Download - 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
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Conformal (Geodesic) Active Contours
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Evolving Space Curves
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Finsler Metrics
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Some Geometry
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Direction-dependent segmentation: Finsler Metrics
positiontangentdirection
localcost
globalcost
direction operatorcurve
localcost
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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)
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Minimization:Gradient flow (2)
The effect of the new term is to align the curve
with the preferred direction
preferreddirection
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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
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Minimization:Dynamic programming (2)
Optimal trajectories can be recovered from thecharacteristics of :
Then, is globally C-minimal between t0 and S.
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Vessel Detection: Dynamic Programming-I
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Vessel Detection: Noisy Images
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Vessel Detection: Curve Evolution
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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
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[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
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Application:Diffusion MRI tractography (3)
2-d axial slice ofdiffusion data S(,kI0
)
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Application:Diffusion MRI tractography (4)
proposedtechnique
streamline technique(based on tensor field)
2-d axial slide of tensor field (based on S/S0)
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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
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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
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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
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The Tangential Component is Important
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Curve Shortening as Semilinear Diffusion-I
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Curve Shortening as Semilinear Diffusion-II
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Curve Shortening as Semilinear Diffusion-III
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Nonconvex Curves
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Stochastic Interpretation-I
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Stochastic Interpretation-II
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Stochastic Interpretation-III
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Example of Stochastic Segmentation