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

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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.

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Introduction Active Contour (Snake Model) –Kass, Witkin and Terzopoulos [Kas88] – - tension – - rigidity –E ext – external energy –Problem is: inf x E + Fast evaluation - But difficult to handle topological changes

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

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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

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

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Chan and Vese model position Chan and Vese model Energy functionals for image segmentation Representation/optimization Contour based gradient descent Level set based gradient descent 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

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

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Chan and Vese model c 1 and c 2 are the average intensity levels inside and outside of the contour Experiments:

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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(u 0 in/out) –C is the CV active contour Cartoon model

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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

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

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Replacing C with Φ The intensity terms

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

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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:

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Approximation of the Curvature

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The algorithm Initialization n=0 repeat –n++ –Computing c 1 and c 2 –Evolving the level-set function until the solution is stationary, or n>n max

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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

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Computing c 1 and c 2 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);

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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

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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!

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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!

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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( )

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Narrow band Initialization n=0 repeat –n++ –Determination of the narrow band –Computing c 1 and c 2 –Evolving the level-set function on the narrow band –Re-initialization until the solution is stationary, or n>n max

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

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

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Demonstration of the program Thanks for your attention

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

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