Alpha Complexes

Lecture 6 January 25, 2016

Today

»  Shape of a point set »  Delaunay complex »  Weighted Voronoi diagram »  Weighted Delaunay complex »  Alpha complex

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Shape of a Point Set

»  Given a point set, what is its shape? »  How do we define shape at multiple

levels (scales) and at different regions?

»  Weights assigned to points? »  How to modify weights using a single

parameter?

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Shape of a Point Set

Source : Edelsbrunner and Mücke, 1994

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Alpha Complex

Union of balls Voronoi decomposition

Overlay Dual complex Source : Edelsbrunner et al., 1983

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Properties

»  Alpha complex and union of balls are topologically (homotopy) equivalent

»  Union expressed as an alternating sum of ball intersections

»  Similar formulae for surface area and length

»  Subset of the Delaunay triangulation

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Applications

»  Surface reconstruction

»  Molecular structure modeling

»  Grid / Mesh generation

Surface Reconstruction

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Source : Tamal Dey Dey et al., Proc. Symp. Geometry Processing (SGP), 2009

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Molecular Structure

Gramicidin

GRAMICIDIN A

Afra Zomorodian – CS 468 Lecture 8 - Page 12

Source : Afra Zomorodian