1. Geodesic Active Contours in a Finsler Geometry Eric 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
preferred direction

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
water molecules

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