Alpha shapes Reporter: Lincong Fang 10th Jan, 2007

What is the shape?

What are alpha shapes? Alpha Controls the desired level of detail.

What are alpha shapes?

Used for Shape Modelling Creates shapes out of point sets Gives a hierarchy of shapes. Has been used for detecting pockets in proteins. For reverse engineering

Papers H. Edelsbrunner, D. G. Kirkpatrick and R.Seidel. On the shape of a set of points in the plane. IEEE ToIT H. Edelsbrunner and E. P. Mucke. Three dimensional alpha shapes, ToG Kaspar Fischer, Introduction to alpha shapes.

Authors Herbert Edelsbrunner: Chief technical consultant for Geomagic Arts & Science Professor of Computer Science at Duke University The only computer scientist ever to be awarded the prestigious National Science Foundation Alan T. Waterman Award

Authors David G. Kirkpatrick Professor, University of British Columbia Raimund Seidel Professor, Chair of Theoretical Computer Science, Saarland University Ernst P Mucke Univ. of Illinois, Urbana-Champaign Kaspar Fischer ETH Zurich, Switzerland

How to get alpha shapes The space generated by point pairs that can be touched by an empty disc of radius alpha.

General position No 4 points of S lie on a common plane No 5 points lie on a common sphere For any fixed alpha, the smallest sphere through any 2, 3 or 4 points of S has a radius different from alpha.

k-simplex

Simplicial Complex Delaunay triangulations are simplicial complexes.

Delaunay Triangulations

Alpha-ball and alpha exposed Alpha exposed Not alpha exposed

Definition of alpha shapes

Alpha shape

Observations

Definition of alpha complex

Alpha complex

Observations

Algorithm

Best alpha? Alpha Controls the desired level of detail.

Limitations Best alpha? Maybe there is no satisfying alpha.

Papers Baining Guo, Jai Menon, Brian Willette, Surface reconstruction using alpha shapes, Computer Graphics M. Teichmann and M. Capps., Surface reconstruction with anisotropic density-scaled alpha-shapes, IEEE Visualization ’ 98. Xiaolong Xu, Koichi Harada, Automatic surface reconstruction with alpha-shape method, Visual Computer 2003

Surface reconstruction using alpha shapes Baining Guo, Jai Menon, Brian Willette Computer Graphics 1997

Authors Baining Guo The research manager of the graphics group at Microsoft Research Asia. Jai Menon IBM Fellow Brian Willette Department of Computer Science, University of Colorado, USA

3D alpha shape

Outer shells of alpha shapes

Exterior face enumeration

Simplicial surface generation

Surface reconstruction with anisotropic density-scaled alpha- shapes M. Teichmann and M. Capps. IEEE Visualization ’ 98.

Authors Marek Teichmann Lab. for Computer Science, MIT Michael Capps President of Epic Games

Anisotropic alpha shapes

Density determination

Scaling algorithm Reduces the size of the alpha ball in areas where point density is high. Undesired effect of passing very large convex hull triangles.

Scaling algorithm Removes triangles which connect high-density and low density areas. Undesirable effects in non-nuiformly distributed point-sets

Automatic surface reconstruction with alpha-shape method Xiaolong Xu, Koichi Harada Visual Computer 2003

Authors Xiaolong Xu Ph.D candidate of Hiroshima, University, Japan Koichi Harada Professor for Hiroshima University

Density estimation

Scaling algorithm

Surface reconstruction

Weighted alpha shapes Herbert Edelsbrunner Technical report 1992

Weighted alpha shapes

Alpha exposed p q x

Weighted alpha shape

Benefit In biology and chemistry, modeling molecular structures. In reconstructing a surface from point data not uniform. Enforce certain edges or faces.

A surface reconstruction algorithm using weighted alpha shapes Si Hyung Park, Seoung Soo Lee, Jong Hwa Kim FSKD 2005

Authors Si Hyung Park Voronoi Diagram Research Center, Hanyang University, Korea Seoung Soo Lee, Jong Hwa Kim CAESIT, Konkuk University, Korea

Calculation of weights

Constant and weighted alpha

Peel off

The conformal alpha shape filtration Frederic Cazals, Joachim Giesen, Mark Pauly, Afra Zomorodian Visual Computer 2006

Authors Frederic Cazals Research Director, INRIA Sophia-Antipolis Joachim Giesen, Mark Pauly Computer Science, ETH Zurich, Switzerland Afra Zomorodian Assistant Professor in the Department of Computer Science at Dartmouth College

Conformal alpha shapes

Surface reconstruction