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

5. 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  -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(n 3 ). 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

51. Characteristic hull: 0.40 ≤ l < 0.51

52. Characteristic hull: 0.39 ≤ l < 0.40

53. Characteristic hull: 0.34 ≤ l < 0.39

54. Characteristic hull: 0.28 ≤ l < 0.34

55. Characteristic hull: 0.25 ≤ l < 0.28

56. Characteristic hull: 0.23 ≤ l < 0.25

57. Characteristic hull: 0.22 ≤ l < 0.23

58. Characteristic hull: 0.00 ≤ l < 0.22

59. Properties of footprints obtained by the Characteristic Hull method No outliers Points on the boundary May not be topologically non-regular May not be disconnected Always polygonal May have large empty spaces May not have non-Jordan boundary

60. Properties of footprints obtained by the Characteristic Hull method Complexity is reported as O(n log n), but relies on regularity constraints See Duckham, Kulik, Galton, Worboys (in prep). Draft at http://www.duckham.org

61. General properties of Delaunay methods DT constrains solution space substantially more than SA and CP methods Lower bound of O(n log n) on DT methods Extensions to three dimensions may be problematic

62. Discussion “Correct” footprint is necessarily application specific, but some general properties can be identified Axiomatic definition of a hull operator does not accord well with these shapes Footprint formation and clustering are often conflated in methods