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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)

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active contours without edges

Active Contours without Edges

Tony Chan

Luminita Vese

Peter Horvath – University of Szeged 29/09/2006

introduction
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.
introduction3
Introduction
  • Active Contour (Snake Model)
    • Kass, Witkin and Terzopoulos [Kas88]
    •  - tension
    •  - rigidity
    • Eext – external energy
    • Problem is: infxE

+ Fast evaluation

- But difficult to handle topological changes

introduction4
Introduction
  • A typical external energy coming from the image:
    • Positive on homogeneous regions
    • Near zero on the sharp edges
intoduction
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 
introduction6
Introduction
  • The curve is moving with an F speed:
  • The geometric active contour, based on a mean curvature (length) motion:
chan and vese model position
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 basedgradient descent

Level set basedgradient descent

chan and vese model
Chan and Vese model
  • The model is based on trying to separate the image into regions based on intensities
  • The minimization problem:
chan and vese model9
Chan and Vese model
  • c1 and c2 are the average intensity levels inside and outside of the contour
  • Experiments:
relation with the mumford shah functional
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
level set formulation
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
replacing c with
Replacing C with Φ
  • Introducing the Heaviside (sign) and Dirac (PSF) functions
replacing c with13
Replacing C with Φ
  • The intensity terms
average intensities
Average intensities
  • We can calculate the average intensities using the step function
level set formulation of the model
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:

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

computing c 1 and c 2
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);

finite differences
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

curvature
Curvature

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!

force
Force

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!

narrow band
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.

narrow band24
Narrow band
  • Initialization n=0
  • repeat
    • n++
    • 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
re initialization
Re-initialization
  • Optional step

H is a normalizing term recommended between 0.1 and 2.

deltaT time step see above!

stop criteria
Stop criteria
  • Stop the iterations if:
    • The maximum iteration number were reached
    • Stationary solution:
      • The energy is not changing
      • The contour is not moving
demonstration of the program
Demonstration of the program
  • Thanks for your attention
matlab tutorial
MATLAB tutorial
  • Imread, imwrite
  • .*, .^, ./
  • Find
  • Size, length, max, min, mod, sum, zeros, ones
  • A(x:y, z:v)
  • Visualization: figure, plot, surf, imagesc