Presentation on theme: "Active Contours without Edges"— Presentation transcript:
1 Active Contours without Edges Tony ChanLuminita VesePeter 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 - rigidityEext – external energyProblem is: infxE+ Fast evaluation- But difficult to handle topological changes
4 Introduction A typical external energy coming from the image: Positive on homogeneous regionsNear 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 functionImplicit 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 segmentationImportant to distinguish the model and the representationModel: describing problems from the real world with equationsRepresentation: type of the descriptionOptimization: solving the equationsChan and Vese modelRepresentation/optimizationContour based gradient descentLevel set based gradient descent
8 Chan and Vese modelThe model is based on trying to separate the image into regions based on intensitiesThe minimization problem:
9 Chan and Vese modelc1 and c2 are the average intensity levels inside and outside of the contourExperiments:
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 formulationConsidering the disadvantages of the active contour representation the model is solved using level set formulationlevel set form -> no explicit contour
12 Replacing C with ΦIntroducing the Heaviside (sign) and Dirac (PSF) functions
14 Average intensitiesWe 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 descentThe corresponding Euler-Lagrange equation:
17 The algorithm Initialization n=0 repeat Computing c1 and c2Evolving the level-set functionuntil the solution is stationary, or n>nmax
18 Initialization We set the values of the level set function outside = -1inside = 1Any shape can be the initialization shapeinit()for all (x, y) in Phiif (x, y) is insidePhi(x, y)=1;elsePhi(x, y)=-1;fi;end for
19 Computing c1 and c2The mean intensity of the image pixels inside and outsidecolors()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 bandIt is useful to compute the level set function not on the whole image domain but in a narrow band near to the contour. Abs()<dDecreasing the computational complexity.
24 Narrow band Initialization n=0 repeat Determination of the narrow bandComputing c1 and c2Evolving the level-set function on the narrow bandRe-initializationuntil 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 reachedStationary solution:The energy is not changingThe contour is not moving…
27 Demonstration of the program Thanks for your attention