Lecture 6 : Level Set Method
Introduction
• Developed by
– 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
Evolving Curves and Surfaces
Geometry Representation
Explicit Techniques for Evolution
Explicit Techniques - Drawbacks
Implicit Geometries
Discretized Implicit Geometries
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
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
Implicit Curve Evolution
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
Equation for Level Set Evolution
• Indirectly move C by manipulating
Level set equation
where F is the speed function normal to the curve
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
Example: an expanding circle
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
Level Set Segmentation
• We may think of
function
as signed distance
– 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
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
Segmentation Example
• Arterial tree segmentation
Discretization
• Use upwinded finite difference
approximations (first order)
Acceleration Techniques
• Acceleration for fast level set method
– Narrow band level set method
– Fast marching method
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)
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
Fast Marching Algorithm
Fast Marching Algorithm
Fast Marching Method
Applications
• Segmentation
• Level Set Surface Editing Operators
• Surface Reconstruction
Segmetation
• 2D
• 3D
Level Set Surface Editing Operators
• SIGGRAPH 2002
Level Set Surface Editing Operators
Surface Reconstruction
• zhao, osher, and fedkiw 2001
A painting interface for interactive
surface deformations