1. Contour Shape Analysis Using Crystalline Flow 2001.Nov.7

2. Crystalline Flow Evolution of a polygon

3. Crystalline Flow Outward Normal Velocity depends on Nonlocal Weighted Curvature

4. Curvature Inscribing Circle Wulff Shape

5. A Convex m-Polygon

6. Wulff Shape A Set of Unit Vectors:

7. Admissible Crystal A simple Polygon All outward normals belong to The normal of adjacent facet is parallel to the normal adjacent in the Wulff Shape Admissible Wulff Shape Not Admissible Jump!

8. Crystalline Flow : Nondecreasing in 2nd variable: Transition Number 0+1 : Length of facet of Wulff Shape: Length of i-th facet Nonlocal Curvature

9. Crystalline Flow Facet Disappearing at t = T* Case A: The polygon becomes convex near T* and all facet disappear at T*. All facets disappear at t = T*.

10. Crystalline Flow Facet Disappearing at t = T* Case B: Two parallel facets meet together.

11. Crystalline Flow is locally Lipschitz on is nondecreasing on for all Case B does not occur if

12. Crystalline Flow Facet Disappearing at t = T* Case C: At most two consecutive facets disappear.

13. Crystalline Flow 2 2 2 10

14. Crystalline Flow

15. Chain Coded Contour 0 1 2 3 4 5 6 7 0000 1 2 2 44444 2 2 7 000 3 3 1 0 Make given polygon Admissible

16. Crystalline Flow

17. Scale Space Analysis Facet Number in Original Time White: Convex Black: Concave

18. Facet Extraction Trace concave(convex) facets back to t=0. Extraction Scale Extracted Facets

19. Facet Extraction As the extraction scale increases, more important facets are extracted.

20. Facet Extraction

23. Conclusions Crystalline for Contour Shape Analysis Wulff Shape Selection!