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

2 Talk Outline Two classes of optimization problems in shape approximation: 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 Best outer approximation

3 1.Motivation and summary of results 2.2D Approximation algorithms 1.Longest stick (line segment) 2.Max-area convex bodies (“potatoes”) 3.Max-area rectangles (“French fries), triangles 4.Max-area ellipses within well sampled curves 3.3D Heuristics Part I: Inner Approximations Joint work with O. Hall-Holt, M. Katz, P. Kumar, A. Sityon (SODA’06) Joint work with O. Hall-Holt, M. Katz, P. Kumar, A. Sityon (SODA’06)

4 1.Natural Optimization Problems 2.Shape Approximation 3.Visibility Culling for Computer Graphics Motivation

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 Divide and conquer Use balanced cuts (Chazelle) Use balanced cuts (Chazelle)

14

15

16

17 1.Compute weak visibility region from anchor edge (diagonal) e. 2.p has combinatorial type (u,v) 3.Optimize for each of the O(n) elementary intervals. Theorem: One can compute a ½-approximation for longest stick in a simple polygon in O(nlogn) time. 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. Open Problem: Can we get O(1)-approx in O(n) time? Approximating the Longest Stick

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

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 Goal: Find max-area e-anchored triangle

30 Approx Biggest Triangular Potato

31 Big FAT Triangles: PTAS

32 Big FAT Triangular Potatos

33 Big FAT Triangles: PTAS

34 Sampling Approach

35 Max-Area Triangle Using Sampling

36 Max-Area Triangle: Sampling Difficulty

37 Max-Area Ellipse Inside Sampled Curves

38 The Set of Maximal Empty Ellipses

39 3D Hueristics “Grow” k-dops from selected seed points: collision detection (QuickCD), response “Grow” k-dops from selected seed points: collision detection (QuickCD), response

40 Summary

41 Open Problems

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

43 Bounding Volume Hierarchy BV-tree: Level 0 k-dops

44 BV-tree: Level 1 26-dops 14-dops 6-dops 18-dops

45 BV-tree: Level 2

46 BV-tree: Level 5

47 BV-tree: Level 8

48 QuickCD: Collision Detection

49 The 2-Box Cover Problem Given set S of n points/polygons Given set S of n points/polygons Compute 2 boxes, B 1 and B 2, to minimize the combined measure, f(B 1,B 2 ) Compute 2 boxes, B 1 and B 2, to minimize the combined measure, f(B 1,B 2 ) Measures: volume, surface area, diameter, width, girth, etcMeasures: volume, surface area, diameter, width, girth, etc Choice of f: Choice of f: Min-Sum, Min-Max, Min-Union Min-Sum, Min-Max, Min-Union

50 Related Work Min-max 2-box cover in d-D in time O(n log n + n d-1 ) [Bespamyatnik & Segal] Min-max 2-box cover in d-D in time O(n log n + n d-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 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] 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] Min-size k-clustering: min sum of radii [Bilo+’05,Lev-Tov&Peleg’05,Alt+’05] k=2, 2D exact in O(n 2 /log log n) [Hershberger]k=2, 2D exact in O(n 2 /log log n) [Hershberger]

51 Lower Bound

52 Simple Exact Algorithm

53 Simple Grid-Based Solution Look at N occupied voxels Look at N occupied voxels Solve 2-box cover exactly on them, exploiting special structure Solve 2-box cover exactly on them, exploiting special structure

54 Bad Case for Grid-Based Solution

55 Minimizing Surface Area For surface area, grids do well For surface area, grids do well If OPT is separable, solve easily If OPT is separable, solve easily Otherwise, use following Lemma: Otherwise, use following Lemma:

56 Cases Separable Edge In Crossing Vertex In Piercing

57 Separable Case Sweep in each of d directions, O(n log n) Sweep in each of d directions, O(n log n) Swap each hit point from one box to the other Swap each hit point from one box to the other Update: O(log n)/pt Update: O(log n)/pt

58 Nonseparable Case Key idea: one of the boxes is “large” (at least ½) in at least half of its extents: Key idea: one of the boxes is “large” (at least ½) in at least half of its extents:

59 (x,y)-Projection Cases

60 Discretizing in (x,y) B 1 is “large” in (x,y) B 1 is “large” in (x,y) Can afford to round it out to grid Can afford to round it out to grid How to determine its z-extent? How to determine its z-extent?

61 Varying the z-Extent

62 Improvement for the Min-Max Case

63 Higher Dimensions

64 Min-Union

65 Covering Polygons/Polyhedra

66 Cardinaltiy Constraints: Balancing the Partition For building hierarchies, we want to control the cardinalities of how many objects are covered by each of the 2 boxes: For building hierarchies, we want to control the cardinalities of how many objects are covered by each of the 2 boxes: OPEN: Can we do the volume measure in near-linear time?

67 Bounding Volume Hierarchies

68 Open Problems