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

positiontangentdirection

localcost

globalcost

direction operatorcurve

localcost

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)

Minimization:Gradient flow (2)

The effect of the new term is to align the curve

with the preferred direction

preferreddirection

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

Minimization:Dynamic programming (2)

Optimal trajectories can be recovered from thecharacteristics 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”

EM gradient

neuron’smembrane

watermolecules

[Pichon, Westin & Tannenbaum, MICCAI 2005]

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 functione.g., f(x)=x3

Application:Diffusion MRI tractography (3)

2-d axial slice ofdiffusion data S(,kI0

)

Application:Diffusion MRI tractography (4)

proposedtechnique

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