Geodesic Active Contours in a Finsler Geometry Eric Pichon, John Melonakos, Allen Tannenbaum

1. Geodesic Active Contours in a Finsler GeometryEric Pichon, John Melonakos, Allen Tannenbaum

2. Conformal (Geodesic) Active Contours

3. Evolving Space Curves

4. Finsler Metrics

5. Some Geometry

6. Direction-dependent segmentation: Finsler Metrics global cost tangent direction local cost position direction operator curve local cost

7. Minimization:Gradient flow • Computing the first variation of the functional C, the L2-optimal C-minimizing deformation is: • The steady state ∞ is locally C-minimal projection (removes tangential component)

8. Minimization:Gradient flow (2) The effect of the new term is to align the curve with the preferred direction preferreddirection

9. Minimization:Dynamic programming Consider a seed region S½Rn, define for all target points t2Rn the value function: It satisfies the Hamilton-Jacobi-Bellman equation: curves between S and t

10. Minimization:Dynamic programming (2) Optimal trajectories can be recovered from the characteristics of : Then, is globally C-minimal between t0 and S.

11. Vessel Detection: Dynamic Programming-I

12. Vessel Detection: Noisy Images

13. Vessel Detection: Curve Evolution

14. Application:Diffusion MRI tractography • Diffusion MRI measures the diffusion of water molecules in the brain • Neural fibers influence water diffusion • Tractography: “recovering probable neural fibers from diffusion information” neuron’s membrane EM gradient watermolecules

15. Application:Diffusion MRI tractography (2) • Diffusion MRI dataset: • Diffusion-free image: • Gradient directions: • Diffusion-weighted images: • We choose: ratio = 1 if no diffusion < 1 otherwise Increasing function e.g., f(x)=x3 [Pichon, Westin & Tannenbaum, MICCAI 2005]

16. Application:Diffusion MRI tractography (3) 2-d axial slice of diffusion data S(,kI0)

17. Application:Diffusion MRI tractography (4) proposed technique streamline technique (based on tensor field) 2-d axial slide of tensor field (based on S/S0)

18. Interacting Particle Systems-I • Spitzer (1970): “New types of random walk models with certain interactions between particles” • Defn: Continuous-time Markov processes on certain spaces of particle configurations • Inspired by systems of independent simple random walks on Zd or Brownian motions on Rd • Stochastic hydrodynamics: the study of density profile evolutions for IPS

19. Interacting Particle Systems-II • Exclusion process: a simple interaction, precludes multiple occupancy --a model for diffusion of lattice gas • Voter model: spatial competition --The individual at a site changes opinion at a rate proportional to the number of neighbors who disagree • Contact process: a model for contagion --Infected sites recover at a rate while healthy sites are infected at another rate • Our goal: finding underlying processes of curvature flows

20. Motivations • Do not use PDEs • IPS already constructed on a discrete lattice (no discretization) • Increased robustness towards noise and ability to include noise processes in the given system

21. The Tangential Component is Important

22. Curve Shortening as Semilinear Diffusion-I

23. Curve Shortening as Semilinear Diffusion-II

24. Curve Shortening as Semilinear Diffusion-III

25. Nonconvex Curves

26. Stochastic Interpretation-I

27. Stochastic Interpretation-II

28. Stochastic Interpretation-III

29. Example of Stochastic Segmentation