wthrmngr
Threshold settings manager
Syntax
Description
wthrmngr returns a global threshold or
level-dependent thresholds for wavelet-based denoising and compression. The
function derives thresholds from the wavelet coefficients in a wavelet
decomposition.
The thresholds are used by Wavelet Toolbox™ denoising and compression tools, such as command-line functions and the Wavelet Analyzer app.
thr = wthrmngr(opt,method,C,L,alpha)[
wavelet decomposition threshold using the sparsity parameter
C,L]alpha. For signals,
[
is the output of C,L]wavedec. For
images,
[
is the output of C,L]wavedec2.
To learn more about alpha, see wdcbm or
wdcbm2 for
compression, and wbmpen for
denoising.
thr = wthrmngr(opt,method,C,L,scale)[
wavelet decomposition threshold using the type of multiplicative
threshold rescaling specified in C,L]scale. For
signals,
[
is the output of C,L]wavedec. For
images,
[
is the output of C,L]wavedec2.
The 'rigrsure', 'heursure', and
'minimaxi' denoising methods are only
applicable to signals.
To learn more about multiplicative threshold rescaling, see wden.
thr = wthrmngr(opt,method,swtdec,alpha)swtdec, of the signal or image to
denoise. alpha specifies the sparsity parameter
(see wbmpen). For
signals, swtdec is the output of swt. For images,
swtdec is the output of swt2.
Thresholds are derived from a subset of the coefficients in the stationary wavelet decomposition. For more information, see Coefficient Selection.
thr = wthrmngr(opt,method,swtdec,scale)scale. For signals,
swtdec is the output of swt. For images,
swtdec is the output of swt2.
Thresholds are derived from a subset of the coefficients in the stationary wavelet decomposition. For more information, see Coefficient Selection.
The 'rigrsure', 'heursure', and
'minimaxi' denoising methods apply only to
signals.
To learn more about multiplicative threshold rescaling, see wden.
thr = wthrmngr(opt,'rem_n0',X)X, using the specified wavelet option and
method 'rem_n0'.
If opt is 'dw1dcompGBL' or
'dw2dcompGBL', thresholds are based on the
finest-scale wavelet coefficients obtained using the Haar wavelet. If
opt is 'wp1dcompGBL' or
'wp2dcompGBL', thresholds are based on the
finest-scale wavelet packet coefficients obtained using the Haar
wavelet.
Examples
Global Threshold — Discrete Wavelet Decomposition
Load and plot a noisy signal.
load noisdopp plot(noisdopp) grid on title('Noisy Signal')
Generate a level 5 wavelet decomposition of the noisy signal using the order 4 Daubechies wavelet. Plot the coefficients.
[c,l] = wavedec(noisdopp,5,'db4'); plot(c) grid on title('Wavelet Coefficients')
Determine a global threshold for compressing the signal.
thr = wthrmngr('dw1dcompGBL','bal_sn',c,l);
The index of the first wavelet detail coefficient in c is l(1)+1. Apply the threshold to all the detail coefficients. Plot the thresholded coefficients. Observe that most of the coefficients have been set to 0.
c(l(1)+1:end) = c(l(1)+1:end).*(c(l(1)+1:end)>thr); plot(c) grid on title('Thresholded Coefficients')
Reconstruct the signal from the thresholded coefficients. Plot the reconstruction.
xrec = waverec(c,l,'db4'); plot(xrec) grid on title('Compressed Signal')
Image Compression — Birgé-Massart Thresholds
Compress an image using the Birgé-Massart strategy.
Load an image and add white Gaussian noise. For purposes of reproducibility, set the random seed to the default value.
rng default load sinsin x = X+18*randn(size(X));
Obtain the 2-D discrete wavelet transform down to level 3 using the Daubechies least-asymmetric wavelet with 4 vanishing moments. Obtain the compression thresholds using the Birgé-Massart strategy with sparsity parameter, alpha, equal to 2.
[C,L] = wavedec2(x,3,'sym4'); alpha = 2; THR = wthrmngr('dw2dcompLVL','scarcehi',C,L,alpha);
Compress the image and display the result.
xd = wdencmp('lvd',x,'sym4',3,THR,'s'); image(X) title('Original Image')
figure
image(x)
title('Noisy Image')
figure
image(xd)
title('Compressed Image')
Level-Dependent Threshold — Stationary Wavelet Transform
This example uses a level-dependent threshold derived from the wavelet coefficients at each scale to implement hard thresholding with the stationary wavelet transform.
Load the noisy blocks signal. Obtain the stationary wavelet transform down to level 5 by using the Haar wavelet.
load noisbloc L = 5; swc = swt(noisbloc,L,'haar');
Make a copy of the wavelet transform coefficients. Determine the Donoho-Johnstone universal threshold based on the detail coefficients for each scale. Using the 'mln' option, wthrmngr returns a 1-by-L vector, with every element equal to the universal threshold for the corresponding scale.
swcnew = swc; ThreshML = wthrmngr('sw1ddenoLVL','sqtwolog',swc,'mln');
Use the universal thresholds to implement hard thresholding. The thresholds are applied in a scale-dependent manner.
for jj = 1:L swcnew(jj,:) = wthresh(swc(jj,:),'h',ThreshML(jj)); end
Invert the stationary wavelet transform on the thresholded coefficients, swcnew. Plot the original signal and the denoised signal for comparison.
noisbloc_denoised = iswt(swcnew,'haar'); plot(noisbloc) hold on plot(noisbloc_denoised,'r','linewidth',2) legend('Original','Denoised')
Global Threshold — Wavelet Packet Decomposition
Denoise a noisy signal by applying a global threshold to a wavelet packet decomposition structure.
Load and plot a noisy signal.
load noisdopp plot(noisdopp) grid on title('Noisy Signal')
Generate a level 3 wavelet packet decomposition of the noisy signal using the order 4 Daubechies wavelet.
T = wpdec(noisdopp,3,'db4');Determine a global threshold for denoising the signal.
thr = wthrmngr('wp1ddenoGBL','sqtwologuwn',T);
Obtain the leaves from the wavelet packet decomposition tree T and apply the threshold to the leaves. Use hard thresholding.
T1 = T; sorh = 'h'; cfs = read(T,'data'); cfs = wthresh(cfs,sorh,thr); T1 = write(T1,'data',cfs);
Reconstruct the denoised signal from the thresholded coefficients. Plot the reconstruction.
xrec = wprec(T1); plot(xrec) grid on title('Denoised Signal')
Level-Independent Threshold — Stationary Wavelet Transform
This example uses a level-independent threshold based on the finest-scale wavelet coefficients to implement hard thresholding with the stationary wavelet transform.
Load the noisy blocks signal. Obtain the stationary wavelet transform down to level 5 by using the Haar wavelet.
load noisbloc L = 5; swc = swt(noisbloc,L,'haar');
Make a copy of the wavelet transform coefficients. Determine the Donoho-Johnstone universal threshold based on the first-level detail coefficients. Using the 'sln' option, wthrmngr returns a 1-by-L vector, with every element equal to the same value. Take the mean of the vector to obtain a scalar threshold.
swcnew = swc; ThreshSL = mean(wthrmngr('sw1ddenoLVL','sqtwolog',swc,'sln'));
Use the universal threshold to implement hard thresholding. The same threshold is applied to the wavelet coefficients at every level.
for jj = 1:L swcnew(jj,:) = wthresh(swc(jj,:),'h',ThreshSL); end
Invert the stationary wavelet transform on the thresholded coefficients, swcnew. Plot the original signal and the denoised signal for comparison.
noisbloc_denoised = iswt(swcnew,'haar'); plot(noisbloc) hold on plot(noisbloc_denoised,'r','linewidth',2) legend('Original','Denoised')
Input Arguments
Output Arguments
Tips
To denoise 1-D signals, consider using the Wavelet Signal Denoiser. The app visualizes and denoises real-valued 1-D signals using default parameters. You can also compare results. In addition, you can also recreate the denoised signal in your workspace by generating a MATLAB® script, which uses the
wdenoisefunction.
Algorithms
References
[1] Birgé, L., and P. Massart. “From Model Selection to Adaptive Estimation.” Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics (E. Torgersen, D. Pollard, and G. Yang, eds.). New York: Springer-Verlag, 1997, pp. 55–88.