1. T-Snake Reference: Tim McInerney, Demetri Terzopoulos, T-snakes: Topology adaptive snakes, Medical Image Analysis, No.4 2000,pp73-91

2. Snake (review) Deformable model Continuous geometric models Make use of a priori knowledge of object shape Parametric representations of the models Support highly intuitive interaction mechanism Snake is classical deformable model

3. Snake (review) Strength T he ability to design energy functions or force functions to constrain and interactively guide the model. The ability to incorporate a priori knowledge.

4. Snake (review) Limitations Sensitive to initial conditions Topology of the object of interest must be known in advance. Fixed geometric parameterization with the internal deformation energy constraint limit the flexibility

5. T-Snake ACD based framework Affine Cell Decomposition

6. Overview T-snakes model A closed 2D contour consisting of a set of nodes connected in series. A discrete approximation to a conventional parametric snakes model.

7. Overview The set of nodes and interconnecting elements of a T-snake does not remain constant during its evolution. Decompose the image domain into a grid of discrete cells Reparameterize the model with a new set of nodes and elements by computing the intersection points with the grid. Keep track of the interior region of the model

8. How to convert… Conversion to the traditional parametric snakes model: Disable the ACD grid at any time during the evolution process

9. Model Description A T-snake is defined as a set of N nodes, indexed by I=0,…,N-1, connected in series by a set of N edges or elements Associate with the nodes time varying positions  i (t) = [x i (t), y i (t)], along with tensile forces  i (t), flexural forces  i (t), inflationary forces  i (t) and external forces  i (t).

10. Model Description First-order ordinary differential equations of motion  ’ i (t) + a  i (t) + b  i (t) =  i +  i Explicit first order Euler method

11. Affine Cell Decomposition A space decomposition subdivides space into a collection of disjoint connected subsets. Two types of ACD methods: nonsimplicial - rectangular simplicial – triangular

12. Simplicial approximation Freudenthal triangulation Partition: interior, exterior, boundary points Simplex classification test the ‘sign’ of the vertices Boundary cells line segment approximating the contour

13. Interactive reparameterization T-snake is reparameterized every M time steps of the numerical time integration. The entire T-snake is set to either expand or shrink during one deformation step. Two-phase reparameterization algorithm

14. Phase I Local search and intersection test of each T-snake element with the grid cell edges. Every intersection point is given a sign Compare against the existing intersection point ( if any ) of a grid cell edge Queue for Phase II.

15. Phase II we dequeue there vertices and check their corresponding grid cell edge data structures. If the grid vertex is off, we can turn it on. Entropy condition: Once a grid vertex is turned on, it remains on. region fill algorithm

16. Topological Transformation When: Collide with itself or another T-snake break into two or more parts Disconnecting or reconnecting nodes? Grid and the reparameterization process perform the reconnections.

17. T-snake Algorithm 1. For M time steps: (a) compute the external forces and internal forces acting on T-snake nodes and elements (b) update the node position 2. Phase I 3. Phase II

18. T-snake Algorithm 4. For all T-snake elements, check valid or not Valid if corresponding grid cell is still a boundary cell; Invalid T-snake elements and unused nodes are discarded 5. Use the grid vertices turned on in Phase II above ( if any) to determine new boundary cells and hence new T-snake nodes and elements

19. Termination When all of its elements have been inactive for a user-specified number of deformation steps. ACD grid is deactivated. The model run as a standard parametric snake