Wavelet Toolbox    

Automatic 1-D de-noising using wavelets



wden is a one-dimensional de-noising function.

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

[XD,CXD,LXD] = wden(X,TPTR,SORH,SCAL,N,'wname') returns a de-noised version XD of input signal X obtained by thresholding the wavelet coefficients.

Additional output arguments [CXD,LXD] are the wavelet decomposition structure (see wavedec for more information) of the de-noised signal XD.

TPTR string contains the threshold selection rule:

'rigrsure' use the principle of Stein's Unbiased Risk

'heursure' is an heuristic variant of the first option

'sqtwolog' for universal threshold

'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 string containing the name of the desired orthogonal wavelet (see wmaxlev and wfilters for more information).

[XD,CXD,LXD] = wden(C,L,TPTR,SORH,SCAL,N,'wname') returns the same output arguments, using the same options as above, but obtained directly from the input wavelet decomposition structure [C,L] of the signal to be de-noised, at level N and using 'wname' orthogonal wavelet.

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

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 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:

  1. Decomposition. Choose a wavelet, and choose a level N. Compute the wavelet decomposition of the signal s at level N.
  2. Detail coefficients thresholding. For each level from 1 to N, select a threshold and apply soft thresholding to the detail coefficients.
  3. 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 can be found in the "De-noising" section of Chapter 6, "Advanced Concepts", in the User's Guide, and in the help of the thselect function. Let us point out that

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.


See Also
thselect, wavedec, wdencmp, wfilters, wthresh


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, 41, 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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