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.