Advanced Wavelet Thresholdings

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This numerical tour present some advanced method for denoising that makes use of some exoting 1D thresholding functions, that in some cases give better results than soft or hard thresholding.

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Generating a Noisy Image

Here we use an additive Gaussian noise.

First we load an image.

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n = 256;
name = 'hibiscus';
M0 = load_image(name,n);
M0 = rescale( sum(M0,3) );

Noise level.

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sigma = .08;

Then we add some Gaussian noise to it.

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M = M0 + sigma*randn(size(M0));

Compute a 2D orthogonal wavelet transform.

In [6]:
Jmin = 3;
MW = perform_wavelet_transf(M,Jmin,+1);

Semi-soft Thresholding

Hard and soft thresholding are two specific non-linear diagonal estimator, but one can optimize the non-linearity to capture the distribution of wavelet coefficient of a class of images.

Semi-soft thresholding is a familly of non-linearities that interpolates between soft and hard thresholding. It uses both a main threshold |T| and a secondary threshold |T1=mu*T|. When |mu=1|, the semi-soft thresholding performs a hard thresholding, whereas when |mu=infty|, it performs a soft thresholding.

In [7]:
T = 1; % threshold value
v = linspace(-5,5,1024);
plot(v, perform_thresholding(v,T,'hard'), 'b--');
plot(v, perform_thresholding(v,T,'soft'), 'r--');
plot(v, perform_thresholding(v,[T 2*T],'semisoft'), 'g');
plot(v, perform_thresholding(v,[T 4*T],'semisoft'), 'g:');
legend('hard', 'soft', 'semisoft, \mu=2', 'semisoft, \mu=4');

Exercise 1

Compute the denoising SNR for different values of |mu| and different value of |T|. Important: to get good results, you should not threshold the low frequency residual. list/sigma, mulist,

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In [9]:
%% Insert your code here.

One can display, for each |mu|, the optimal SNR (obtained by testing many different |T|). For |mu=1|, one has the hard thresholding, which gives the worse SNR. The optimal SNR is atained here for |mu| approximately equal to 6.

In [10]:
err_mu = compute_min(err, 2);
plot(mulist, err_mu, '.-');
set_label('\mu', 'SNR');

Soft and Stein Thresholding

Another way to achieve a tradeoff between hard and soft thresholding is to use a soft-squared thresholding non-linearity, also named a Stein estimator.

We compute the thresholding curves.

In [11]:
T = 1; % threshold value
v = linspace(-4,4,1024);

hard thresholding

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v_hard = v .* (abs(v)>T);

soft thresholding

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v_soft = v .* max( 1-T./abs(v), 0 );

Stein thresholding

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v_stein = v .* max( 1-(T^2)./(v.^2), 0 );

We display the classical soft/hard thresholders.

In [15]:
plot(v, v_hard, 'b');
plot(v, v_soft, 'r');
plot(v, v_stein, 'k--');
legend('Hard', 'Soft', 'Stein');

Exercise 2

Compare the performance of Soft and Stein thresholders, by determining the best threshold value.

In [16]:
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%% Insert your code here.