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) 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.

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

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

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 

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 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

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 model c1 and c2 are the average intensity levels inside and outside of the contour Experiments:

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 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 Φ Introducing the Heaviside (sign) and Dirac (PSF) functions

Replacing C with Φ The intensity terms

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

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:

Approximation of the Curvature

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

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 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 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 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!

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!

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 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

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

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 Thanks for your attention

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