What is the Region Occupied by a Set of Points?

1. What is the Region Occupied by a Set of Points? Antony Galton University of Exeter, UK Matt Duckham University of Melbourne, Australia

2. The General Problem To assign a region to a set of points, in order to represent the location or configuration of the points as an aggregate, abstracting away from the individual points themselves.

3. Example: Generalisation

4. Example: Generalisation

5. Example: Clustering

6. Example: Clustering

7. Evaluation Criteria

8. Are outliers allowed?

9. Must the points lie in the interior?

10. Can the region be topologically non-regular?

11. Can the region be disconnected?

12. Can the boundary be curved?

13. Can the boundary be non-Jordan?

14. How much ‘empty space’ is allowed?

15. Questions about method • How easily can the method be generalised to three (or more) dimensions? • What is the computational complexity of the algorithm?

16. Other criteria • Perceptual • Cognitive • Aesthetic • … We do not consider these!

17. Why not use the Convex Hull?

18. The ‘C’ shape is lost!

19. A non-convex region is better

20. Another Example

21. Convex hull is connected

22. Non-convex shows two ‘islands’

23. Edelsbrunner’s a-shape • H. Edelsprunner, D. Kirkpatrick and R. Seidel, ‘On the Shape of a Set of Points in the Plane’, IEEE Transactions on Information Theory, 1983.

24. A-Shape • M. Melkemi and M. Djebali, ‘Computing the shape of a planar points set’, Pattern Recognition, 2000.

25. DSAM Method • H. Alani, C. B. Jones and D. Tudhope,‘Voronoi-based region approximation for geographical information retrieval with gazeteers’, IJGIS, 2001

26. The Swinging Arm Method

27. A set of points …

28. Their convex hull …

29. The swinging arm

30. Non-convex hull: r = 2

31. Non-convex hull: r = 3

32. Non-convex hull: r = 4

33. Non-convex hull: r = 5

34. Non-convex hull: r = 6

35. Non-convex hull: r = 6(Anticlockwise)

36. Non-convex hull: r = 7

37. Non-convex hull: r = 7(anticlockwise)

38. Non-convex hull: r = 8

39. Convex Hull (r=17.117…)

40. Properties of footprints obtained by the swinging arm method • No outliers • Points on the boundary • May be topologically non-regular • May be disconnected • Always polygonal (possibly degenerate) • May have large empty spaces • May have non-Jordan boundary

41. Properties of the swinging arm method • Does not generalise straightforwardly to 3D (must use a ‘swinging flap’). • Complexity could be as high as O(n3). • Essentially the same results can be obtained by the ‘close pairs’ method (see paper).

42. Delaunay triangulation methods

43. Characteristic hull: 0.98 ≤ l ≤ 1.00

44. Characteristic hull: 0.91 ≤ l < 0.98

45. Characteristic hull: 0.78 ≤ l < 0.91

46. Characteristic hull: 0.64 ≤ l < 0.78

47. Characteristic hull: 0.63 ≤ l < 0.64

48. Characteristic hull: 0.61 ≤ l < 0.63

49. Characteristic hull: 0.56 ≤ l < 0.61

50. Characteristic hull: 0.51 ≤ l < 0.56