1. Deformable Models Segmentation methods until now (no knowledge of shape: Thresholding Edge based Region based Deformable models Knowledge of the shape of the object

2. Deformable Models Initial shape (curve or surface) Move the shape according: Internal forces (curve/surface properties) E.g.: Curvature to keep the object smooth External forces (image properties) E.g.: To track the object to the boundary 2D Snakes Kass, Witkin and Terzopoulos 1987

3. Example Animation with a 2D countour adapting to the edge

4. Deformable Models Various names for the same: 2D snakes, active contours, and deformable contours... 3D ballons, active surfaces, and deformable surfaces... Two main groups Parametric deformable models (based on parametric form of models) Geometric deformable models (curve evolution or level sets)

5. Parametric Contour CurveExample: Internal Energy Functional (internal forces) Potential Energy Functional (external forces) Moves to minimize the energy functional:

6. Internal Energy Functional Elasticity, avoids stretching Curvature are often defined as constants

7. Potential Energy Functional Energy based on features of the image Potential field based on borders. We want small values close to borders, e.g.:

8. Potential Field Example Image smoothed by convolving with a Gaussian

9. Solution Find that is local minimum With previous definition of forces. Variational calculus problem solve by Euler-Lagrange equation: We want to obtain a force balance

10. Potential field

11. Parametric Deformable Models

12. Parametric Deformable Model Curve moves in time Solve by gradient descent Find local minima

13. Dynamic Force Formulation Formulation as a dynamic problem Allows definition of extra external forces Based on Newton’s second law :

14. External Forces Combination of different forces Pressure Force (ballon)

15. Extending Attraction Range Multiscale Gaussian potential forces Distance Potential Force

16. Problem with Cavities Distance Potential Force

17. Problems with Cavities Gradient Vector Flow Diffuse the vector field according to strength of the edges Dynamic Distance Force Use of signed distances Interactive Variations User interaction modeled as force terms

18. Implementation We work with in a discrete form: The derivatives need to be approximated by, for example, finite differences

19. Problems of Active Contours Initial contour needs to be close to final result Reparameterization is needed (3D is difficult) Topological adaptation (3D is complicated) Forces definition, parameter setting