Discussion :

[hamidakhemila]

j'ai une image IRM, je veux modéliser son histogramme par une mélange gaussienne. sachant que l'histogramme présente trois pics, c a d il faut trois gaussiennes pouar la modélisation. svp vous pouvez m'aider comment débuter le code. merci

[Gooby]

Bonjour,

Le sujet n'est pas des plus clair. Peux tu expliquer plus en détail ? Eventuellement avec des schémas?

Qu'est-ce que l'histogramme d'un IRM?
Qu'est-ce que tu appelles mélange gaussienne?
Comment peux tu savoir que l'histogramme présente 3 pics alors que c'est ce que tu désires modéliser?

[hamidakhemila]

bonjour
j'ai deux images cérébrales IRM , je veux les fusionner par la méthode probabiliste, pour cela je commence par prétraiter ces images(filtrage et recalage) ensuite, aprés segmenation par kmeans, je veux modéliser ses histogrammes par la mélange des gaussiennes.
par exemple: l'histogramme de IRM1 présente deux bosses, je veux les modéliser par la somme de deux gaussiennes.j'utilise ce code:

Code :

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function  [em_thr,em_thr_behavior,P,Mean,Std,pdf_x,xx,pdf_xx,cdf_xx,P_0] = em_1dim(x,nr_groups,fig_nr,P_0,max_nr_it);
 
% [COLOR="#FF0000"][COLOR="#FF0000"][COLOR="#FF0000"][COLOR="#FF0000"] Syntax:  
%
%  [mixture_parameters,mixture_density] = em_1dim(x,nr_groups,fig_nr,P_0,max_nr_it);
%
%  EM algo for 1-dimensional Gaussian Mixture Model
%
%  INPUT:
% -------
%
% x =  data  with  [nr_pts,nr_dim] = size(x)
%
% nr_groups = number of Gaussians
%
% P_0  =  (OPTIONAL) initial soft assignment of data points to different clusters
%          (matrix of size = (nr_pts,k)
%          If omitted, we use either 
%              * k-means to divide group in approx equal parts
%              * or home-made function (see end-of-file) to divide RANGE in
%                equal parts (this should be beneficial for distributions
%                that have long tails).
%
%  OUTPUT:
%  -------
%
%    mixture_parameters = structure that contains the Gaussian Mixture parameters:
%       mixture_parameters.props  =  priors for each gaussian (size = [nr_gauss,1]);
%       mixture_parameters.means  =  means for each gaussian (size = [nr_gauss,1]);
%       mixture_parameters.stds   =  std for each gaussian (size = [nr_gauss,1]);
%
%    mixture_density = structure that contains various density components for plotting:
%
%       mixture_density.xplot :  x-values for plotting
%       mixture_density.pdf :    pdf values for plotting;
%       mixture_density.cdf :    cdf values for plotting;
%
%
%
% History:
%  THIS IS SPECIALIZATION OF THE GENERAL  "em_ndim.m" algorithm FOR 1-DIM
%  DATA   (CREATED:   26 Nov 2003, EP)
% 
%  8 feb 04:  eliminated potential "divide by zero" problem when
%             renormalizing P  (EP)
%  11 may 04:  streamlined output (more extensive original version:  em_1dim_extra.m)
%
 
em_thr = 0;
em_thr_behavior = 0;
 
plot_ind = 0;
 
% Define default values wherever needed.
 
if size(x,1) < size(x,2), x = x'; end   % turn into column if need be
 
nr_pts = length(x);
 
if nargin < 3,  fig_nr = 0;  end      %  default: no plotting 
 
if nargin < 4  % construct initial assignment -----(-5)
 
   if 0   %nr_groups > 1  % use K-means to divide dataset in equal GROUPS
     P_0 = zeros(nr_pts,nr_groups);  
     disp('Creating initial assignment based on k-means clustering')
     [cluster_thr,cluster_label] = k_means_cluster_1dim(x,nr_groups);  
     for j = 1:nr_groups
         zz = find(cluster_label==j);
         P_0(zz,j) = 1;
     end
   elseif nr_groups >1  % divide RANGE in equal parts
 
     P_0 = initialize_em_range_based(x,nr_groups);    % See end of this file
 
   else % only one group: trivial assignment
       P_0 = ones(nr_pts,1);   
   end
 
end %----------------------------------------(-5)
 
if nargin < 5
  max_nr_it = 1000;
end
 
 
P = P_0;
Mean = zeros(nr_groups,1);
Std = ones(nr_groups,1);
 
green_light = 1;
 
nr_it = 0;
 
while (green_light == 1) & (nr_it < max_nr_it)  %  begin---------------(100)
 
   nr_it  = nr_it + 1;
 
   P_new = zeros(size(P));
 
   for k = 1 : nr_groups  %--------------------------(50)
 
       PP = P(:,k);
 
       D = x.* PP;    %  Data weighted with P-matrix
 
       if sum(P(:,k)) ~=0   % there are datapoints assigned to this group
 
           mean_grp = sum(D)/sum(PP);
           var_grp = sum(((x - mean_grp).^2).*PP)/sum(PP);    %  should this be sum(PP)-1 ??
	       std_grp = sqrt(var_grp);
        else
           mean_grp = 0;
           std_grp = 1;
        end
 
       F =  normpdf(x,mean_grp,std_grp);
 
       Mean(k,:) = mean_grp;
       Std(k,:) = std_grp(:)';
 
       P_new(:,k) = F;
 
    end  %--------------------------(50)
 
    P_old = P;
    P = P_new;
 
    %  Renormalize
 
    P_sum = sum(P,2);  PP_sum = P_sum *ones(1,nr_groups);
 
    % Precautions to avoid "divide by zero"
 
    u_zero = find(P_sum < 10^(-200)); %
 
    if ~isempty(u_zero)
        % create uniform distribution
        Q = zeros(nr_pts,nr_groups);  Q(u_zero,:) = 1/nr_groups;
        N  = ones(nr_pts,nr_groups);  N(u_zero,:)=0;
        PP_sum(u_zero,:) = 1;
        P = (P./PP_sum).*N  + Q;
    else
          P = P./(sum(P,2)*ones(1,nr_groups));   % 
    end 
 
  if  sum(sum(abs(P-P_old))) < 0.001*nr_pts
 
   %    disp(['Convergence occurred after '  int2str(nr_it)  ' iterations'])
       green_light = 0;
   end
 
end   % end -------------------------------------------(100)
 
% Compute stuff to SHOW DENSITY if requested
%===========================================
 
r = range(x);   
dx = r/250;
 
xx = (min(x)-0.05*r:dx:max(x)+0.05*r);
 
gx = zeros(nr_groups,length(x));
g = zeros(nr_groups,length(xx));
G = zeros(nr_groups,length(xx));
s = sum(P,1);
s = s/sum(s);
 
for i = 1:nr_groups
 
   g(i,:) = s(i)*(normpdf(xx,Mean(i),Std(i)));
   G(i,:) = s(i)*(normcdf(xx,Mean(i),Std(i)));
   gx(i,:) = s(i)*(normpdf(x',Mean(i),Std(i)));
end
 
 
g_tot = sum(g,1);
gx_tot = sum(gx,1);
G_tot = sum(G,1)/nr_pts;
 
pdf_x = gx_tot;
pdf_xx = g_tot;
cdf_xx = G_tot;
 
if fig_nr > 0 
	figure(fig_nr), clf
	hold on
	for k = 1:nr_groups
        plot(xx,g(k,:),'b')
	end
	plot(xx,pdf_xx,'r'),title('EM-based Gaussian Mixture density')
	hold off
	drawnow
end 
 
% 
% PACKAGE RESULTS FOR EXPORT
%===========================
 
% P is matrix of size [nr_pts,nr_gauss] such that each row contains the
% soft assignment of the corresponding point to the different Gaussian components. 
% By summing over the rows, we get the proportional contribution of each
% Gaussian. 
 
props = sum(P,1);
props = props'/nr_pts;
 
 
mixture_parameters.props = props;
mixture_parameters.means = Mean;
mixture_parameters.stds = Std;
 
% 
% thresholds.vals = em_thr;
% thresholds.behavior = em_thr_behavior;
% 
% local_mins.xvals = xmin;
% local_mins.fvals = fmin;
% local_mins.quality = qmin;
 
mixture_density.xplot = xx;
mixture_density.pdf = pdf_xx;
mixture_density.cdf = cdf_xx;
 
 
%  PLOT RESULTING DENSITY ON REQUEST
 
if plot_ind == 1
    [nbin,xbin]=hist(x,30);   dxbin = xbin(2)-xbin(1);
    total_mass = sum(nbin).*dxbin;
    figure(fignr), 
    hist(x,30);
    hold on
    plot(xx,total_mass*pdf_xx,'r')
    title('Proposed GMM density')
    hold off
    drawnow
end
 
 
return
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
function P_0 = initialize_em_range_based(x,nr_groups)
 
% divide the range up in equal parts (rather than the dataset itself, as is
% done by k-means).
% This should favour the creation of groups in long and slender tails
 
nr_pts = length(x);
P_raw = zeros(nr_pts,nr_groups);
 
xrange = range(x);
dx = xrange/(nr_groups-1);
ss = 0.5*xrange/(2*(nr_groups-1));
 
xc = min(x) + dx*(0:nr_groups-1);
 
for j = 1:nr_groups
 
    P_raw(:,j) = normpdf(x,xc(j),ss);  
end
 
%  make sure each row sums to 1:
 
P_0 = P_raw./(sum(P_raw,2)*ones(1,nr_groups));
 
return
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% TEST SUITE
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
n = 300;      % sample size
x1 = randn(n,1);
x2 = 0.5*randn(n,1)+3;
 
x = [x1; x2];   % mixture of two gaussians
 
figure(1), hist(x,30);
title('Input data')
drawnow
 
nr_gauss = 4
 
[mixture_parameters,mixture_density] = em_1dim(x,nr_gauss,77);
 
disp(['Extracted number of Gaussians: '  int2str(nr_gauss)])

mais je peux pas l'appliquer sur un histogramme d'une image, pouvez vous m'aider svp.