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# wden

Automatic 1-D denoising

wden is no longer recommended. Use wdenoise instead.

## Syntax

XD = wden(X,TPTR,SORH,SCAL,N,wname)
XD = wden(C,L,___)
XD = wden(W,'modwtsqtwolog',SORH,'mln',N,wname)
[XD,CXD] = wden(___)
[XD,CXD,LXD] = wden(___)
[XD,CXD,LXD,THR] = wden(___)
[XD,CXD,THR] = wden(___)

## Description

example

XD = wden(X,TPTR,SORH,SCAL,N,wname) returns a denoised version 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.

XD = wden(C,L,___) returns a denoised version XD of the signal X using the same options as in the previous syntax, but obtained directly from the wavelet decomposition structure [C,L] of X. [C,L] is the output of wavedec.

XD = wden(W,'modwtsqtwolog',SORH,'mln',N,wname) returns the denoised signal XD obtained by operating on the maximal overlap discrete wavelet transform (MODWT) matrix W, where W is the output of modwt. You must use the same orthogonal wavelet in both modwt and wden.

[XD,CXD] = wden(___) returns the denoised wavelet coefficients. For discrete wavelet transform (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 of CXD 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 denoised 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 the 'modwtsqtwolog' input argument.

## Examples

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This example shows how to apply three different denoising techniques to a noisy signal. It compares the results with plots and the threshold values produced by each technique.

First, to ensure reproducibility of results, set a seed that will be used to generate the random noise.

rng('default')

Create a signal consisting of a 2 Hz sine wave with transients at 0.3 and 0.72 seconds. Add randomly generated noise to the signal and plot the result.

N = 1000;
t = linspace(0,1,N);
x = 4*sin(4*pi*t);
x = x - sign(t-0.3) - sign(0.72-t);
sig = x + 0.5*randn(size(t));
plot(t,sig)
title('Signal')
grid on

Using the sym8 wavelet, perform a level 5 wavelet decomposition of the signal and denoise it by applying three different threshold selection rules to the wavelet coefficients: SURE, minimax, and Donoho and Johnstone's universal threshold with level-dependent estimation of the noise. In each case, apply hard thresholding.

lev = 5;
wname = 'sym8';
[dnsig1,c1,l1,threshold_SURE] = wden(sig,'rigrsure','h','mln',lev,wname);
[dnsig2,c2,l2,threshold_Minimax] = wden(sig,'minimaxi','h','mln',lev,wname);
[dnsig3,c3,l3,threshold_DJ] = wden(sig,'sqtwolog','h','mln',lev,wname);

Plot and compare the three denoised signals.

subplot(3,1,1)
plot(t,dnsig1)
title('Denoised Signal - SURE')
grid on
subplot(3,1,2)
plot(t,dnsig2)
title('Denoised Signal - Minimax')
grid on
subplot(3,1,3)
plot(t,dnsig3)
title('Denoised Signal - Donoho-Johnstone')
grid on

Compare the thresholds applied at each detail level for the three denoising methods.

threshold_SURE
threshold_SURE = 1×5

0.9592    0.6114    1.4734    0.7628    0.4360

threshold_Minimax
threshold_Minimax = 1×5

1.1047    1.0375    1.3229    1.1245    1.0483

threshold_DJ
threshold_DJ = 1×5

1.8466    1.7344    2.2114    1.8798    1.7524

This example denoises a signal using the DWT and MODWT. It compares the results with plots and the threshold values produced by each technique.

First, to ensure reproducibility of results, set a seed that will be used to generate random noise.

rng('default')

Create a signal consisting of a 2 Hz sine wave with transients at 0.3 and 0.72 seconds. Add randomly generated noise to the signal and plot the result.

N = 1000;
t = linspace(0,1,N);
x = 4*sin(4*pi*t);
x = x - sign(t-0.3) - sign(0.72-t);
sig = x + 0.5*randn(size(t));
plot(t,sig)
title('Signal')
grid on

Using the db2 wavelet, perform a level 3 wavelet decomposition of the signal and denoise it using Donoho and Johnstone's universal threshold with level-dependent estimation of the noise. Obtain denoised versions using DWT and MODWT, both with soft thresholding.

wname = 'db2';
lev = 3;
[xdDWT,c1,l1,threshold_DWT] = wden(sig,'sqtwolog','s','mln',lev,wname);
[xdMODWT,c2,threshold_MODWT] = wden(sig,'modwtsqtwolog','s','mln',lev,wname);

Plot and compare the results.

subplot(2,1,1)
plot(t,xdDWT)
grid on
title('DWT Denoising')
subplot(2,1,2)
plot(t,xdMODWT)
grid on
title('MODWT Denoising')

Compare the thresholds applied in each case.

threshold_DWT
threshold_DWT = 1×3

1.7783    1.6876    2.0434

threshold_MODWT
threshold_MODWT = 1×3

1.2760    0.6405    0.3787

This example denoises a blocky signal using the Haar wavelet with DWT and MODWT denoising. It compares the results with plots and metrics for the original and denoised versions.

First, to ensure reproducibility of results, set a seed that will be used to generate random noise.

rng('default')

Generate a signal and a noisy version with the square root of the signal-to-noise ratio equal to 3. Plot and compare each.

[osig,nsig] = wnoise('blocks',10,3);
plot(nsig,'r')
hold on
plot(osig,'b')
legend('Noisy Signal','Original Signal')

Using the Haar wavelet, perform a level 6 wavelet decomposition of the noisy signal and denoise it using Donoho and Johnstone's universal threshold with level-dependent estimation of the noise. Obtain denoised versions using DWT and MODWT, both with soft thresholding.

wname = 'haar';
lev = 6 ;
[xdDWT,c1,l1] = wden(nsig,'sqtwolog','s','mln',lev,wname);
[xdMODWT,c2] = wden(nsig,'modwtsqtwolog','s','mln',lev,wname);

Plot and compare the original, noise-free version of the signal with the two denoised versions.

figure
plot(osig,'b')
hold on
plot(xdDWT,'r--')
plot(xdMODWT,'k-.')
legend('Original','DWT','MODWT')
hold off

Calculate the L2 and L-infinity norms of the difference between the original signal and the two denoised versions.

L2norm_original_DWT = norm(abs(osig-xdDWT),2)
L2norm_original_DWT = 36.1194
L2norm_original_MODWT = norm(abs(osig-xdMODWT),2)
L2norm_original_MODWT = 14.5987
LInfinity_original_DWT = norm(abs(osig-xdDWT),Inf)
LInfinity_original_DWT = 4.7181
LInfinity_original_MODWT = norm(abs(osig-xdMODWT),Inf)
LInfinity_original_MODWT = 2.9655

## Input Arguments

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Input data to denoise, specified as a real-valued vector.

Data Types: double

Wavelet expansion coefficients of the data to be denoised, specified as a real-valued vector. C is the output of wavedec.

Example: [C,L] = wavedec(randn(1,1024),3,'db4')

Data Types: double

Size of wavelet expansion coefficients of the signal to be denoised, specified as a vector of positive integers. L is the output of wavedec.

Example: [C,L] = wavedec(randn(1,1024),3,'db4')

Data Types: double

Maximal overlap wavelet decomposition structure of the signal to denoise, specified as a real-valued matrix. W is the output of modwt. You must use the same orthogonal wavelet in both modwt and wden.

Data Types: double

Threshold selection rule to apply to the wavelet decomposition structure of X:

• 'rigsure' — Use the principle of Stein's Unbiased Risk.

• 'heursure' — Use a heuristic variant of Stein's Unbiased Risk.

• 'sqtwolog — Use the universal threshold $\sqrt{2\mathrm{ln}\left(\text{length}\left(x\right)\right)}.$

Type of thresholding to perform:

• 's' — Soft thresholding

• 'h' — Hard thresholding

Multiplicative threshold rescaling:

• 'one' — No rescaling

• 'sln' — Rescaling using a single estimation of level noise based on first-level coefficients

• 'mln' — Rescaling using a level-dependent estimation of level noise

Level of wavelet decomposition, specified as a positive integer. Use wmaxlev to ensure that the wavelet coefficients are free from boundary effects. If boundary effects are not a concern in your application, a good rule is to set N less than or equal to fix(log2(length(X))).

Name of wavelet, specified as a character array, to use for denoising. For DWT denoising, the wavelet must be orthogonal or biorthogonal. For MODWT denoising, the wavelet must be orthogonal. Orthogonal and biorthogonal wavelets are designated as type 1 and type 2 wavelets, respectively, in the wavelet manager, wavemngr.

• Valid built-in orthogonal wavelet families begin with haar, dbN, fkN, coifN, or symN, where N is the number of vanishing moments for all families except fk. For fk, N is the number of filter coefficients.

• Valid biorthogonal wavelet families begin with 'biorNr.Nd' or 'rbioNd.Nr', where Nr and Nd are the number of vanishing moments in the reconstruction (synthesis) and decomposition (analysis) wavelet.

Determine valid values for the vanishing moments by using waveinfo with the wavelet family short name. For example, enter waveinfo('db') or waveinfo('bior'). Use wavemngr('type',wname) to determine if a wavelet is orthogonal (returns 1) or biorthogonal (returns 2).

## Output Arguments

collapse all

Denoised data, returned as a real-valued vector.

Data Types: double

Denoised wavelet coefficients, returned as a real-valued vector or matrix. 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.

Data Types: double

Size of denoised wavelet coefficients by level for DWT denoising, returned as a vector of positive integers (see wavedec). The LXD output is not supported for MODWT denoising. [CXD,LXD] is the wavelet decomposition structure of the denoised signal XD.

Data Types: double

Denoising thresholds by level, returned as a length N real-valued vector.

Data Types: double

## Algorithms

The most general model for the noisy signal has the following form:

$s\left(n\right)=f\left(n\right)+\sigma e\left(n\right),$

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:

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 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 CD1 (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 σlev level by level. This estimation is implemented in the file wnoisest, which handles the wavelet decomposition structure of the original signal s directly.

## References

[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.