geodesic active contours in a finsler geometry eric pichon, john melonakos, allen tannenbaum

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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. - PowerPoint PPT Presentation

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  • 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 Metricspositiontangentdirectionlocalcostglobalcostdirection operatorcurvelocalcost

  • Minimization:Gradient flowComputing the first variation of the functional C,the L2-optimal C-minimizing deformation is:

    The steady state is locally C-minimalprojection (removes tangential component)

  • Minimization:Gradient flow (2)The effect of the new term is to align the curvewith the preferred directionpreferred direction

  • Minimization:Dynamic programmingConsider a seed region SRn, 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 tractographyDiffusion MRI measures the diffusion of water molecules in the brainNeural fibers influence water diffusionTractography: recovering probable neural fibers from diffusion informationEM gradientneuronsmembranewater molecules

  • Application:Diffusion MRI tractography (2)Diffusion MRI dataset:Diffusion-free image:

    Gradient directions:

    Diffusion-weighted images:

    We choose: [Pichon, Westin & Tannenbaum, MICCAI 2005]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)proposedtechniquestreamline technique(based on tensor field)2-d axial slide of tensor field (based on S/S0)

  • Interacting Particle Systems-ISpitzer (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-IIExclusion 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