Level Set Methods An Overview

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Level Set Methods(An Overview)

• Amer Abufadel
• November 10, 2000

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Propagating Interfaces

• Ocean Waves
• Burning Flames
• Material Boundaries
• Shapes against background
• Handwriting recognition
• Image Contours and segmentation

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Overview

• Interface Propagation
• Boundary Value Problem
• Initial Value Problem
• Applications
• Results

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Interface Propagation
Inside
Outside
Outside
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Interface Propagation

• Speed Function F depends on
• L Local Properties (curvature, normal
  direction)
• G Global Properties (pressure, heat source)
• I Independent Properties (fluid velocity)

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One extra dimension
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Boundary Value Formulation
T
T 2
Y
T 1
T 0
T 1
T 0
T 2
X
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Boundary Value Formulation

• Assume F gt 0
• Front Moving Outward
• Crossing Time at
• (x,y) for expanding front F gt 0

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Initial Value Formulation
F(x,y,t2)
F0
F(x,y,t1)
F0
F(x,y,t0)
F0
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Initial Value Formulation

• F positive or negative
• Boundary can cross point more than once
• T(x,y) is not single valued
• Make the initial position as a zero level of a
  higher-dimensional function f
• Track evolution of f and determine the zero level
  set

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Advantages

• Unchanged in higher dimensions
• Topological changes are handled naturally
• Accurately approximated by computational schemes
• Intrinsic geometric properties are easily
  calculated
• Efficient

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Advantages

• Immune to topological changes

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Level Set Approach

• Boundary considered as the zero value of
  hypersurface
• As hypersurface moves under a speed function, the
  boundary moves as well
• Image based speed function will stop evolution
  near boundaries

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Speed Function

• FA is called the advection speed function
• FG is called the geometric speed function and is
  dependent on curvature of the surface

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Advection
t
A
x
B

• The domain of dependence of point A is point B
• The domain of influence of point B is the set of
  points on line A,B

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Sample Speed Function
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Speed Function Extension

• Speed function derived from the image is only
  valid at the boundary,i.e. where the function has
  zero value.
• What do we do about other points on the surface?
  Extend the speed function

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Speed Function Extension
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Speed Function Extension

• For every point P on the grid
• Find closest point on boundary Q
• Assign the value of the image based speed of the
  point Q to point P

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The Level Set Algorithm

• Start with an initial condition (position of the
  front at t0)
• Repeat until done
• Calculate the extension velocity
• Calculate the surface at time n1 from the
  current time n
• Calculate the level set at zero

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Level Set Method

• Update equation

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Level Set Method
The update equation can be simplified for easy
programming
where
Here
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Practical Issues

• Algorithm is very computationally intensive
• Can be done in a more efficient manner
• Narrow Band Level Set Method
• Fast Marching Methods

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Narrow Band Level Set Method

• The efficiency comes from updating the speed
  function.
• We do not need to update the function over the
  whole image or volume.
• Update over a narrow band (2D) or tube (3D)

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Narrow Band Level Set Method

• Update over a narrow band (2D) or tube (3D)

Update Band
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Results

• http//math.lbl.gov/malladi/

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My Research Interest

• Apply level sets to PET and SPECT reconstructions
• Isolate the blood pool
• Accurate ejection fraction calculation

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References

• Sethian J.A. Level Set Methods and Fast Marching
  Methods, Cambridge University Press, Cambridge
  UK, 1999
• Malladi R., Sethian J.A., and Vemuri B.C., Shape
  Modeling with Front Propagation A Level Set
  Approach IEEE Trans. On Pattern Analysis and
  Machine Intelligence, 17, , pp.158-175, 1995
• http//math.lbl.gov/malladi/