Comp 775: Deformable models:
snakes and active contours
Marc Niethammer, Stephen Pizer
Department of Computer Science
University of North Carolina, Chapel Hill
Deformable Models Motivating Example
Vessel Segmentation
If solution cannot easily be computed directly, iterative refine
a solution guess (e.g., by gradient descent). Methods based
on edge information, region information, statistics, etc.
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Deformable Models Motivating Example
Heart Segmentation
Image: Angenent et al.
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Deformable Models Motivating Example
Boundary Curve/Boundary Surface
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Deformable Models Parameterizations
Pixel/Voxel vs. Boundary Representation
Classifying individual pixels versus finding an optimal separating
curve/surface between object and background.
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Deformable Models Parameterizations
Parameterized Curve Evolution
Evolution should be geometric
Arclength is a special
parameterization, traversing
the curve with unit speed
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Deformable Models Parameterizations
Geometric Curve Evolution
The closed curve C evolves according to
influences the curve’s
shape
moves “particles” along
the curve
How is the speed
determined?
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Deformable Models Variational Approach
Curve Evolution through Energy Minimization
Find curve that minimizes a given energy
Static curve evolution
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Deformable Models Variational Approach
Curve Evolution
Minimize
using the functionals
Kass snake (parametric)
Geodesic active contour (geometric)
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Deformable Models Parameterizations
Curve Evolution
Minimizing
leads to the Euler-Lagrange equations
Kass snake (parametric)
Geodesic active contour (geometric)
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Deformable Models Variational Approach
Curve Evolution
Minimizing
results in the gradient descent flow
Kass snake (parametric)
Geodesic active contour (geometric)
is an artificial time parameter
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Deformable Models Variational Approach
Active Contour
Minimizing
leads to
Active contour (geometric)
is an artificial time parameter to solve a
static problem by gradient descent!
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Deformable Models Implementation
Particle-based approach
The curve is represented by a finite number of particles
Advantages
• easy to implement
• fast
Disadvantages
• topological changes
• particle spacing
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Deformable Models Parameterizations
Level Set Method
The curve is described implicitly as the zero level set of
a higher dimensional function
The curve is described by
The level set function evolves as
Only works for closed curves or surfaces of codimension one.
Osher, Sethian, "Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,"
Journal of Computational Physics, vol. 79, pp. 12-49, 1988.
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Deformable Models Parameterizations
Transporting Information
Flow information subject to the velocity field v.
Velocity field can for example be the curve evolution velocity.
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Deformable Models Implementation
Level Set Method
Image from http://math.berkeley.edu/~sethian/
Advantages
• topological changes
• higher dimensions
Disadvantages
• computational complexity (narrow-banding)
• velocity field extension
• restriction to the evolution of closed curves
or surfaces of codimension one
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Deformable Models Implementation
Level Set Method
Zero level set of the level set function Φ corresponds to
a curve in the plane.
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Deformable Models Implementation
Level Set Method: Some Evolution Examples
Curve evolutions with (left) and without (right) image information.
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Deformable Models Advanced Models
Mumford-Shah
[Image: Mumford]
The Mumford Shah model is a method that yields a
segmentation and at the same time a (piece-wise
smooth) image reconstruction.
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Deformable Models Advanced Models
Chan-Vese (=Otsu thresholding w/ spatial regularity)
Specialization of the Mumford-Shah model
• two segments (foreground/background)
• piecewise constant image models
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