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

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

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

4. Invariant arclength should be 1.Re-parameterization invariant 2.Invariant under the group of transformations Geometric measure Transform

5. Euclidean arclength qLength is preserved, thus,

6. Curvature flow qEuclidean geometric heat equation flow Euclidean transform

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

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

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

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

11. Constant flow qOffset curves qLevel sets of distance map qEqual-height contours of the distance transform qEnvelope of all disks of equal radius centered along the curve (Huygens principle)

12. Constant flow qOffset curves Change in topology Shock Cusp

13. Area inside C qArea is defined via

14. So far we defined qConstant flow qCurvature flow qEqui-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 color movies Goldenberg, Kimmel, Rivlin, Rudzsky, IEEE T-IP 2001

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

22. Surface qA surface, qFor example, in 3D qNormal qArea element qTotal area

23. Surface evolution qTangential velocity has no influence on the geometry qMean curvature flow, area minimizing

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

25. Conclusions qConstant flow, geometric heat equations uEuclidean uEqui-affine uOther data dependent flows qSurface evolution www.cs.technion.ac.il/~ron