timetable.

If

`X`

is a matrix,`wdenoise`

denoises each column of`X`

.If

`X`

is a timetable,`wdenoise`

must contain real-valued vectors in separate variables, or one real-valued matrix of data.`X`

is assumed to be uniformly sampled.If

`X`

is a timetable and the timestamps are not linearly spaced,`wdenoise`

issues a warning.

specifies options using name-value pair arguments in addition to any of the
input arguments in previous syntaxes.`XDEN`

= wdenoise(___,`Name,Value`

)

`[`

returns the denoised wavelet and scaling coefficients in the cell array
`XDEN`

,`DENOISEDCFS`

] = wdenoise(___)`DENOISEDCFS`

. The elements of
`DENOISEDCFS`

are in order of decreasing resolution. The
final element of `DENOISEDCFS`

contains the approximation
(scaling) coefficients.

`[`

returns the original wavelet and scaling coefficients in the cell array
`XDEN`

,`DENOISEDCFS`

,`ORIGCFS`

] = wdenoise(___)`ORIGCFS`

. The elements of `ORIGCFS`

are in order of decreasing resolution. The final element of
`ORIGCFS`

contains the approximation (scaling)
coefficients.

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 is 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.

[1] Abramovich, F., Y. Benjamini,
D.L. Donoho, I.M. Johnstone (2006), “Adapting to Unknown Sparsity by Controlling
the False Discovery Rate,” *Ann. Statist.*, 34(2), pp.
584–653.

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

[3] Cai, T.T. (2002), “On
Block Thresholding in Wavelet Regression: Adaptivity, Block size, and Threshold
Level,” *Statistica Sinica*, 12, pp. 1241–1273.

[4] 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.

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

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

[7] 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.

[8] Johnstone, I.M., B.W.
Silverman (2004), “Needles and Straw in Haystacks: Empirical Bayes Estimates of
Possibly Sparse Sequences,” *Ann. Statist.* 32(4), pp.
1594-1649.

Wavelet Signal
Denoiser | `thselect`

| `waveinfo`

| `wavemngr`

| `wmaxlev`

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