1 Outline Previous work on geometric and solid modeling Multiresolution models based on tetrahedral meshes for volume data analysis Current work on non-manifold multiresolution modeling

2 Non-Manifold Multiresolution Modeling A mathematical framework for describing non-manifold d- dimensional objects as assembly of simpler (quasi- manifold ) components. Topological data structures for non-manifold meshes in three (and higher) dimensions Multiresolution models for meshes with a non-manifold and non-regular domain for CAD applications: Data structures Query algorithms (extract topological data structures)

3 Non-Manifold Multiresolution Modeling: Why Non-Manifold? Need to represent and manipulate objects which combine wire-frames, surfaces, and solid parts: Boolean operators are not closed in the manifold domain. Sweeping, or offset operations may generate parts of different dimensionalities. Non-manifold topologies are required in different product development phases. Complex spatial objects described by meshes with a non- manifold and non-regular domain.

4 Non-Manifold Multiresolution Modeling A d-manifold M is a subset of the Euclidean space such that the neighborhood of any point of M is locally equivalent to a d- dimensional open ball. Spatial objects which do not fulfill the above condition are called non-manifolds. Spatial objects composed of parts of different dimensionalities are called non-regular.

5 Non-Manifold Multiresolution Modeling: Why Multiresolution? Availability of CAD models of large size Need for a multiresolution representation to be able to extract selectively refined meshes Our aim: multiresolution modeling not only for view-dependent rendering, but also for extracting adaptive meshes with a complete topological description (to support efficient mesh navigation though adjacencies)

6 Non-Manifold Multiresolution Modeling: Issues Non-manifolds are not well understood and classified from a mathematical point of view. Non-manifold cell complexes are difficult to encode and manipulate. Topological data structures have been proposed only for two- dimensional complexes, but do not scale well with the degree of “non-manifoldness” of the complex. Decomposing a non-manifold object into manifold components is possible only in two dimensions since the class of manifolds is not decidable in higher dimensions.

7 Non-Manifold Multiresolution Modeling A mathematical framework for describing non-manifold simplicial complexes in three and higher dimensions as assembly of simpler quasi-manifold components (DGCI,2002). An algorithm for decomposing a d-complex into a natural assembly of quasi-manifolds of dimension h<=d. A dimension-independent data structure for representing the decomposition (on-going work).

8 Pseudo-manifolds Let V be a finite set of vertices. An abstract simplicial complex on V is a subset  of the set of the non-empty part of V such that {v}   for every v  V, if   V is an element of , then every subset of  is also an element of . A d-complex in which all cells are maximal is called a regular complex. Let  be a (d-1)-cell of a regular d-complex.  is a manifold cell iff there exist at most two d-cells incident into  A regular complex which has only manifold cells is called a combinatorial pseudo-manifold.

9 Pseudo-manifolds A 2-pseudo-manifold An abstract simplicial complex which is not a pseudo-manifold A regular adjacent simplicial 1-complex is a regularly adjacent complex. A regular abstract simplicial complex is regularly adjacent iff the link of each of its vertices is a connected regularly adjacent (d-1)-complex.

10 Quasi-manifolds A complex is a quasi-manifold iff it is both a pseudo-manifold and a regularly adjacent complex. A 3-quasi-manifold (which is not a combinatorial 3-manifold) A 3-pseudo-manifold which is not regularly adjacent Quasi-manifolds  Manifolds in 2D

11 An example of a decomposition in the 2D case

12 Non-Manifold Multi-Triangulation (NMT) Extension of a Multi-Triangulation to deal with simplicial meshes having a non-manifold, non-regular domain Dimension-independent and application-independent definition of a modification An example of a modification

13 Non-Manifold Multi-Triangulation: An Example

14 Non-Manifold Multi-Triangulation (NMT) A compact data structure for a specific instance of the NMT in which each modification is a vertex expansion (vertex expansion = inverse of vertex-pair contraction) A topological data structure for non-regular, non-manifold 2D simplicial complexes, which scales to the manifold case with a small overhead Algorithms for performing vertex-pair contraction and vertex expansion (basic ingredients for performing selective refinement) on the topological representation of the complex.