1 pde methods are not necessarily level set methods allen tannenbaum georgia institute of technology...
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PDE Methods are Not Necessarily
Level Set Methods
Allen TannenbaumGeorgia Institute of Technology
Emory University
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PDE Methods in Computer Vision and ImagingImage Enhancement
Segmentation Edge DetectionShape-from-ShadingObject RecognitionShape TheoryOptical FlowVisual TrackingRegistration
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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. thesisSigurd 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
moves “particles” along the curve
influences the curve’s shape
How is the speed determined?
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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
Kass snake (parametric)
Geodesic active contour (geometric)
using the functionals
Minimize
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leads to the Euler-Lagrange equations
Minimizing
Static Approaches
Kass snake (parametric)
Geodesic active contour (geometric)
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Static Approaches
Kass snake (parametric)
Geodesic active contour (geometric)
results in the gradient descent flow
Minimizing
is an artificial time parameter
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Dynamic Approach
Minimizethe actionintegral
where is the Lagrangian,
is the kinetic energy and is the potential energy.
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Dynamic Approach
using the functional
Minimizing
Terzopoulos and Szeliski (parametric)
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results in the Euler-Lagrange equation
Terzopoulos and Szeliski (parametric)
Minimizing
Dynamic Approach
Here, is physical time
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Dynamic Approach
But what about a geometric formulation?
results in the Euler-Lagrange equation
Terzopoulos and Szeliski (parametric)
Minimizing
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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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PDE’s 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
Calculus of variationsStart with initial curveDeform to minimize energySteady state is locally optimum
Dynamic programmingChoose seed point sFor any point t, determine globally
optimal curve t s
Registration,Atlas-basedsegmentation
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, W=N
Particle configuration space: N TN
Markov generator:
)()()( 12 fLfLNLf o
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Birth/Death Zero Range Processes-II
Markov generator:
)()()( 12 fLfLNLf o
)](2)()())[((2
1)( 1,1,
0 fffigfL ii
Ti
ii
N
elsej
iijj
iijj
jii
),(
0)(,,1)(
0)(,11)(
)(1,
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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:
)()()( 12 fLfLNLf o
NTi
ii ffidffibfL )]()())[(()]()())[(()( ,,1
elsej
ijjji
)(
1)()(,
elsej
iijjji
)(
,0)(,1)()(,
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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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ConclusionsLevel 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.