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Planar Curve Evolution

Explore the concepts of planar curve evolution, including re-parameterization invariance, length preservation, curvature flow, equi-affine flow, and geodesic active contours. Learn about their applications in geometric image processing and surface evolution.

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Planar Curve Evolution

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  1. Computer Science Department • Technion-Israel Institute of Technology Planar Curve Evolution Ron Kimmel www.cs.technion.ac.il/~ron • Geometric Image Processing Lab

  2. C =tangent p Planar Curves • C(p)={x(p),y(p)}, p [0,1] C(0.1) C(0.2) C(0.7) C(0) C(0.4) C(0.8) C(0.95) y C(0.9) x

  3. Arc-length and Curvature s(p)= | |dp C

  4. Geometric measure Invariant arclength should be • Re-parameterization invariant • Invariant under the group of transformations Transform

  5. Euclidean arclength • Length is preserved, thus ,

  6. Curvature flow • Euclidean geometric heat equation Euclidean transform flow

  7. Curvature flow Given any simple planar curve • Takes any simple curve into a circular point in finite time proportional to the area inside the curve • Embedding is preserved (embedded curves keep their order along the evolution). Grayson Vanish at a Circular point First becomes convex Gage-Hamilton

  8. Important property • Tangential components do not affect the geometry of an evolving curve

  9. re-parameterization invariance Reminder: Equi-affine arclength • Area is preserved, thus

  10. Affine heat equation • Special (equi-)affine heat flow Given any simple planar curve Sapiro Affine transform First becomes convex Vanish at an elliptical point flow

  11. Constant flow • Offset curves • Level sets of distance map • Equal-height contours of the distance transform • Envelope of all disks of equal radius centered along the curve (Huygens principle)

  12. Cusp Shock Constant flow • Offset curves Change in topology

  13. Area inside C • Area is defined via

  14. So far we defined • Constant flow • Curvature flow • Equi-affine flow We would like to explore evolution properties of measures like curvature, length, and area

  15. For Length Area Curvature

  16. Constant flow ( ) Length Area Curvature The curve vanishes at Riccati eq. Singularity (`shock’) at

  17. Curvature flow ( ) Length Area Curvature The curve vanishes at

  18. Equi-Affine flow ( ) Length Area Curvature

  19. Geodesic active contours Goldenberg, Kimmel, Rivlin, Rudzsky, IEEE T-IP 2001

  20. Tracking in colormovies Goldenberg, Kimmel, Rivlin, Rudzsky, IEEE T-IP 2001

  21. From curve to surface evolution • It’s a bit more than invariant measures…

  22. Surface • A surface, • For example, in 3D • Normal • Area element • Total area

  23. Surface evolution • Tangential velocity has no influence on the geometry • Mean curvature flow, area minimizing

  24. Segmentation in 3D Change in topology Caselles,Kimmel, Sapiro, Sbert, IEEE T-PAMI 97

  25. Conclusions • Constant flow, geometric heat equations • Euclidean • Equi-affine • Other data dependent flows • Surface evolution www.cs.technion.ac.il/~ron

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