A Decomposition-based Approach to Modeling and Understanding Arbitrary Shapes

Modeling and understanding complex non-manifold shapes is a key issue in shape analysis and retrieval. The topological structure of a non-manifold shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we consider a representation for arbitrary shapes, that we call ManifoldConnected Decomposition (MC-decomposition), which is based on a unique decomposition of the shape into nearly manifold parts. We present efficient and powerful two-level representations for non-manifold shapes based on the MC-decomposition and on an efficient and compact data structure for encoding the underlying components. We describe a dimension-independent algorithm to generate such decomposition. We also show that the MC-decomposition provides a suitable basis for geometric reasoning and for homology computation on nonmanifold shapes. Finally, we present a comparison with existing representations for arbitrary shapes.

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

[2]  Luiz Velho,et al.  CHF: A Scalable Topological Data Structure for Tetrahedral Meshes , 2005, XVIII Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI'05).

[3]  Leila De Floriani,et al.  A Dimension-Independent Data Structure for Simplicial Complexes , 2010, IMR.

[4]  Leila De Floriani,et al.  A scalable data structure for three-dimensional non-manifold objects , 2003, Symposium on Geometry Processing.

[5]  Leila De Floriani,et al.  A multi-resolution topological representation for non-manifold meshes , 2002, SMA '02.

[6]  Leila De Floriani,et al.  IA*: An adjacency-based representation for non-manifold simplicial shapes in arbitrary dimensions , 2011, Comput. Graph..

[7]  Jarek Rossignac,et al.  3D compression made simple: Edgebreaker with ZipandWrap on a corner-table , 2001, Proceedings International Conference on Shape Modeling and Applications.

[8]  Gabriel Taubin,et al.  Converting sets of polygons to manifold surfaces by cutting and stitching , 1998, Proceedings Visualization '98 (Cat. No.98CB36276).

[9]  Sang Hun Lee,et al.  Partial entity structure: a compact non-manifold boundary representation based on partial topological entities , 2001, SMA '01.

[10]  Jarek Rossignac,et al.  SQuad: Compact Representation for Triangle Meshes , 2011, Comput. Graph. Forum.

[11]  Neil F. Stewart,et al.  An extension of manifold boundary representations to the r-sets , 1992, TOGS.

[12]  Hélio Lopes,et al.  A stratification approach for modeling two-dimensional cell complexes , 2004, Comput. Graph..

[13]  Bianca Falcidieno,et al.  Two‐manifold cell‐decomposition of r‐sets. , 1992, Comput. Graph. Forum.

[14]  Leila De Floriani,et al.  An iterative algorithm for homology computation on simplicial shapes , 2011, Comput. Aided Des..

[15]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[16]  Leila De Floriani,et al.  Data structures for simplicial complexes: an analysis and a comparison , 2005, SGP '05.

[17]  Leila De Floriani,et al.  A data structure for non-manifold simplicial d-complexes , 2004, SGP '04.

[18]  Jarek Rossignac,et al.  SOT: compact representation for tetrahedral meshes , 2009, Symposium on Solid and Physical Modeling.

[19]  Leila De Floriani,et al.  A two-level topological decomposition for non-manifold simplicial shapes , 2007, Symposium on Solid and Physical Modeling.

[20]  Hélio Lopes,et al.  Structural operators for modeling 3-manifolds , 1997, SMA '97.

[21]  Sara McMains,et al.  Out-of-core build of a topological data structure from polygon soup , 2001, SMA '01.

[22]  H. Whitney.  Local Properties of Analytic Varieties , 1992 .

[23]  Leila De Floriani,et al.  Computing and Visualizing a Graph-Based Decomposition for Non-manifold Shapes , 2009, GbRPR.

[24]  Carlo Cattani,et al.  Dimension-independent modeling with simplicial complexes , 1993, TOGS.