Signal and Image Noise Models¶

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This numerical tour show several models for signal and image noise. It shows how to estimate the noise level for a Gaussian additive noise on a natural image. It also shows the relevance of thresholding to remove Gaussian noise contaminating sparse data.

In [35]:

warning off
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('solutions/denoisingsimp_1_noise_models')
warning on

Additive Gaussian Noise Model¶

The simplest noise model consist in adding a realization of a zero mean random vector to a clean signal or image.

Load a clean image.

In [3]:

N = 128;
name = 'boat';
M0 = load_image(name,256);
M0 = rescale(crop(M0,N));

Load a clean signal.

In [4]:

n = 1024;
name = 'piece-regular';
f0 = rescale( load_signal(name,n) );

The simplest noise model is Gaussian white noise. Here we generate a noisy signal or image.

In [5]:

sigma = .1;
M = M0 + randn(N,N)*sigma;
f = f0 + randn(n,1)*sigma;

Display the signals.

In [6]:

clf;
subplot(3,1,1);
plot(f0); axis([1 n 0 1]);
title('Clean signal');
subplot(3,1,2);
plot(f-f0); axis([1 n -3*sigma 3*sigma]);
title('Noise');
subplot(3,1,3);
plot(f); axis([1 n 0 1]);
title('Noisy signal');

Display the images.

In [7]:

clf;
imageplot(M0, 'Clean image', 1,3,1);
imageplot(M-M0, 'Noise', 1,3,2);
imageplot(clamp(M), 'Noisy image', 1,3,3);

Display the statistics of the noise

In [8]:

nbins = 51;
[h,t] = hist( M(:)-M0(:), nbins ); h = h/sum(h);
subplot(3,1,2);
bar(t,h);
axis([-sigma*5 sigma*5 0 max(h)*1.01]);

Additive Uniform Noise¶

A slightly different kind of noise is uniform in a given interval.

Generate noisy data with uniform noise distribution in |[-a,a]|, with |a| chosen so that the variance is sigma.

In [9]:

a = sqrt(3)*sigma;
M = M0 + 2*(rand(N,N)-.5)*a;
f = f0 + 2*(rand(n,1)-.5)*a;

Display the signals.

In [10]:

clf;
subplot(3,1,1);
plot(f0); axis([1 n 0 1]);
title('Clean signal');
subplot(3,1,2);
plot(f-f0); axis([1 n -3*sigma 3*sigma]);
title('Noise');
subplot(3,1,3);
plot(f); axis([1 n 0 1]);
title('Noisy signal');

Display the images.

In [11]:

clf;
imageplot(M0, 'Clean image', 1,3,1);
imageplot(M-M0, 'Noise', 1,3,2);
imageplot(clamp(M), 'Noisy image', 1,3,3);

Display the statistics of the noise

In [12]:

nbins = 51;
[h,t] = hist( M(:)-M0(:), nbins ); h = h/sum(h);
subplot(3,1,2);
bar(t,h);
axis([-sigma*5 sigma*5 0 max(h)*1.01]);

Impulse Noise¶

A very different noise model consist in sparse impulsions, generate by a random distribution with slowly decaying probability.

Generate noisy image with exponential distribution, with variance |sigma|.

In [13]:

W = log(rand(N,N)).*sign(randn(N,N)); 
W = W/std(W(:))*sigma;
M = M0 + W;

Generate noisy signal with exponential distribution, with variance |sigma|.

In [14]:

W = log(rand(n,1)).*sign(randn(n,1)); 
W = W/std(W(:))*sigma;
f = f0 + W;

Display the signals.

In [15]:

clf;
subplot(3,1,1);
plot(f0); axis([1 n 0 1]);
title('Clean signal');
subplot(3,1,2);
plot(f-f0); axis([1 n -3*sigma 3*sigma]);
title('Noise');
subplot(3,1,3);
plot(f); axis([1 n 0 1]);
title('Noisy signal');

Display the images.

In [16]:

clf;
imageplot(M0, 'Clean image', 1,3,1);
imageplot(M-M0, 'Noise', 1,3,2);
imageplot(clamp(M), 'Noisy image', 1,3,3);