Computing Inner and Outer Shape Approximations

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

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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 (SODA06)

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Motivation

• Natural Optimization Problems
• Shape Approximation
• Visibility Culling for Computer Graphics

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Max-Area Convex Potato
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Max-Area Ellipse Inside Smooth Closed Curves
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Biggest French Fry
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Longest Stick
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Related Work Largest Inscribed Bodies
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Convex Polygons on Point Sets
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Related Work Longest Stick
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Our Results
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Approximating the Longest Stick

• Divide and conquer
• Use balanced cuts (Chazelle)

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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?

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Improved Approximation

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

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Pixels and the visibility problem
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Pixels and the visibility problem
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Visibility between pixels
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Visibility between pixels
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Visibility between pixels (cont)
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Approximating the Longest Stick
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Big Potatoes
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Big FAT Potatoes
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Approx Biggest Convex Potato
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Approx Biggest Convex Potato
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Approx Biggest Convex Potato

• Goal Find max-area e-anchored triangle

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Approx Biggest Triangular Potato
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Big FAT Triangles PTAS
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Big FAT Triangular Potatos
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Big FAT Triangles PTAS
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Sampling Approach
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Max-Area Triangle Using Sampling
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Max-Area Triangle Sampling Difficulty
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Max-Area Ellipse Inside Sampled Curves
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The Set of Maximal Empty Ellipses
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3D Hueristics

• Grow k-dops from selected seed points
  collision detection (QuickCD), response

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Summary
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Open Problems
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Part II Outer Approximation

• Joint work with E. Arkin, G. Barequet (SoCG06)

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Bounding Volume Hierarchy
BV-tree Level 0
k-dops
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BV-tree Level 1
14-dops
6-dops
26-dops
18-dops
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BV-tree Level 2
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BV-tree Level 5
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BV-tree Level 8
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QuickCD Collision Detection
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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

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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
  Bilo05,Lev-TovPeleg05,Alt05
• k2, 2D exact in O(n2/log log n) Hershberger

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Lower Bound
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Simple Exact Algorithm
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Simple Grid-Based Solution

• Look at N occupied voxels
• Solve 2-box cover exactly on them, exploiting
  special structure

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Bad Case for Grid-Based Solution
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Minimizing Surface Area

• For surface area, grids do well
• If OPT is separable, solve easily
• Otherwise, use following Lemma

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Cases
Separable
Piercing
Crossing
Edge In
Vertex In
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Separable Case

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

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Nonseparable Case

• Key idea one of the boxes is large (at least
  ½) in at least half of its extents

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

• B1 is large in (x,y)
• Can afford to round it out to grid
• How to determine its z-extent?

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Varying the z-Extent
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Improvement for the Min-Max Case
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Higher Dimensions
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Min-Union
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Covering Polygons/Polyhedra
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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

OPEN Can we do the volume measure in
near-linear time?
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Bounding Volume Hierarchies
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Open Problems