Non-Manifold Multi-Resolution Modeling

1. Non-Manifold Multi-Resolution Modeling Geometric Modeling and Computer Graphics Group Department of Computer and Information Sciences University of Genova, Genova (Italy) http://www.disi.unige.it/person/DeflorianiL/

2. What does Non-Manifold Mean? • D-manifold:the neighborhood of any point is locally equivalent to a d-dimensional open ball. • Non-manifold:not a d-manifoldfor any d. • Non-regular:composed of parts of different dimensionalities.

3. Why Non-Manifold? • 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 non-manifold and non-regular meshes.

4. Why Multi-Resolution? • 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)

5. 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.

6. Current Work • 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).

7. Pseudo-manifolds Pseudo-manifold: • uniformly d-dimensional • each (d-1)-cell has at most two incident d-cells A 2-pseudo-manifold Not a pseudo-manifold

8. Quasi-manifolds Quasi-manifold: • pseudo-manifold • for each vertex p, the cells incident in p are connected through (d-1)-faces A 3-quasi-manifold Not a pseudo-manifold • Quasi-manifoldsManifolds in 2D

9. An example of a decomposition in 2D

10. 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

11. Non-Manifold Multi-Triangulation: An Example

12. Non-Manifold Multi-Triangulation (NMT) • 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. • A compact data structure for a specific instance of the NMT in which each modification is a vertex expansion / vertex-pair contraction.