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 tangent direction local cost position direction operator curve local cost

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 preferred direction

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 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 EM gradient 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