Active contours with open ends - matlab

Question

Does anyone know how to make an open active contour? I'm familiar with closed contours and I have several Matlab programs describing them. I've tried to change the matrix P but that was not enough: My Matlab code is as follows:
% Read in the test image
GrayLevelClump=imread('cortex.png');
cortex_grad=imread('cortex_grad.png');
rect=[120 32 340 340];
img=imcrop(GrayLevelClump,rect);
grad_mag=imcrop(cortex_grad(:,:,3),rect);
% Draw the initial snake as a line
a=300;b=21;c=21;d=307;
snake = brlinexya(a,b,c,d); % startpoints,endpoints
x0=snake(:,1);
y0=snake(:,2);
N=length(x0);
x0=x0(1:N-1)';
y0=y0(1:N-1)';
% parameters for the active contour
alpha = 5;
beta = 10;
gamma = 1;
kappa=1;
iterations = 50;
% Make a force field
[u,v] = GVF(-grad_mag, 0.2, 80);
%disp(' Normalizing the GVF external force ...');
mag = sqrt(u.*u+v.*v);
px = u./(mag+(1e-10)); py = v./(mag+(1e-10));
% Make the coefficient matrix P
x=x0;
y=y0;
N = length(x0);
a = gamma*(2*alpha+6*beta)+1;
b = gamma*(-alpha-4*beta);
c = gamma*beta;
P = diag(repmat(a,1,N));
P = P + diag(repmat(b,1,N-1), 1) + diag( b, -N+1);
P = P + diag(repmat(b,1,N-1),-1) + diag( b, N-1);
P = P + diag(repmat(c,1,N-2), 2) + diag([c,c],-N+2);
P = P + diag(repmat(c,1,N-2),-2) + diag([c,c], N-2);
P = inv(P);
d = gamma * (-alpha);
e = gamma * (2*alpha);
% Do the modifications to the matrix in order to handle OAC
P(1:2,:) = 0;
P(1,1) = -gamma;
P(2,1) = d;
P(2,2) = e;
P(2,3) = d;
P(N-1:N,:) = 0;
P(N-1,N-2) = d;
P(N-1,N-1) = e;
P(N-1,N) = d;
P(N,N) = -gamma;
x=x';
y=y';
figure,imshow(grad_mag,[]);
hold on, plot([x;x(1)],[y;y(1)],'r');
plot(x([1 end]),y(([1 end])), 'b.','MarkerSize', 20);
for ii = 1:50
% Calculate external force
vfx = interp2(px,x,y,'*linear');
vfy = interp2(py,x,y,'*linear');
tf=isnan(vfx);
ind=find(tf==1);
if ~isempty(ind)
vfx(ind)=0;
end
tf=isnan(vfy);
ind=find(tf==1);
if ~isempty(ind)
vfy(ind)=0;
end
% Move control points
x = P*(x+gamma*vfx);
y = P*(y+gamma*vfy);
%x = P * (gamma* x + gamma*vfx);
%y = P * (gamma* y + gamma*vfy);
if mod(ii,5)==0
plot([x;x(1)],[y;y(1)],'b')
end
end
plot([x;x(1)],[y;y(1)],'r')
I've modified the coefficient matrix P in order to handle cases with open ended active contours, but that is clearly not enough.
My image: