Cedric Roux
56c66c09aa
Pre-ci test passed git-svn-id: http://svn.eurecom.fr/openair4G/trunk@4000 818b1a75-f10b-46b9-bf7c-635c3b92a50f
147 lines
5.9 KiB
Matlab
147 lines
5.9 KiB
Matlab
%% Rice/Rician Distribution Demo
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% Demonstrate ricernd, ricepdf, and ricestat, in the context of simulating
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% Rician distributed noise for Magnetic Resonance Imaging magnitude data.
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%% Example: Rician noise vs (additive) Gaussian noise
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% "Rician noise" depends on the data itself, it is not additive, so to
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% "add" Rician noise to data, what we really mean is make the data
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% Rician distributed.
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clear, close('all'), clc
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data = load('mri'); % MRI dataset (magnitude image) distributed with MATLAB
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im = double(data.D(:,:,1,16)); % (particular slice through the brain)
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s=5; % noise level (NB actual Rician stdev depends on signal, see ricestat)
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im_g = im + 5 * randn(size(im)); % *Add* Gaussian noise
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im_r = ricernd(im, s); % "Add" Rician noise (make Rician distributed)
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% Note that the Gaussian noise has made some data go negative.
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min_o = round(min(im(:))); max_o = round(max(im(:)));
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min_g = round(min(im_g(:))); max_g = round(max(im_g(:)));
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min_r = round(min(im_r(:))); max_r = round(max(im_r(:)));
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% Show each image with the same color scaling limits
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clim = [min_g max(max_g, max_r)];
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figure('Position', [30 500 800 300]); colormap(data.map);
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subplot(1,3,1); imagesc(im, clim); axis image;
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title('Original');
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xlabel(['Range: (' num2str(min_o) ', ' num2str(max_o) ')'])
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subplot(1,3,2); imagesc(im_g, clim); axis image;
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title('Gaussian noise');
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xlabel(['Range: (' num2str(min_g) ', ' num2str(max_g) ')'])
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subplot(1,3,3); imagesc(im_r, clim); axis image;
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title('Rician noise')
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xlabel(['Range: (' num2str(min_r) ', ' num2str(max_r) ')'])
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%% The Rician PDF
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x = linspace(0, 8, 100);
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figure;
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subplot(3, 1, 1)
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plot(x, ricepdf(x, 0, 1), x, ricepdf(x, 1, 1),...
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x, ricepdf(x, 2, 1), x, ricepdf(x, 4, 1))
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title('Rice PDF with s = 1')
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legend('v=0', 'v=1', 'v=2', 'v=4')
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subplot(3,1,2)
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plot(x, ricepdf(x, 1, 0.25), x, ricepdf(x, 1, 0.50),...
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x, ricepdf(x, 1, 1.00), x, ricepdf(x, 1, 2.00))
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title('Rice PDF with v = 1')
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legend('s=0.25', 's=0.50', 's=1.00', 's=2.00')
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subplot(3,1,3)
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plot(x, ricepdf(x, 0.5, 0.25), x, ricepdf(x, 1, 0.50),...
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x, ricepdf(x, 2.0, 1.00), x, ricepdf(x, 4, 2.00))
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title('Rice PDF with v/s = 2')
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legend('s=0.25', 's=0.50', 's=1.00', 's=2.00')
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%% Approximations to the Rician PDF
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% At high SNR (e.g. v/s > 3), Rician data is approximately Gaussian, though
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% note that v is not the mean of the Rician distribution, nor of the
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% Gaussian approximation (though for very high SNR it tends towards it).
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v = 1.75; s = 0.5; x = linspace(0, 4, 100);
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mu1 = v;
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gpdf1 = (2*pi*s^2)^(-0.5) * exp(-0.5*(x-mu1).^2/s^2);
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mu2 = ricestat(v, s); % (mu2 is approximately sqrt(v^2 + s^2))
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gpdf2 = (2*pi*s^2)^(-0.5) * exp(-0.5*(x-mu2).^2/s^2);
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figure; plot(x, ricepdf(x, v, s), x, gpdf1, x, gpdf2);
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title('High SNR');
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legend('Rician', 'Gaussian, naive mean', 'Gaussian, corrected mean')
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%%
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% At very low SNR, Rician data is approximately Rayleigh distributed
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% (with no signal, the distributions are exactly equivalent)
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v = 0.25; s = 0.5; x = linspace(0, 3, 100);
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raypdf = (x / s^2) .* exp(-x.^2 / (2*s^2));
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figure; plot(x, ricepdf(x, v, s), x, raypdf);
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title('Low SNR'); legend('Rician', 'Rayleigh')
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%%
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% At low-medium SNR, neither Gaussian nor Rayleigh is a great approximation
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v = 0.75; s = 0.5; x = linspace(-1, 3, 100);
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mu = ricestat(v, s);
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gpdf = (2*pi*s^2)^(-0.5) * exp(-0.5*(x-mu).^2/s^2);
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raypdf = (x / s^2) .* exp(-x.^2 / (2*s^2)); raypdf(x <= 0) = 0;
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figure; plot(x, ricepdf(x, v, s), x, gpdf, x, raypdf);
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title('Medium SNR'); legend('Rician', 'Gaussian', 'Rayleigh')
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%% A pitfal: Adding approximate Rician noise
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% Since the Rician distribution with zero signal is equivalent to the
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% Rayleigh, and with high SNR is approximated by a Gaussian, it is
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% tempting to add Rayleigh or Gaussian noise (depending on SNR) to existing
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% data, to simulate Rician distributed data (aka "adding Rician noise").
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v = 75; s = 50; % medium SNR => Add Rayleigh noise
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% Generate Rayleigh samples
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N = 10000;
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gsamp = s * randn(2, N);
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rsamp = sqrt(sum(gsamp .^ 2));
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%%
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% Compare data with added Rayleigh samples to expected Rician distribution
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r = v + rsamp; % Add Rayleigh noise to (constant) signal
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c = linspace(0, 250, 20);
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w = c(2); % histogram bin-width
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h = histc(r, c);
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figure; hold on; bar(c, h, 'histc');
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xl = xlim; x = linspace(xl(1), xl(2), 100);
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yl = ylim; % (for second plot)
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plot(x, N*w*ricepdf(x, v, s), 'r');
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title('Histogram of data, vs desired Rician PDF')
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legend('Data with added Rayleigh noise', 'Rician distribution')
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%%
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% Compare Rician samples to expected Rician distribution
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r = ricernd(v*ones(1, N), s);
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c = linspace(0, 250, 20);
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w = c(2); % histogram bin-width
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h = histc(r, c);
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figure; hold on; bar(c, h, 'histc');
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xl = xlim; x = linspace(xl(1), xl(2), 100);
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ylim(yl);
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plot(x, N*w*ricepdf(x, v, s), 'r');
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title('Histogram of data, vs desired Rician PDF')
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legend('Rician sampled data', 'Rician distribution')
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%%
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% We see that the additive Rayleigh result is very poor (with medium SNR).
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%
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% With high noise levels the appearance of results can differ dramatically:
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s = 0.15 * max_o;
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% Approximate additive Rayleigh/Gaussian noise:
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im_r = im;
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for i=1:numel(im_r)
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if im_r(i) <= 3*s
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im_r(i) = im_r(i) + s*norm(randn(2,1)); % Add Rayleigh noise
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else
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im_r(i) = im_r(i) + s*randn; % Add Gaussian noise
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end
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end
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% "Add" Rician noise (make Rician distributed):
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im_R = ricernd(im, s);
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% Show each image with the same color scaling limits
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min_r = round(min(im_r(:))); max_r = round(max(im_r(:)));
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min_R = round(min(im_R(:))); max_R = round(max(im_R(:)));
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clim = [min_r max(max_r, max_R)];
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figure('Position', [30 500 800 300]); colormap(data.map);
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subplot(1,3,1); imagesc(im, clim); axis image;
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title('Original');
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xlabel(['Range: (' num2str(min_o) ', ' num2str(max_o) ')'])
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subplot(1,3,2); imagesc(im_r, clim); axis image;
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title('Rayleigh/Gaussian noise')
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xlabel(['Range: (' num2str(min_r) ', ' num2str(max_r) ')'])
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subplot(1,3,3); imagesc(im_R, clim); axis image;
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title('Rician data');
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xlabel(['Range: (' num2str(min_R) ', ' num2str(max_R) ')'])
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%%
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% Copyright 2007 Ged Ridgway (Ged at cantab dot net)
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