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Lecture 6 : Level Set Method
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Introduction Developed by Books Stanley Osher (UCLA)
J. A. Sethian (UC Berkeley) Books J.A. Sethian: Level Set Methods and Fast Marching Methods, 1999 S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces , 2002
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Evolving Curves and Surfaces
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Geometry Representation
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Explicit Techniques for Evolution
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Explicit Techniques - Drawbacks
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Implicit Geometries
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Discretized Implicit Geometries
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Level Set Method: Overview
Generic numerical method for evolving fronts in an implicit form Handles topological changes of the evolving interface Define problem in 1 higher dimension Use an implicit representation of the contour C as the zero level set of higher dimensional function the level set function
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Level Set Method: Overview
Move the level set function, so that it deforms in the way the user expects contour = cross section at z=t
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Implicit Curve Evolution
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Level Set Evolution Define a speed function F, that specifies how contour points move in time Based on application-specific physics such as time, position, normal, curvature, image gradient magnitude Build an initial level set curve Adjust over time Current contour is defined as
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Equation for Level Set Evolution
Indirectly move C by manipulating where F is the speed function normal to the curve Level set equation
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Example: an expanding circle
Level Set representation of a circle Setting F=1 causes the circle to expand uniformly Observe everywhere We obtain Explicit solution: meaning the circle has radius r+t at time t
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Example: an expanding circle
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Motion under curvature
Complicated shapes? Each piece of the curve moves perpendicular to the curve with speed proportional to the curvature Since curvature can be either positive or negative , some parts of the curve move outwards while others move inwards Example movie file Setting F = curvature
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Level Set Segmentation
We may think of as signed distance function Negative inside the curve Positive outside the curve Distance function has unit gradient almost everywhere and smooth By choosing a suitable speed function F, we may segment an object in an image
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Level Set Segmentation
Evolving Geometry : F(X,t)=0 Intuitively, move a lot on low intensity gradient area and move little on high intensity gradient area along normal direction F : speed function , k : curvature , I : intensity
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Segmentation Example Arterial tree segmentation
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Discretization Use upwinded finite difference approximations (first order)
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Acceleration Techniques
Acceleration for fast level set method Narrow band level set method Fast marching method
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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 3D)
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Fast Marching Method Assume the front (level set) propagates always outward or always inward Compute T(x,y)=time at which the contour crosses grid point (x,y) At any height T, the surface gives the set of points reached at time T
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Fast Marching Algorithm
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Fast Marching Algorithm
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Fast Marching Method
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Applications Segmentation Level Set Surface Editing Operators
Surface Reconstruction
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Segmetation 2D 3D
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Level Set Surface Editing Operators
SIGGRAPH 2002
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Level Set Surface Editing Operators
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Surface Reconstruction
zhao, osher, and fedkiw 2001
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A painting interface for interactive surface deformations
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