1. Discrete Approach to Curve and Surface Evolution
Longin Jan Latecki
Dept. of Computer and Information Science
Temple University
Philadelphia

2. Discrete Curve Evolution P=P0,
Discrete Curve Evolution P=P0, ..., Pm Pi+1 is obtained from Pi by deleting the vertices of Pi that have minimal relevance measure K(v, Pi) = K(u,v,w) = |d(u,v)+d(v,w)-d(u,w)| v v w w u u

3. Discrete Curve Evolution: Preservation of position, no blurring

4. Discrete Curve Evolution: robustness with respect to noise

5. Discrete Curve Evolution: extraction of linear segments

6. Parts of Visual Form (Siddiqi, Tresness, and Kimia 1996) = maximal convex arcs

7. Discete Cureve Evolution is used in shape similarity retrieval in image databases

8. Shape similarity measure based on correspondence of visual parts

9. A video sequence is mapped to a trajectory in a high dimensional space, e.g. by mapping each frame to a feature vector in R37 Discrete curve evolution allows us to determine key frames

10. Trajectory Simplification
2379 vertices
20 vertices

11. Mr. Bean

12. The 10 most relevant frames in Mr. Bean www.videokeyframes.de

13. Detection of unpredictable events in videos: Mov3.mpg

14. Alarm threshold = avg rel + 0.1*max. rel

15. Detection of unpredictable events in videos: seciurity1.mpg

16. Two most unpredictable frames extracted from Mov3.mpg

17. Alarm threshold = avg rel + 0.1*max. rel

18. Two most unpredictable frames extracted from security1.mpg

19. Discrete Surface Evolution: repeated removal of least relevant vertices

23. (Lyche and Morken in late 80s): Surface patch f:R2 -> R is represented with radial base splines S given a set of knots T: ||f – G(T)(f)|| = min{||f - g||: g in S} Surface evolution by knot removal Relevance measure of the knot: r(t) = ||f – G(T – {t})(f)||