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Computing Inner and Outer Shape Approximations

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## Computing Inner and Outer Shape Approximations

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**Computing Inner and Outer Shape Approximations**Joseph S.B. Mitchell Stony Brook University**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**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)**Motivation**• Natural Optimization Problems • Shape Approximation • Visibility Culling for Computer Graphics**Approximating the Longest Stick**• Divide and conquer • Use balanced cuts (Chazelle)**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?**Improved Approximation**Algorithm: Bootstrap from the O(1)-approx, discretize search space more finely, reduce to a visibility problem, and apply efficient data structures**Approx Biggest Convex Potato**Goal: Find max-area e-anchored triangle**3D Hueristics**• “Grow” k-dops from selected seed points: collision detection (QuickCD), response**Part II: Outer Approximation**• Joint work with E. Arkin, G. Barequet (SoCG’06)**Bounding Volume Hierarchy**BV-tree: Level 0 k-dops**BV-tree: Level 1**14-dops 6-dops 26-dops 18-dops**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**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]