[MatLab] EM – Expectation Maximization reconstruction technique implementation

EM – Expectation Maximization is an iterative algorithm used in tomographic images (as in CT) reconstruction, very useful when the FBP – Filtered Back Projection technique is not applicable.

Basic formula is:

f_k+1 = (f_k / alpha) (At (g / (A f_k)))

where:

• f_k is solution (the resulting reconstructed image) at k-th iteration, at first iteration it is our guess
• g is image sinogram (what we get from the scanning)
• A f_k is Radon transform of f_k
• At is inverse Radon
• alpha is inverse Radon of a sinogram with all values 1 which represents our scanning machine

%EM - Expectation Maximization
%Formula is:
%f_k+1 = (f_k / alpha) (At (g / (A f_k)))
%f_k is solution at k-th iteration, at first iteration it is our guess
%g is image sinogram
%A f_k is Radon transform of f_k
%At is inverse Radon of its argument
%alpha is inverse Radon of a sinogram with all values 1 which represents our scanning machine

%clear matlab environment
clear all;
close all;
theta=[0:89]; %limited projection angle 90 degrees instead of 180
F=phantom(64); %create test phantom 64x64 pixels aka alien
figure, imshow(F,[]),title('Alien vs Predator'); %show alien
S1 = sum(sum(F));%calculate pixels sum on F
R = radon(F,theta); %apply Radon aka scan the alien to create a sinogram
figure, imshow(R,[]),title('Sinoalien');%show sinogram
%set all values <0 to 0
index=R<0;
R(index)=0;
n = 2000;%iterations
Fk=ones(64);%our initial guess
%create alpha aka pixels sum of the projection matrix (our scanning machine)
sinogramma1=ones(95,90);
alpha=iradon(sinogramma1, theta,'linear', 'none', 1,64);
%calculate relative error
relerrfact=sum(sum(F.^2));
for  k=1:n
     Afk = radon(Fk,theta);%create sinogram from current step solution
     %calculate g / A f_k=Noised./(A f_k+eps); aka initial sinogram / current step sinogram
     %eps is matlab thing to prevent 0 division
     GsuAFK=R./(Afk+eps);
     retro=iradon(GsuAFK, theta, 'linear', 'none', 1,64);%At (g / (A f_k))
     %multiply current step with previous step result and divide for alpha updating f_k
     ratio=Fk./alpha;
     Fk=ratio.*retro;
     %normalize
     St = sum(sum(Fk));
     Fk = (Fk/St)*S1;
	 %calculate step improvement
     Arrerr(k) = sum(sum((F - Fk).^2))/relerrfact;
	 %stop when there is no more improvement
     if((k>2) &&(Arrerr(k)>Arrerr(k-1)))
        break;
     end     
end
figure, imshow(Fk,[]),title('Fk');%show reconstructed alien
figure, plot(Arrerr),title('Arrerr');%show error improvement over all iterations

%compare our result with the one we would have had using the FBP - Filtered Back Projection
easy=iradon(R,theta, 'Shepp-Logan',1,64);
figure, imshow(easy,[]),title('FBP');

%calculate error between EM and FBP - with limited image size and projection degree FBP is bad!
FBPerr=sum(sum((F - easy).^2))/relerrfact;