Fast Marching Algorithm Minimal Paths Vida Movahedi Elder Lab, February 2010.
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Transcript of Fast Marching Algorithm Minimal Paths Vida Movahedi Elder Lab, February 2010.
Fast Marching Algorithm Fast Marching Algorithm & &
Minimal PathsMinimal Paths
Vida MovahediVida Movahedi
Elder Lab, February 2010Elder Lab, February 2010
ContentsContents
• Level Set Methods
• Fast Marching Algorithm
• Minimal Path Problem
Level set MethodsLevel set Methods• Problem: Finding the location of a moving
interface• For example: ‘edge of a forest fire’
Figure adapted from [2]
Level set MethodsLevel set Methods• Adding an extra dimension, “trade a moving
boundary problem for one in which nothing moves at all!”
• z= distance from (x,y) to the interface at t=0• Red: level set function, Blue: zero level set=
initial interface
Figure adapted from [2]
Level set MethodsLevel set Methods
0
0sinsincoscos
00),(),(
F
FFtt
yyt
xx
ttytx
t
t
02/122 yxt F
Figures adapted from [2]
Fast Marching MethodFast Marching Method• Special case of a front moving with speed F>0
everywhere• Fast marching algorithm is a numerical
implementation of this special case• Does not suffer from digitization bias, and is
guaranteed to converge to the true solution as the grid is refined
Figure adapted from [1]
Minimal PathMinimal Path• Inputs:
– Two key points– A potential function
to be minimized along the path
• Output:– The minimal path
Minimal Path- problem formulationMinimal Path- problem formulation• Global minimum of the active contour energy:
C(s): curve, s: arclength, L: length of curve
• Surface of minimal action U: minimal energy integrated along a path between p0 and p
Ap0,p : set of all paths between p0 and p
],0[
))((~)(L
dssCPCE
dssCPCEpUpoppop ΑΑ
)(~inf)(inf)(,,
Solving Minimal Path with Level Set methodsSolving Minimal Path with Level Set methods
Assume• initial interface= infinitesimal circle around Po • • Then U(p)= time the interface reaches p• • •
PF ~
1
PU ~
),(~~, yjxiPP ji
Fast Marching AlgorithmFast Marching Algorithm• Computing U by frontpropagation: evolving a front
starting from an infinitesimal circle around p0 until each point in image is reached
adapted from [5]
SummarySummary• Level Set Methods can be used to find the
location of moving interfaces• When F>0, Fast Marching Algorithm is a fast
numerical implementation for the Level Set Method
• In the Minimal Path Problem, U(p) (the surface of minimal energy) can be modeled as the time an infinitesimal interface around po reaches p– Fast Marching Algorithm can be used to find U
ReferencesReferences
[1] http://math.berkeley.edu/~sethian/2006/level_set.html[2] J.A. Sethian (1996), “Level Set Method: An Act of Violence“, American Scientist. [3] J.A. Sethian (1996) “A Fast Marching Level Set Method for Monotonically Advancing Fronts”, Proc. National Academy of Sciences, 93, 4, pp.1591-1595. [4] L.D. Cohen and R. Kimmel (1996), “Global Minimum for Active Contour Models: A Minimal Path Approach”, Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'96).[5] Laurent D. Cohen (2001), “Multiple Contour Finding and Perceptual Grouping using Minimal Paths”, Journal of Mathematical Imaging and Vision, vol. 14, pp. 225-236.