Computing Inner and Computing Inner and Outer Shape Outer Shape

ApproximationsApproximations

Joseph S.B. MitchellStony Brook University

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Talk OutlineTalk Outline Two classes of optimization problems Two classes of optimization problems

in shape approximation: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 approximationBest outer approximation

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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 ApproximationsPart I: Inner Approximations

Joint work with O. Hall-Holt, M. Katz, P. Kumar, A. Joint work with O. Hall-Holt, M. Katz, P. Kumar, A. Sityon (SODA’06) Sityon (SODA’06)

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1. Natural Optimization Problems

2. Shape Approximation

3. Visibility Culling for Computer Graphics

MotivationMotivation

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Max-Area Convex “Potato”Max-Area Convex “Potato”

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Max-Area Ellipse Inside Smooth Max-Area Ellipse Inside Smooth Closed CurvesClosed Curves

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Biggest French FryBiggest French Fry

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Longest StickLongest Stick

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Related Work: Largest Inscribed Related Work: Largest Inscribed BodiesBodies

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Convex Polygons on Point SetsConvex Polygons on Point Sets

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Related Work: Longest StickRelated Work: Longest Stick

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Our ResultsOur Results

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Approximating the Longest StickApproximating the Longest Stick Divide and conquerDivide and conquer Use balanced cuts (Chazelle)Use balanced cuts (Chazelle)

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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 StickApproximating the Longest Stick

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Algorithm:

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

Improved ApproximationImproved Approximation

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Pixels and the visibility problemPixels and the visibility problem

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Pixels and the visibility problemPixels and the visibility problem

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Visibility between pixelsVisibility between pixels

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Visibility between pixelsVisibility between pixels

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Visibility between pixels (cont)Visibility between pixels (cont)

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Approximating the Longest StickApproximating the Longest Stick

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Big PotatoesBig Potatoes

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Big FAT PotatoesBig FAT Potatoes

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Approx Biggest Convex PotatoApprox Biggest Convex Potato

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Approx Biggest Convex PotatoApprox Biggest Convex Potato

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Approx Biggest Convex PotatoApprox Biggest Convex Potato Goal: Find max-area Goal: Find max-area ee-anchored triangle-anchored triangle

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Approx Biggest Triangular PotatoApprox Biggest Triangular Potato

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Big FAT Triangles: PTASBig FAT Triangles: PTAS

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Big FAT Triangles: PTASBig FAT Triangles: PTAS

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Sampling ApproachSampling Approach

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Max-Area Triangle Using SamplingMax-Area Triangle Using Sampling

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Max-Area Triangle: Sampling Max-Area Triangle: Sampling DifficultyDifficulty

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Max-Area Ellipse Inside Sampled Max-Area Ellipse Inside Sampled CurvesCurves

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The Set of Maximal Empty EllipsesThe Set of Maximal Empty Ellipses

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3D Hueristics3D Hueristics ““Grow” k-dops from selected seed Grow” k-dops from selected seed

points: collision detection (QuickCD), points: collision detection (QuickCD), responseresponse

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SummarySummary

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Open ProblemsOpen Problems

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Part II: Outer ApproximationPart II: Outer Approximation

Joint work with E. Arkin, G. Barequet Joint work with E. Arkin, G. Barequet (SoCG’06)(SoCG’06)

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Bounding Volume HierarchyBounding Volume Hierarchy

BV-tree: BV-tree: Level 0Level 0 k-dopsk-dops

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BV-tree: Level 1BV-tree: Level 1

26-dops26-dops

14-dops14-dops6-dops6-dops

18-dops18-dops

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BV-tree: Level 2BV-tree: Level 2

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BV-tree: Level 5BV-tree: Level 5

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BV-tree: Level 8BV-tree: Level 8

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QuickCD: Collision DetectionQuickCD: Collision Detection

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The 2-Box Cover ProblemThe 2-Box Cover Problem Given set S of n points/polygonsGiven set S of n points/polygons Compute 2 boxes, Compute 2 boxes, BB11 and and BB22, to , to

minimize the combined measure, minimize the combined measure, f(Bf(B11,B,B22))• Measures: volume, surface area, Measures: volume, surface area,

diameter, width, girth, etcdiameter, width, girth, etc Choice of Choice of ff: :

Min-Sum, Min-Max, Min-UnionMin-Sum, Min-Max, Min-Union

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Related WorkRelated Work Min-max 2-box cover in d-D in time Min-max 2-box cover in d-D in time O(n log O(n log

n + nn + nd-1d-1)) [Bespamyatnik & Segal][Bespamyatnik & Segal] Clustering: Clustering: kk-center (min-max radius), -center (min-max radius), kk--

median (min-sum of dist), median (min-sum of dist), kk-clustering, -clustering, min-size min-size kk-clustering, core sets for approx-clustering, core sets for approx

Rectilinear 2-center: cover with 2 cubes of Rectilinear 2-center: cover with 2 cubes of min-max size: O(n) [LP-type]min-max size: O(n) [LP-type]

Min-size Min-size kk-clustering: min sum of radii -clustering: min sum of radii [Bilo+’05,Lev-Tov&Peleg’05,Alt+’05][Bilo+’05,Lev-Tov&Peleg’05,Alt+’05]• kk=2, 2D exact in =2, 2D exact in O(nO(n22/log log n)/log log n) [Hershberger][Hershberger]

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Lower BoundLower Bound

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Simple Exact AlgorithmSimple Exact Algorithm

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Simple Grid-Based SolutionSimple Grid-Based Solution

Look at Look at NN occupied voxels occupied voxels Solve 2-box cover exactly on them, Solve 2-box cover exactly on them,

exploiting special structureexploiting special structure

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Bad Case for Grid-Based SolutionBad Case for Grid-Based Solution

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

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CasesCases

Separable

Edge In

Crossing

Vertex In

Piercing

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Separable CaseSeparable Case Sweep in each of d Sweep in each of d

directions, O(n log n)directions, O(n log n) Swap each hit point Swap each hit point

from one box to the from one box to the otherother

Update: O(log n)/ptUpdate: O(log n)/pt

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Nonseparable CaseNonseparable Case Key idea: one of the boxes is “large” Key idea: one of the boxes is “large”

(at least ½) in at least half of its (at least ½) in at least half of its extents:extents:

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(x,y)-Projection Cases(x,y)-Projection Cases

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Discretizing in Discretizing in (x,y)(x,y) BB11 is “large” in (x,y) is “large” in (x,y) Can afford to round it out Can afford to round it out

to gridto grid How to determine its How to determine its zz--

extent?extent?

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Varying the Varying the zz-Extent-Extent

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Improvement for the Min-Max CaseImprovement for the Min-Max Case

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Higher DimensionsHigher Dimensions

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Min-UnionMin-Union

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Covering Polygons/PolyhedraCovering Polygons/Polyhedra

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Cardinaltiy Constraints: Balancing Cardinaltiy Constraints: Balancing the Partitionthe Partition

For building hierarchies, we want to For building hierarchies, we want to control the cardinalities of how many control the cardinalities of how many objects are covered by each of the 2 objects are covered by each of the 2 boxes:boxes:

OPEN: Can we do the volume measure in near-linear time?

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Bounding Volume HierarchiesBounding Volume Hierarchies

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Open ProblemsOpen Problems