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Wavelet signal denoising

`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 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.

[1] Abramovich, F., Y. Benjamini,
D. L. Donoho, and I. M. Johnstone. “Adapting to Unknown Sparsity by Controlling
the False Discovery Rate.” *Annals of Statistics*, Vol. 34,
Number 2, pp. 584–653, 2006.

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

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

[4] 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.

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

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

[7] Donoho, D. L., I. M.
Johnstone, G. Kerkyacharian, and D. Picard. “Wavelet Shrinkage:
Asymptopia?” *Journal of the Royal Statistical Society*,
*series B*, Vol. 57, No. 2, pp. 301–369, 1995.

[8] Johnstone, I. M., and B. W.
Silverman. “Needles and Straw in Haystacks: Empirical Bayes Estimates of Possibly
Sparse Sequences.” *Annals of Statistics*, Vol. 32, Number 4,
pp. 1594–1649, 2004.