How to find coordinates of corners in Matlab?

Question

How can I find the coordinates of the four corners in this "square" in Matlab? I tried with corner, regionprops('Extrema') and detectMinEigenFeatures with no luck. The problem is the blue square is not a perfect square, so I need some kind of sharpening edge technique or a more "intelligent" find-corner-algorithm to find them.

Answer 1

Because I'm a nice guy, I've translated the Tasos's explainations into code, check the comments :

%Open your image
I = imread('square.png');
I = im2bw(I(:,:,3));

%Compute the centroid
centroid = round(size(I)/2);

%Find the pixels that create the square
[x,y] = find(I);

%Change the origin
X = [y,x]-centroid;

%Sort the data
X = sortrows(X,[1 2]);

%Cartesian to polar coordinates
[theta,rho] = cart2pol(X(:,1),X(:,2));

%sort the polar coordinate according to the angle.
[POL,index] = sortrows([theta,rho],1);

%Smoothing, using a convolution
len = 15 %the smoothing factor
POL(:,2) = conv(POL(:,2),ones(len ,1),'same')./conv(ones(length(POL(:,2)),1),ones(len ,1),'same');

%Find the peaks
pfind = POL(:,2);
pfind(pfind<mean(pfind)) = 0;
[~,pos] = findpeaks(pfind);

%Change (again) the origin
X = X+centroid;


%Plot the result
plot(POL(:,1),POL(:,2))
hold on
plot(POL(pos,1),POL(pos,2),'ro')
figure
imshow(I)
hold on
plot(X(index(pos),1),X(index(pos),2),'ro')

RESULTS:

The distance (y-axis) vs angle (x-axis) plot:

The final detection: