Alpha-Beta Witness Complexes

Dominique Attali (1) , Herbert Edelsbrunner (2) , John Harer (3) , Yuriy Mileyko (2)

1 GIPSA-GPIG - GIPSA - Géométrie, Perception, Images, Geste
2 Department of Computer Science
3 Department of Mathematics [Durham]

Building on the work of Martinetz, Schulten and de Silva, Carlsson, we introduce a 2-parameter family of witness complexes and algorithms for constructing them. This family can be used to determine the gross topology of point cloud data in Rd or other metric spaces. The 2-parameter family is sensitive to differences in sampling density and thus amenable to detecting patterns within the data set. It also lends itself to theoretical analysis. For example, we can prove that in the limit, when the witnesses cover the entire domain, witness complexes in the family that share the first, scale parameter have the same homotopy type.

Mots clés: spheres; indicator functions; discrete geometry; dual complexes; balls; Combinatorial topology

Domaines: Géométrie algorithmique [cs.CG]

Dominique Attali, Herbert Edelsbrunner, John Harer, Yuriy Mileyko. Alpha-Beta Witness Complexes. Workshop on Algorithms and Data Structures, Aug 2007, Halifax, Canada. pp.386-397, ⟨10.1007/978-3-540-73951-7⟩. ⟨hal-00200996⟩