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Computing Inner and Outer Shape Approximations
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Computing Inner and Outer Shape Approximations

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  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)

  14. 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?

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

  16. Pixels and the visibility problem

  17. Pixels and the visibility problem

  18. Visibility between pixels

  19. Visibility between pixels

  20. Visibility between pixels (cont)

  21. Approximating the Longest Stick

  22. Big Potatoes

  23. Big FAT Potatoes

  24. Approx Biggest Convex Potato

  25. Approx Biggest Convex Potato

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

  27. Approx Biggest Triangular Potato

  28. Big FAT Triangles: PTAS

  29. Big FAT Triangles: PTAS

  30. Sampling Approach

  31. Max-Area Triangle Using Sampling

  32. Max-Area Triangle: Sampling Difficulty

  33. Max-Area Ellipse Inside Sampled Curves

  34. The Set of Maximal Empty Ellipses

  35. 3D Hueristics • “Grow” k-dops from selected seed points: collision detection (QuickCD), response

  36. Summary

  37. Open Problems

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

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

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

  41. BV-tree: Level 2

  42. BV-tree: Level 5

  43. BV-tree: Level 8

  44. QuickCD: Collision Detection

  45. 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

  46. 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]

  47. Lower Bound