Non-manifold Multiresolution Modeling (some preliminary results)

1. Non-manifold Multiresolution Modeling(some preliminary results) Leila De Floriani DISI University of Genova Genova (Italy)

2. Non-Manifold Multiresolution Modeling • A mathematical framework for describing non-manifold d-dimensional objects as assembly of simpler components (that we call quasi-manifold ). • Topological data structures for non-manifold meshes in arbitrary 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 • A Euclidean d-manifold M [with boundary] is a subspace of the Euclidean space Ed such that the neighborhood of any point of M is homeomorphic to a d-dimensional open ball [intersected with a closed d-dimensional half-space in Ed]. • 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.

4. Non-Manifold Modeling: Issues • Non-manifolds are not well-understood and classified from a mathematical point of view: • The class of combinatorial manifolds is not decidable in high dimensions. • The homeomorphism problem is completely solved only in two dimensions • All 3-manifolds can be triangulated • There is no known algorithm to decide if a complex describes 5-manifold (the problem is recognizing the4-sphere)

5. Non-Manifold Modeling: Issues • Non-manifold cell complexes are difficult to encode and manipulate: • Decomposition of a non-manifold object into manifold components done only in two dimensions • Topological data structures have been proposed for two- dimensional complexes, but do not scale well with the degree of “non-manifoldness” of the complex. • General, but verbose and redundant, representations: incidence graph, and chains of n-Gmaps.

6. Non-Manifold modeling in arbitrary dimensions • A mathematical framework for describing non-manifold simplicial complexes in arbitrary dimensions as assembly of simpler quasi-manifold components (De Floriani et al., 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 resulting decomposition (on-going work).

7. Abstract simplicial complexes • 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 gV is an element of , then every subset of g is also an element of . • Dimension of a cell g = | g|-1. Order d of a complex is the maximum of the dimensions of its cells: simplicial d-complex • A d-complex in which all k-cells, k=0,1,…,d-1, are on the boundary of some d-cell is called a regular complex. An example of a non-regular complex

8. Pseudo-manifolds • A (d-1)-cell g in a regular d-complex is a manifold cell iff there exist at most two d-cells incident at g. • A regular complex which has only manifold cells is called a combinatorial pseudo-manifold. A 2-pseudo-manifold A regular 2-complex which is not a pseudo-manifold

9. Quasi-manifolds • A regular 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 . • A complex is a quasi-manifold iff it is both a pseudo-manifold and a regularly-adjacent complex. A 3-pseudo-manifold which is notregularly-adjacent A 3-quasi-manifold (which is not atriangulation of a 3-manifold)

10. An example of a decomposition Quasi-manifolds  Triangulations of 2-manifolds in E2

11. 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

12. Non-Manifold Multi-Triangulation: An Example

13. 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, i.e., contraction of a pair of vertices to a new vertex). • 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.