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| function [P_u, P_c, mu_u, mu_c, sigma2_u, sigma2_c] = ExpectationMaximizationAlgorithm(imDiff, alpha, NbIter, epsilon, flagDisp)
%
% [P_u, P_c, mu_u, mu_c, sigma2_u, sigma2_c] = ExpectationMaximizationAlgorithm(imDiff, alpha, NbIter, flagDisp)
% performs Expectation Maximization (EM) algorithm on input image imDiff to
% retrieve gaussians parameters
%
% IN:
% - imDiff: grayscale difference image
% - alpha: coefficient to set initial gaussian mixture model
% - NbIter: number of iterations
% - epsilon: difference (between two consecutive log-likelihood
% computations) to reach to exit EM loop
% - flagDisp: '0': no plot / '1': plot
%
% OUT:
% - P_u/P_c: a priori probabilities
% - mu_u (resp. mu_c): gaussians means
% - sigma2_u (resp. sigma2_c): gaussians variances
%
% REFERENCE(S):
% - "Automatic Analysis of the Difference Image for Unsupervised
% Change Detection", Lorenzo Bruzzone, Diego Fernandez Prieto,
% IEEE Transactions on Geoscience and Remote Sensing, VOL.38, NO.3
% May 2000
% - "Mixture densities, maximum likelihood and the EM algorithm",
% A.P. Redner & H.F. Walker, SIAM Review, VOL.26, NO.2, pp. 195-239,
% 1984
% - "Blobworld: Image Segmentation using Expectation-Maximization
% and its Application to Image Querying", Chad Carson & Hayit
% Greenspan, IEEE Transactions on Pattern Analysis and Machine
% Intelligence, VOL.24, NO.8, August 2002
%
% AUTHOR: XXXXXXXXX
imDiff=double(imDiff);
[M,N]=size(imDiff);
if flagDisp
figure
imagesc(imDiff)
title('imDiff')
colormap gray
end
% Compute normalized image histogram
histo=imhist(imDiff/255,256);
distrib_mat=[[0:255].' histo/(M*N)];
distrib_u=distrib_mat;
distrib_c=distrib_mat;
% Initialize values for P(t=0,wn), mu(t=0,wn), sigma^2(t=0,wn) by
% thresholding the previously computed histogram
Md=(max(max(imDiff))-min(min(imDiff)))/2;
Tu=floor(Md*(1-alpha));
Tc=floor(Md*(1+alpha));
% Means of 'changed' and 'unchanged' gaussian distributions
mu_c=sum(distrib_c(Tc:end,1).*distrib_c(Tc:end,2))/sum(distrib_c(Tc:end,2));
mu_u=sum(distrib_u(1:Tu,1).*distrib_u(1:Tu,2))/sum(distrib_u(1:Tu,2));
% Variances of 'changed' and 'unchanged' gaussian distributions
sigma2_c=sum(distrib_c(Tc:end,2).*((distrib_c(Tc:end,1)-mu_c).^2))/sum(distrib_c(Tc:end,2));
sigma2_u=sum(distrib_u(1:Tu,2).*((distrib_u(1:Tu,1)-mu_u).^2))/sum(distrib_u(1:Tu,2));
% Compute P(t+1,wn), mu(t+1,wn), sigma^2(t+1,wn) using
% Expectation-Maximization algorithm
% Reshape imDiff matrix to a M*N vector to avoid extra loop in the EM
% algorithm
imDiffLin=reshape(imDiff,1,M*N);
% Equal prior probabilities
P_u=1/2;
P_c=1/2;
fprintf(' Initial values: mu_c=%.4f, mu_u=%.4f, sigma2_c=%.4f, sigma2_u=%.4f, P_c=%.4f, P_u=%.4f \n',mu_c, mu_u, sigma2_c, sigma2_u, P_c, P_u);
loglike_previous=0;
for k=1:NbIter
% Compute p(X|w_u) and p(X|w_c)
pX_u=(sqrt(2*pi*sigma2_u))^(-1)*exp(-(imDiffLin-mu_u).^2/(2*sigma2_u));
pX_c=(sqrt(2*pi*sigma2_c))^(-1)*exp(-(imDiffLin-mu_c).^2/(2*sigma2_c));
% p(X) is a Gaussian Mixture (GM) model
pX=P_u*pX_u+P_c*pX_c;
% Compute p(w_u|X) and p(w_c|X) thanks to Bayes law
pu_X=P_u*(pX_u./pX);
pc_X=P_c*(pX_c./pX);
% Compute new prior probabilities for all clusters
P_u=(1/(M*N))*sum(pu_X);
P_c=(1/(M*N))*sum(pc_X);
% Compute new means
mu_u=sum(pu_X.*imDiffLin)/sum(pu_X);
mu_c=sum(pc_X.*imDiffLin)/sum(pc_X);
% Compute new variances
sigma2_u=sum(pu_X.*(imDiffLin-mu_u).*(imDiffLin-mu_u))/sum(pu_X);
sigma2_c=sum(pc_X.*(imDiffLin-mu_c).*(imDiffLin-mu_c))/sum(pu_X);
fprintf(' Iteration n°%.0f: mu_c=%.4f, mu_u=%.4f, sigma2_c=%.4f, sigma2_u=%.4f, P_c=%.4f, P_u=%.4f \n',k,mu_c, mu_u, sigma2_c, sigma2_u, P_c, P_u);
% Just to plot
pX_u_plot=(sqrt(2*pi*sigma2_u))^(-1)*exp(-(([0:255])-mu_u).^2/(2*sigma2_u));
pX_c_plot=(sqrt(2*pi*sigma2_c))^(-1)*exp(-(([0:255])-mu_c).^2/(2*sigma2_c));
% Check exiting condition by comparing log-likelihoods from previous
% and current iterations
loglike_current=sum(log(pX));
if abs(loglike_previous-loglike_current)<epsilon
break;
end
loglike_previous=loglike_current;
if flagDisp
figure(29)
plot(P_u*pX_u_plot,'b')
hold on
plot(P_c*pX_c_plot,'r')
hold on
bar(distrib_mat(:,2),'g')
hold off
axis([0 255 0 0.06])
pause(0.3)
end
end
end |
