1. Fast Marching Algorithm & Minimal Paths Vida Movahedi Elder Lab, February 2010

2. Contents Level Set Methods Fast Marching Algorithm Minimal Path Problem

3. Level set Methods Problem: Finding the location of a moving interface For example: ‘edge of a forest fire’ Figure adapted from [2]

4. Level 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]

6. Level set Methods Figures adapted from [2]

7. Fast 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]

8. Minimal Path Inputs: –Two key points – A potential function to be minimized along the path Output: –The minimal path

9. Minimal 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 p 0 and p A p0,p : set of all paths between p0 and p

10. Solving Minimal Path with Level Set methods Assume initial interface= infinitesimal circle around P o Then U(p)= time the interface reaches p

11. Fast 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]

12. Summary 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 p o reaches p –Fast Marching Algorithm can be used to find U

13. References [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.