Active Contours without Edges. Tony Chan Luminita Vese. Peter Horvath – University of Szeged 29/09/2006 . Introduction. Variational approach The main problem is to minimise an integral functional (e.g.): In the case f: , f’=0 gives the extremum(s)
Peter Horvath – University of Szeged 29/09/2006
+ Fast evaluation
- But difficult to handle topological changes
+Automatically handles the topological changes
- Slower evaluation
Energy functionals for image segmentation
Chan and Vese model
Contour basedgradient descent
Level set basedgradient descent
Combining the above presented energy terms we can write the Chan and Vese functional as a function of Φ.
Minimization F wrt. Φ -> gradient descent
The corresponding Euler-Lagrange equation:
for all (x, y) in Phi
if (x, y) is inside
out = find(Phi < 0);
in = find(Phi > 0);
c1 = sum(Img(in)) / size(in);
c2 = sum(Img(out)) / size(out);
for all (x, y)
fx(x, y) = (Phi(x+1, y)-Phi(x-1, y))/(2*delta_s);
fy(x, y) =…
fxx(x, y) =…
fyy(x, y) =…
fxy(x, y) =…
delta_s recommended between 0.1 and 1.0
grad = (fx.^2.+fy.^2);
curvature = (fx.^2.*fyy + fy.^2.*fxx - 2.*fx.*fy.*fxy) ./ (grad.^1.5);
Be careful! Grad can be 0!
gradient_m = (fx.^2.+fy.^2).^0.5;
force = mu * curvature .* gradient_m - nu – lambda1 * (image - c1).^2 + lambda2 * (image - c2).^2;
We should normalize the force. abs(force) <= 1!
deltaT is recommended between 0.01 and 0.9. Be careful deltaT<1!
It is useful to compute the level set function not on the whole image domain but in a narrow band near to the contour. Abs()<d
Decreasing the computational complexity.
H is a normalizing term recommended between 0.1 and 2.
deltaT time step see above!