1 / 33

Lecture 6 : Level Set Method

Lecture 6 : Level Set Method. Introduction. Developed by Stanley Osher (UCLA) J. A. Sethian (UC Berkeley) Books J.A. Sethian: Level Set Methods and Fast Marching Methods, 1999 S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces , 2002. Evolving Curves and Surfaces.

fsawyer
Download Presentation

Lecture 6 : Level Set Method

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lecture 6 : Level Set Method

  2. Introduction • Developed by • Stanley Osher (UCLA) • J. A. Sethian (UC Berkeley) • Books • J.A. Sethian: Level Set Methods and Fast Marching Methods, 1999 • S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces , 2002

  3. Evolving Curves and Surfaces

  4. Geometry Representation

  5. Explicit Techniques for Evolution

  6. Explicit Techniques - Drawbacks

  7. Implicit Geometries

  8. Discretized Implicit Geometries

  9. Level Set Method: Overview • Generic numerical method for evolving fronts in an implicit form • Handles topological changes of the evolving interface • Define problem in 1 higher dimension • Use an implicit representation of the contour C as the zero level set of higher dimensional function - the level set function

  10. Level Set Method: Overview • Move the level set function, so that it deforms in the way the user expects • contour = cross section at z=t

  11. Implicit Curve Evolution

  12. Level Set Evolution • Define a speed function F, that specifies how contour points move in time • Based on application-specific physics such as time, position, normal, curvature, image gradient magnitude • Build an initial level set curve • Adjust over time • Current contour is defined as

  13. Equation for Level Set Evolution • Indirectly move C by manipulating where F is the speed function normal to the curve Level set equation

  14. Example: an expanding circle • Level Set representation of a circle • Setting F=1 causes the circle to expand uniformly • Observe everywhere • We obtain • Explicit solution: • meaning the circle has radius r+t at time t

  15. Example: an expanding circle

  16. Motion under curvature • Complicated shapes? • Each piece of the curve moves perpendicular to the curve with speed proportional to the curvature • Since curvature can be either positive or negative , some parts of the curve move outwards while others move inwards • Example movie file • Setting F = curvature

  17. Level Set Segmentation • We may think of as signed distance function • Negative inside the curve • Positive outside the curve • Distance function has unit gradient almost everywhere and smooth • By choosing a suitable speed function F, we may segment an object in an image

  18. Level Set Segmentation • Evolving Geometry : F(X,t)=0 • Intuitively, move a lot on low intensity gradient area and move little on high intensity gradient area along normal direction • F : speed function , k : curvature , I : intensity

  19. Segmentation Example • Arterial tree segmentation

  20. Discretization • Use upwinded finite difference approximations (first order)

  21. Acceleration Techniques • Acceleration for fast level set method • Narrow band level set method • Fast marching method

  22. Narrow band level set method • The efficiency comes from updating the speed function • We do not need to update the function over the whole image or volume • Update over a narrow band (2D or 3D)

  23. Fast Marching Method • Assume the front (level set) propagates always outward or always inward • Compute T(x,y)=time at which the contour crosses grid point (x,y) • At any height T, the surface gives the set of points reached at time T

  24. Fast Marching Algorithm

  25. Fast Marching Algorithm

  26. Fast Marching Method

  27. Applications • Segmentation • Level Set Surface Editing Operators • Surface Reconstruction

  28. Segmetation • 2D • 3D

  29. Level Set Surface Editing Operators • SIGGRAPH 2002

  30. Level Set Surface Editing Operators

  31. Surface Reconstruction • zhao, osher, and fedkiw 2001

  32. A painting interface for interactivesurface deformations

More Related