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Advanced Numerical Methods for Interface Tracking in Fast-Marching Algorithm

Explore the Fast Marching Algorithm and Minimal Path Problem using Level Set Methods to locate moving interfaces efficiently. The algorithm ensures accuracy and convergence, making it ideal for various scenarios. Learn how to compute the surface of minimal energy and find minimal paths using this advanced approach.

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Advanced Numerical Methods for Interface Tracking in Fast-Marching Algorithm

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  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]

  5. Level set Methods Figures adapted from [2]

  6. 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]

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

  8. 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 p0 and p Ap0,p : set of all paths between p0 and p

  9. Solving Minimal Path with Level Set methods Assume • initial interface= infinitesimal circle around Po • Then U(p)= time the interface reaches p

  10. 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]

  11. 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 po reaches p • Fast Marching Algorithm can be used to find U

  12. 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.

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