Recursive geometry of the flow complex and topology of the flow complex filtration

The flow complex is a geometric structure, similar to the Delaunay tessellation, to organize a set of (weighted) points in R^k. Flow shapes are topological spaces corresponding to substructures of the flow complex. The flow complex and flow shapes have found applications in surface reconstruction, shape matching, and molecular modeling. In this article we give an algorithm for computing the flow complex of weighted points in any dimension. The algorithm reflects the recursive structure of the flow complex. On the basis of the algorithm we establish a topological similarity between flow shapes and the nerve of a corresponding ball set, namely homotopy equivalence.

[1]  Jesse Freeman,et al.  in Morse theory, , 1999 .

[2]  R. Ho.  Algebraic Topology , 2022 .

[3]  Vin de Silva,et al.  Topological approximation by small simplicial complexes , 2003 .

[4]  Tamal K. Dey,et al.  Shape Segmentation and Matching with Flow Discretization , 2003, WADS.

[5]  Kevin Buchin,et al.  Flow Complex: General Structure and Algorithm , 2005, CCCG.

[6]  Tamal K. Dey,et al.  Alpha-shapes and flow shapes are homotopy equivalent , 2003, STOC '03.

[7]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation (Cambridge Monographs on Applied and Computational Mathematics) , 2006 .

[8]  Afra J. Zomorodian,et al.  Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics) , 2005 .

[9]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation , 2001, Cambridge monographs on applied and computational mathematics.

[10]  Herbert Edelsbrunner,et al.  On the Definition and the Construction of Pockets in Macromolecules , 1998, Discret. Appl. Math..

[11]  H. Edelsbrunner.  Surface Reconstruction by Wrapping Finite Sets in Space , 2003 .

[12]  M. Goresky,et al.  Stratified Morse theory , 1988 .

[13]  Joachim Giesen,et al.  The flow complex: a data structure for geometric modeling , 2003, SODA '03.

[14]  D. Siersma.  Voronoi diagrams and Morse theory of the distance function , 1996 .

[15]  G. E. Bredon.  Topology and geometry , 1993 .

[16]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.

[17]  Jeff Cheeger,et al.  Critical points of distance functions and applications to geometry , 1991 .

[18]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[19]  H. Edelsbrunner.  The union of balls and its dual shape , 1995 .

[20]  G. Ziegler.  Lectures on Polytopes , 1994 .

[21]  Karsten Grove,et al.  A generalized sphere theorem , 1977 .

[22]  David G. Kirkpatrick,et al.  On the shape of a set of points in the plane , 1983, IEEE Trans. Inf. Theory.

[23]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[24]  Joachim Giesen,et al.  Computing the Weighted Flow Complex , 2003, VMV.

[25]  K. Grove.  Critical point theory for distance functions , 1993 .

[26]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation , 2001, Cambridge monographs on applied and computational mathematics.

[27]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .