PDE Methods are Not Necessarily Level Set Methods

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PDE Methods are Not NecessarilyLevel Set Methods

• Allen Tannenbaum
• Georgia Institute of Technology
• Emory University

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PDE Methods in Computer Vision and Imaging

• Image Enhancement
• Segmentation
• Edge Detection
• Shape-from-Shading
• Object Recognition
• Shape Theory
• Optical Flow
• Visual Tracking
• Registration

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Scale in Biological Systems
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Micro/Macro Models-Scale I
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Micro/Macro Models-Scale II
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How to Move Curves and Surfaces

• Parameterized Objects methods dominate control
  and visual tracking ideal for filtering and
  state space techniques.
• Level Sets implicitly defined curves and
  surfaces. Several compromises narrow banding,
  fast marching.
• Minimize Directly Energy Functional conjugate
  gradient on triangulated surface (Ken Brakke).

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Level Sets-A History
Independently Peter Olver (1976), Ph.D.
thesis Sigurd Angenent (Leiden University Report,
1982) Mathematical Justification Chen-Giga-Goto
(1991) Evans and Spruck (1991)
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When Do They Work
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Parameterized Curve Description
infinite dimensional
parameterization for derivations only, evolution
should be geometric
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Generic Curve Evolution
The closed curve C evolves according to
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Classification of Curve Evolutions
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Classification of Curve Evolutions
Kass, Witkin, Terzopoulos, "Snakes Active
Contour Models," International Journal of
Computer Vision, pp. 321-331, 1988.
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Classification of Curve Evolutions
Terzopoulos, Szeliski, Active Vision, chapter
Tracking with Kalman Snakes, pp. 3-20, MIT Press,
1992.
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Classification of Curve Evolutions
Kichenassamy, Kumar, Olver, Tannenbaum, Yezzi,
"Conformal curvature flows From phase
transitions to active vision," Archive for
Rational Mechanics and Analysis, vol. 134, no. 3,
pp. 275-301, 1996. Caselles, Kimmel, Sapiro,
"Geodesic active contours," International Journal
of Computer Vision, vol. 22, no. 1, pp. 61-79,
1997.
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Classification of Curve Evolutions
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Static Approaches
Minimize
using the functionals
Kass snake (parametric)
Geodesic active contour (geometric)
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Static Approaches
Kass snake (parametric)
Geodesic active contour (geometric)
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Static Approaches
Minimizing
results in the gradient descent flow
Kass snake (parametric)
Geodesic active contour (geometric)
is an artificial time parameter
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Dynamic Approach
Minimize the action integral
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Dynamic Approach
Minimizing
using the functional
Terzopoulos and Szeliski (parametric)
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Dynamic Approach
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Dynamic Approach
But what about a geometric formulation?
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Geometric Dynamic Approach
Minimize
using the Lagrangian
results in the Euler-Lagrange equation
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Geometric Dynamic Approach
We can write
We then obtain the following two coupled PDEs for
the tangential and the normal velocities
The tangential velocity matters.
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PDEs Without Level Sets Some Examples
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Cortical Surface Flattening-Normal Brain
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White Matter Segmentation and Flattening
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Conformal Mapping of Neonate Cortex
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Surface Warping-Area Preserving
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Flame Morphing
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Anisotropic active contours

• Add directionality

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Curve minimization
Registration, Atlas-basedsegmentation

• Calculus of variations
• Start with initial curve
• Deform to minimize energy
• Steady state is locally optimum
• Dynamic programming
• Choose seed point s
• For any point t, determine globallyoptimal curve
  t ? s

Segmentation
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Synthetic example (3D)
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Stochastic Approximations
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Curvature Driven Flows
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Euclidean and Affine Flows
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Euclidean and Affine Flows
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Birth/Death Zero Range Processes-I

• S discrete torus TN, WN
• Particle configuration space N TN
• Markov generator

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Birth/Death Zero Range Processes-II

• Markov generator

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Birth/Death Zero Range Process-III

• Markov generator
• Each particle configuration defines a positive
  measure on the unit circle
• To make the curve zero barycenter, a corrected
  measure is used
• Reconstruct the curve with

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The Tangential Component is Important
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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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Stochastic Curve Shortening
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Conclusions

• Level sets are a way of implementing curvature
  driven flows.
• Loss of information.
• Modifications are necessary.
• Do not work if no maximum principle.
• Combination with other methods, e.g. Bayesian.