Transcription of The Level Set Method - MIT Mathematics
{{id}} {{{paragraph}}}
Example: quiz answers
The Level Set Method - MIT Mathematics
The Level Set MethodMIT / / Methods for Partial Differential EquationsPer-Olof Persson 8, 2005 Evolving Curves and Surfaces Propagate curve according to speed functionv=Fn Fdepends on space, time, and the curve itself Surfaces in three dimensionsFGeometry RepresentationsExplicit Geometry Parameterized boundaries(x, y) = (x(s), y(s))Implicit Geometry Boundaries given by zero Level set (x, y) = 0 (x, y)<0 (x, y)>0 Explicit Techniques Simple approach: Represent curve explicitly by nodesx(i)and lines Propagate curve by solving ODEsdx(i)dt=v(x(i), t),x(i)(0) =x(i)0, Normal vector, curvature, etc by difference approximations, :dx(i)ds x(i+1) x(i 1)2 s MATLAB DemoExplicit Techniques - Drawbacks Node redistribution required, introduces errors No entropy solution, sharp corners handled incorrectly Need special treatment for topology changes Stability constraints for curvature dependent speed functionsNode distributionSharp cornersTopology changesThe Level Set Method Implicit geometries, evolve interface by solving PDEs Invented in 1988 by Osher and Sethian: Stanley Osher and James A.
The Fast Marching Method • Use the fact that the front propagates outward • Tag known values and update neighboring T values (using the difference approximation) • Pick unknown with smallest T (will not be affected by other unknowns) • Update new neighbors and repeat until all nodes are known • Store unknowns in priority queue, O(nlogn) performance for n nodes
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: