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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

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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)

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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.

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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.

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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)

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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.

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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).

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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.

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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.

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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-manifoldsManifolds in 2D

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An example of a decomposition in the 2D case

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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

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Non-Manifold Multi-Triangulation: An Example

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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.