Parallel loops and image processing in matlab - image

Question

I am going to implement a salient object detection method based on a simple linear feedback control system (LFCS). The control system model is represented as in the following equation:
I've come up with the following program codes but the result would not be what should be. Specifically, the output should be something like the following image:
But the code produces this output:
The codes are as follows.
%Calculation of euclidian distance between adjacent superpixels stores in variable of Euc
A = imread('aa.jpg');
[rows, columns, cnumberOfColorChannels] = size(A);
[L,N] = superpixels(A,400);
%% Determination of adjacent superpixels
glcms = graycomatrix(L,'NumLevels',N,'GrayLimits',[1,N],'Offset',[0,1;1,0]); %Create gray-level co-occurrence matrix from image
glcms = sum(glcms,3); % add together the two matrices
glcms = glcms + glcms.'; % add upper and lower triangles together, make it symmetric
glcms(1:N+1:end) = 0; % set the diagonal to zero, we don't want to see "1 is neighbor of 1"
idx = label2idx(L); % Convert label matrix to cell array of linear indices
numRows = size(A,1);
numCols = size(A,2);
%%Mean color in Lab color space for each channel
data = zeros(N,3);
for labelVal = 1:N
redIdx = idx{labelVal};
greenIdx = idx{labelVal}+numRows*numCols;
blueIdx = idx{labelVal}+2*numRows*numCols;
data(labelVal,1) = mean(A(redIdx));
data(labelVal,2) = mean(A(greenIdx));
data(labelVal,3) = mean(A(blueIdx));
end
Euc=zeros(N);
%%Calculation of euclidian distance between adjacent superpixels stores in Euc
for a=1:N
for b=1:N
if glcms(a,b)~=0
Euc(a,b)=sqrt(((data(a,1)-data(b,1))^2)+((data(a,2)-data(b,2))^2)+((data(a,3)-data(b,3))^2));
end
end
end
%%Creation of Connectivity matrix "W" between adjacent superpixels
W=zeros(N);
W_num=zeros(N);
W_den=zeros(N);
OMG1=0.1;
for c=1:N
for d=1:N
if(Euc(c,d)~=0)
W_num(c,d)=exp(-OMG1*(Euc(c,d)));
W_den(c,c)=W_num(c,d)+W_den(c,c); %
end
end
end
%Connectivity matrix W between adjacent superpixels
for e=1:N
for f=1:N
if(Euc(e,f)~=0)
W(e,f)=(W_num(e,f))/(W_den(e,e));
end
end
end
%%calculation of geodesic distance between nonadjacent superpixels stores in variable "s_star_temp"
s_star_temp=zeros(N); %temporary variable for geodesic distance measurement
W_sparse=zeros(N);
W_sparse=sparse(W);
for g=1:N
for h=1:N
if W(g,h)==0 & g~=h;
s_star_temp(g,h)=graphshortestpath(W_sparse,g,h,'directed',false);
end
end
end
%%Calculation of connectivity matrix for nonadjacent superpixels stores in "S_star" variable"
S_star=zeros(N);
OMG2=8;
for i=1:N
for j=1:N
if s_star_temp(i,j)~=0
S_star(i,j)=exp(-OMG2*s_star_temp(i,j));
end
end
end
%%Calculation of connectivity matrix "S" for measuring connectivity between all superpixels
S=zeros(N);
S=S_star+W;
%% Defining non-isolation level for connectivity matrix "W"
g_star=zeros(N);
for k=1:N
g_star(k,k)=max(W(k,:));
end
%%Limiting the range of g_star and calculation of isolation cue matrix "G"
alpha1=0.15;
alpha2=0.85;
G=zeros(N);
for l=1:N
G(l,l)=alpha1*(g_star(l,l)- min(g_star(:)))/(max(g_star(:))- min(g_star(:)))+(alpha2 - alpha1);
end
%%Determining the supperpixels that surrounding the image boundary
lr = L([1,end],:);
tb = L(:,[1,end]);
labels = unique([lr(:);tb(:)]);
%% Calculation of background likelihood for each superpixels stores in"BgLike"
sum_temp=0;
temp=zeros(1,N);
BgLike=zeros(N,1);
BgLike_num=zeros(N);
BgLike_den=zeros(N);
for m=1:N
for n=1:N
if ismember(n,labels)==1
BgLike_num(m,m)=S(m,n)+ BgLike_num(m,m);
end
end
end
for o=1:N
for p=1:N
for q=1:N
if W(p,q)~=0
temp(q)=S(o,p)-S(o,q);
end
end
sum_temp=max(temp)+sum_temp;
temp=0;
end
BgLike_den(o,o)=sum_temp;
sum_temp=0;
end
for r=1:N
BgLike(r,1)= BgLike_num(r,r)/BgLike_den(r,r);
end
%%%%Calculation of Foreground likelihood for each superpixels stores in "FgLike"
FgLike=zeros(N,1);
for s=1:N
for t=1:N
FgLike(s,1)=(exp(-BgLike(t,1))) * Euc(s,t)+ FgLike(s,1);
end
end
The above codes are prerequisite for the following sections (in fact, they produce necessary data and matrices for the next section. The aforementioned codes provided to make the whole process reproducible).
Specifically, I think that this section did not give the desired results. I'm afraid I did not properly simulate the parallelism using for loops. Moreover, the terminating conditions (employed with for and if statements to simulate do-while loop) are never satisfied and the loops continue until the last iteration (instead terminating when a specified condition occurs). A major concern here is that if the terminating conditions are properly implemented.
The pseudo algorithm for the following code is as the image below:
%%parallel operations for background and foreground implemented here
T0 = 0 ;
Tf = 20 ;
Ts = 0.1 ;
Ti = T0:Ts:Tf ;
Nt=numel(Ti);
Y_Bg=zeros(N,Nt);
Y_Fg=zeros(N,Nt);
P_Back_Bg=zeros(N,N);
P_Back_Fg=zeros(N,N);
u_Bg=zeros(N,Nt);
u_Fg=zeros(N,Nt);
u_Bg_Star=zeros(N,Nt);
u_Fg_Star=zeros(N,Nt);
u_Bg_Normalized=zeros(N,Nt);
u_Fg_Normalized=zeros(N,Nt);
tau=0.1;
sigma_Bg=zeros(Nt,N);
Temp_Bg=0;
Temp_Fg=0;
C_Bg=zeros(Nt,N);
C_Fg=zeros(Nt,N);
%%System Initialization
for u=1:N
u_Bg(u,1)=(BgLike(u,1)- min(BgLike(:)))/(max(BgLike(:))- min(BgLike(:)));
u_Fg(u,1)=(FgLike(u,1)- min(FgLike(:)))/(max(FgLike(:))- min(FgLike(:)));
end
%% P_state and P_input
P_state=G*W;
P_input=eye(N)-G;
% State Initialization
X_Bg=zeros(N,Nt);
X_Fg=zeros(N,Nt);
for v=1:20 % v starts from 1 because we have no matrices with 0th column number
%The first column of X_Bg and X_Fg is 0 for system initialization
X_Bg(:,v+1)=P_state*X_Bg(:,v) + P_input*u_Bg(:,v);
X_Fg(:,v+1)=P_state*X_Fg(:,v) + P_input*u_Fg(:,v);
v=v+1;
if v==2
C_Bg(1,:)=1;
C_Fg(1,:)=1;
else
for w=1:N
for x=1:N
Temp_Fg=S(w,x)*X_Fg(x,v-1)+Temp_Fg;
Temp_Bg=S(w,x)*X_Bg(x,v-1)+Temp_Bg;
end
C_Fg(v-1,w)=inv(X_Fg(w,v-1)+((Temp_Bg)/(Temp_Fg)*(1-X_Fg(w,v-1))));
C_Bg(v-1,w)=inv(X_Bg(w,v-1)+((Temp_Fg)/(Temp_Bg))*(1-X_Bg(w,v-1)));
Temp_Bg=0;
Temp_Fg=0;
end
end
P_Bg=diag(C_Bg(v-1,:));
P_Fg=diag(C_Fg(v-1,:));
Y_Bg(:,v)= P_Bg*X_Bg(:,v);
Y_Fg(:,v)= P_Fg*X_Fg(:,v);
for y=1:N
Temp_sig_Bg=0;
Temp_sig_Fg=0;
for z=1:N
Temp_sig_Bg = Temp_sig_Bg +S(y,z)*abs(Y_Bg(y,v)- Y_Bg(z,v));
Temp_sig_Fg = Temp_sig_Fg +S(y,z)*abs(Y_Fg(y,v)- Y_Fg(z,v));
end
if Y_Bg(y,v)>= Y_Bg(y,v-1)
sign_Bg=1;
else
sign_Bg=-1;
end
if Y_Fg(y,v)>= Y_Fg(y,v-1)
sign_Fg=1;
else
sign_Fg=-1;
end
sigma_Bg(v-1,y)=sign_Bg*Temp_sig_Bg;
sigma_Fg(v-1,y)=sign_Fg*Temp_sig_Fg;
end
%Calculation of P_Back for background and foreground
P_Back_Bg=tau*diag(sigma_Bg(v-1,:));
P_Back_Fg=tau*diag(sigma_Fg(v-1,:));
u_Bg_Star(:,v)=u_Bg(:,v-1)+P_Back_Bg*Y_Bg(:,v);
u_Fg_Star(:,v)=u_Fg(:,v-1)+P_Back_Fg*Y_Fg(:,v);
for aa=1:N %Normalization of u_Bg and u_Fg
u_Bg(aa,v)=(u_Bg_Star(aa,v)- min(u_Bg_Star(:,v)))/(max(u_Bg_Star(:,v))-min(u_Bg_Star(:,v)));
u_Fg(aa,v)=(u_Fg_Star(aa,v)- min(u_Fg_Star(:,v)))/(max(u_Fg_Star(:,v))-min(u_Fg_Star(:,v)));
end
if (max(abs(Y_Fg(:,v)-Y_Fg(:,v-1)))<=0.0118) &&(max(abs(Y_Bg(:,v)-Y_Bg(:,v-1)))<=0.0118) %% epsilon= 0.0118
break;
end
end
Finally, the saliency map will be generated by using the following codes.
K=4;
T=0.4;
phi_1=(2-(1-T)^(K-1))/((1-T)^(K-2));
phi_2=(1-T)^(K-1);
phi_3=1-phi_1;
for bb=1:N
Y_Output_Preliminary(bb,1)=Y_Fg(bb,v)/((Y_Fg(bb,v)+Y_Bg(bb,v)));
end
for hh=1:N
Y_Output(hh,1)=(phi_1*(T^K))/(phi_2*(1-Y_Output_Preliminary(hh,1))^K+(T^K))+phi_3;
end
V_rs=zeros(N);
V_Final=zeros(rows,columns);
for cc=1:rows
for dd=1:columns
V_rs(cc,dd)=Y_Output(L(cc,dd),1);
end
end
maxDist = 10; % Maximum chessboard distance from image
wSF=zeros(rows,columns);
wSB=zeros(rows,columns);
% Get the range of x and y indices who's chessboard distance from pixel (0,0) are less than 'maxDist'
xRange = (-(maxDist-1)):(maxDist-1);
yRange = (-(maxDist-1)):(maxDist-1);
% Create a mesgrid to get the pairs of (x,y) of the pixels
[pointsX, pointsY] = meshgrid(xRange, yRange);
pointsX = pointsX(:);
pointsY = pointsY(:);
% Remove pixel (0,0)
pixIndToRemove = (pointsX == 0 & pointsY == 0);
pointsX(pixIndToRemove) = [];
pointsY(pixIndToRemove) = [];
for ee=1:rows
for ff=1:columns
% Get a shifted copy of 'pointsX' and 'pointsY' that is centered
% around (x, y)
pointsX1 = pointsX + ee;
pointsY1 = pointsY + ff;
% Remove the the pixels that are out of the image bounds
inBounds =...
pointsX1 >= 1 & pointsX1 <= rows &...
pointsY1 >= 1 & pointsY1 <= columns;
pointsX1 = pointsX1(inBounds);
pointsY1 = pointsY1(inBounds);
% Do stuff with 'pointsX1' and 'pointsY1'
wSF_temp=0;
wSB_temp=0;
for gg=1:size(pointsX1)
Temp=exp(-OMG1*(sqrt(double(A(pointsX1(gg),pointsY1(gg),1))-double(A(ee,ff,1)))^2+(double(A(pointsX1(gg),pointsY1(gg),2))-double(A(ee,ff,2)))^2 + (double(A(pointsX1(gg),pointsY1(gg),3))-double(A(ee,ff,3)))^2));
wSF_temp=wSF_temp+(Temp*V_rs(pointsX1(gg),pointsY1(gg)));
wSB_temp=wSB_temp+(Temp*(1-V_rs(pointsX1(gg),pointsY1(gg))));
end
wSF(ee,ff)= wSF_temp;
wSB(ee,ff)= wSB_temp;
V_Final(ee,ff)=V_rs(ee,ff)/(V_rs(ee,ff)+(wSB(ee,ff)/wSF(ee,ff))*(1-V_rs(ee,ff)));
end
end
imshow(V_Final,[]); %%Saliency map of the image

Answer 1

Part of your terminating criterion is this:
max(abs(Y_a(:,t)-Y_a(:,t-1)))<=eps
Say Y_a tends to 2. You are really close... In fact, the closest you can get without subsequent values being identical is Y_a(t)-Y_a(t-1) == 4.4409e-16. If the two values were any closer, their difference would be 0, because this is the precision with which floating-point values can be represented. So you have reached this fantastic level of closeness to your goal. Subsequent iterations are changing the target value by the smallest possible amount, 4.4409e-16. But your test is returning false! Why? Because eps == 2.2204e-16!
eps is short-hand for eps(1), the difference between 1 and the next representable larger value. Because how floating-point values are represented, this difference is half the difference between 2 and the next representable larger value (which is given by eps(2).
However, if Y_a tends to 1e-16, subsequent iterations could double or halve the value of Y_a and you'd still meet the stopping criterion!
Thus, what you need is to come up with a reasonable stopping criterion that is a fraction of the target value, something like this:
max(abs(Y_a(:,t)-Y_a(:,t-1))) <= 1e6 * eps(max(abs(Y_a(:,t))))
Unsolicited advice
You should really look into vectorized operations in MATLAB. For example,
for y=1:N
Temp_sig_a=0;
for z=1:N
Temp_sig_a = Temp_sig_a + abs(Y_a(y,t)- Y_a(z,t));
end
sigma_a(t-1,y)= Temp_sig_a;
end
can be written as
for y=1:N
sigma_a(t-1,y) = sum(abs(Y_a(y,t) - Y_a(:,t)));
end
which in turn can be written as
sigma_a(t-1,:) = sum(abs(Y_a(:,t).' - Y_a(:,t)));
Avoiding loops is not only usually more efficient, but it also leads to shorter code that is easier to read.
Also, this:
P_FB_a = diag(sigma_a(t-1,:));
u_a(:,t) = u_a(:,t-1) + P_FB_a * Y_a(:,t);
is the same as
u_a(:,t) = u_a(:,t-1) + sigma_a(t-1,:).' .* Y_a(:,t);
but of course creating a diagonal matrix and doing a matrix multiplication with so many zeros is much more expensive than directly computing an element-wise multiplication.