Computing Inner and Outer Shape Approximations

1. Computing Inner and Outer Shape Approximations Joseph S.B. Mitchell Stony Brook University

2. Talk Outline • Two classes of optimization problems in shape approximation: • Finding “largest” subset of a body B of specified type • Best inner approximation • Finding “smallest” (tightest fitting) pair of bounding boxes • Best outer approximation

3. Part I: Inner Approximations • Motivation and summary of results • 2D Approximation algorithms • Longest stick (line segment) • Max-area convex bodies (“potatoes”) • Max-area rectangles (“French fries), triangles • Max-area ellipses within well sampled curves • 3D Heuristics • Joint work with O. Hall-Holt, M. Katz, P. Kumar, A. Sityon (SODA’06)

4. Motivation • Natural Optimization Problems • Shape Approximation • Visibility Culling for Computer Graphics

5. Max-Area Convex “Potato”

6. Max-Area Ellipse Inside Smooth Closed Curves

7. Biggest French Fry

8. Longest Stick

9. Related Work: Largest Inscribed Bodies

10. Convex Polygons on Point Sets

11. Related Work: Longest Stick

12. Our Results

13. Approximating the Longest Stick • Divide and conquer • Use balanced cuts (Chazelle)

17. Approximating the Longest Stick • Compute weak visibility region from anchor edge (diagonal) e. • p has combinatorial type (u,v) • Optimize for each of the O(n) elementary intervals. Algorithm: At each level of the recursive decomposition of P, compute longest anchored sticks from each diagonal cut: O(n) per level. Longest Anchored stick is at least ½ the length of the longest stick. Theorem: One can compute a ½-approximation for longest stick in a simple polygon in O(nlogn) time. Open Problem: Can we get O(1)-approx in O(n) time?

18. Improved Approximation Algorithm: Bootstrap from the O(1)-approx, discretize search space more finely, reduce to a visibility problem, and apply efficient data structures

19. Pixels and the visibility problem

20. Pixels and the visibility problem

21. Visibility between pixels

22. Visibility between pixels

23. Visibility between pixels (cont)

24. Approximating the Longest Stick

25. Big Potatoes

26. Big FAT Potatoes

27. Approx Biggest Convex Potato

28. Approx Biggest Convex Potato

29. Approx Biggest Convex Potato Goal: Find max-area e-anchored triangle

30. Approx Biggest Triangular Potato

31. Big FAT Triangles: PTAS

32. Big FAT Triangles: PTAS

33. Sampling Approach

34. Max-Area Triangle Using Sampling

35. Max-Area Triangle: Sampling Difficulty

36. Max-Area Ellipse Inside Sampled Curves

37. The Set of Maximal Empty Ellipses

38. 3D Hueristics • “Grow” k-dops from selected seed points: collision detection (QuickCD), response

39. Summary

40. Open Problems

41. Part II: Outer Approximation • Joint work with E. Arkin, G. Barequet (SoCG’06)

42. Bounding Volume Hierarchy BV-tree: Level 0 k-dops

43. BV-tree: Level 1 14-dops 6-dops 26-dops 18-dops

44. BV-tree: Level 2

45. BV-tree: Level 5

46. BV-tree: Level 8

47. QuickCD: Collision Detection

48. The 2-Box Cover Problem • Given set S of n points/polygons • Compute 2 boxes, B1 and B2, to minimize the combined measure, f(B1,B2) • Measures: volume, surface area, diameter, width, girth, etc • Choice of f: • Min-Sum, Min-Max, Min-Union

49. Related Work • Min-max 2-box cover in d-D in time O(n log n + nd-1)[Bespamyatnik & Segal] • Clustering: k-center (min-max radius), k-median (min-sum of dist), k-clustering, min-size k-clustering, core sets for approx • Rectilinear 2-center: cover with 2 cubes of min-max size: O(n) [LP-type] • Min-size k-clustering: min sum of radii [Bilo+’05,Lev-Tov&Peleg’05,Alt+’05] • k=2, 2D exact in O(n2/log log n)[Hershberger]

50. Lower Bound