Alpha Complexes -...
Transcript of Alpha Complexes -...
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