1 / 28

Active Contours without Edges

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)

jana
Download Presentation

Active Contours without Edges

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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) • 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 basedgradient descent Level set basedgradient 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 • n++ • 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 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 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 • 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

  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

More Related