1. Active Contours without Edges
Tony Chan
Luminita Vese
Peter Horvath – University of Szeged 29/09/2006

2. Introduction Variational approach
The main problem is to minimise an integral functional (e.g.):
In the case f:, f’=0 gives the extremum(s)
In the case of functionals similary F’=0, where F’=(F/u) is the first variation.
Most of the cases the solution is analyticly hard, in these cases we use gradient descent to optimise.

3. Introduction Active Contour (Snake Model) + Fast evaluation
Kass, Witkin and Terzopoulos [Kas88]
 - tension
 - rigidity
Eext – external energy
Problem is: infxE
+ Fast evaluation
- But difficult to handle topological changes

4. Introduction A typical external energy coming from the image:
Positive on homogeneous regions
Near zero on the sharp edges

5. Intoduction Level Set methods S.Osher and J. Sethian [Set89]
Embed the contour into a higher dimensional space
+Automatically handles the topological changes
- Slower evaluation
(., t) level set function
Implicit contour (=0)
The contour is evolved implicitly by moving the surface 

6. Introduction The curve is moving with an F speed:
The geometric active contour, based on a mean curvature (length) motion:

7. Chan and Vese model position
Energy functionals for image segmentation
Important to distinguish the model and the representation
Model: describing problems from the real world with equations
Representation: type of the description
Optimization: solving the equations
Chan and Vese model
Representation/optimization
Contour based gradient descent
Level set based gradient descent

8. Chan and Vese model
The model is based on trying to separate the image into regions based on intensities
The minimization problem:

9. Chan and Vese model
c1 and c2 are the average intensity levels inside and outside of the contour
Experiments:

10. Relation with the Mumford-Shah functional
The Chan and Vese model is a special case of the Mumford Shah model (minimal partition problem)
=0 and 1=2=
u=average(u0 in/out)
C is the CV active contour
“Cartoon” model

11. Level set formulation
Considering the disadvantages of the active contour representation the model is solved using level set formulation
level set form -> no explicit contour

12. Replacing C with Φ
Introducing the Heaviside (sign) and Dirac (PSF) functions

13. Replacing C with Φ
The intensity terms

14. Average intensities
We can calculate the average intensities using the step function

15. Level set formulation of the model
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:

16. Approximation of the Curvature

17. The algorithm Initialization n=0 repeat
Computing c1 and c2
Evolving the level-set function
until the solution is stationary, or n>nmax

18. Initialization We set the values of the level set function
outside = -1
inside = 1
Any shape can be the initialization shape
init()
for all (x, y) in Phi
if (x, y) is inside
Phi(x, y)=1;
else
Phi(x, y)=-1;
fi;
end for

19. Computing c1 and c2
The mean intensity of the image pixels inside and outside
colors()
out = find(Phi < 0);
in = find(Phi > 0);
c1 = sum(Img(in)) / size(in);
c2 = sum(Img(out)) / size(out);

20. Finite differences 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

21. Curvature Be careful! Grad can be 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!

22. Force We should normalize the force. abs(force) <= 1! Main step:
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!
Main step:
Phi=Phi+deltaT*force;
deltaT is recommended between 0.01 and 0.9. Be careful deltaT<1!

23. Narrow band
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.

24. Narrow band Initialization n=0 repeat
Determination of the narrow band
Computing c1 and c2
Evolving the level-set function on the narrow band
Re-initialization
until the solution is stationary, or n>nmax

25. Re-initialization Optional step
H is a normalizing term recommended between 0.1 and 2.
deltaT time step see above!

26. Stop criteria Stop the iterations if:
The maximum iteration number were reached
Stationary solution:
The energy is not changing
The contour is not moving …

27. Demonstration of the program
Thanks for your attention

28. MATLAB tutorial Imread, imwrite .*, .^, ./ Find
Size, length, max, min, mod, sum, zeros, ones
A(x:y, z:v)
Visualization: figure, plot, surf, imagesc