Transcript Core1.tannenbaum.north.carolina

Geodesic Active Contours in a Finsler
Geometry
Eric Pichon, John Melonakos, Allen Tannenbaum
Conformal (Geodesic) Active Contours
Evolving Space Curves
Finsler Metrics
Some Geometry
Direction-dependent segmentation: Finsler
Metrics
global
cost
curve
local
cost
local
cost
position
tangent
direction
direction
operator
Minimization:
Gradient flow
Computing the first variation of the functional C,
the L2-optimal C-minimizing deformation is:

projection (removes
tangential component)

The steady state ∞ is
locally C-minimal
Minimization:
Gradient flow (2)
The effect of the new term is to align the
curve
with the preferred direction
preferred
direction
Minimization:
Gradient flow (3)
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Evolution to Minkowski Spheres
Minimization:
Dynamic programming
Consider a seed region S½Rn, define
for all target points t2Rn the value function:
curves between S and t
It satisfies the Hamilton-Jacobi-Bellman equation:
Minimization:
Dynamic programming (2)
Optimal trajectories can be recovered from the
characteristics of :
Then,
is globally C-minimal between t0 and S.
Vessel Detection: Dynamic Programming-I
Vessel Detection: Noisy Images
Vessel Detection: Curve Evolution
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
water
molecules
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]
Application:
Diffusion MRI tractography (3)
2-d axial slice of
diffusion data S(,kI0)
Application:
Diffusion MRI tractography (4)
proposed
technique
streamline technique
(based on tensor field)
2-d axial slide of tensor
field (based on S/S0)
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
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
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
The Tangential Component is Important
Curve Shortening as Semilinear Diffusion-I
Curve Shortening as Semilinear Diffusion-II
Curve Shortening as Semilinear Diffusion-III
Nonconvex Curves
Stochastic Interpretation-I
Stochastic Interpretation-II
Stochastic Interpretation-III
Example of Stochastic Segmentation