Solid Modeling 2002

A Multi-resolution Topological A Multi-resolution Topological Representation for Non-manifold Representation for Non-manifold MeshesMeshes

Leila De Floriani, Paola Magillo,

Enrico Puppo, Davide Sobrero

University of Genova

Genova (Italy)

Solid Modeling 2002

Non-Manifold Multiresolution Modeling: Non-Manifold Multiresolution Modeling: Why Non-Manifold?Why Non-Manifold?

Need to represent and manipulate objects which combine wire-frames, surfaces, and solid parts in CAD/CAM applications

Complex spatial objects are described by meshes with a non-manifoldnon-manifold and non-regularnon-regular domain

Non-manifold and non-regular meshes are generated by topology-modifying simplificationtopology-modifying simplification algorithms

Solid Modeling 2002

Non-Manifold Multiresolution Modeling: Non-Manifold Multiresolution Modeling: Why Multiresolution?Why Multiresolution?

Availability of CAD models of large sizelarge size

Need for a multiresolution representation to be able to extract selectively refinedselectively refined meshes

Our aim:Our aim: multiresolution modeling for extracting adaptiveadaptive meshesmeshes with a complete topological descriptioncomplete topological description (to support efficient mesh navigation through adjacencies)

Solid Modeling 2002

Our contributionOur contribution

Non-Manifold Multi-Tessellation (NMT)

A compact data structure for a 2D instance of the NMT

A new scalable topological data structure for 2D simplicial complexes

Algorithms for selective refinement on the previous data structures

Solid Modeling 2002

Related workRelated work

Topology-modifying simplification algorithms:Topology-modifying simplification algorithms: e.g., Rossignac and Borrel, 1990; Garland and Heckbert, 1997; Popovic and Hoppe, 1997; El-Sana and Varshney, 1998.

Non-manifold data structures: Non-manifold data structures: Weiler, 1988; Gursoz et al., 1990; Yamaguchi and Kimura, 1995; Rossignac and O’Connor, 1990; Campagna et al., 1999; Lee and Lee, 2001.

Multiresolution models for unstructured triangle meshes:Multiresolution models for unstructured triangle meshes: Hoppe, 1997; De Floriani et al., 1997; Xia et al., 1997; Gueziec et al., 1998; Luebke and Erikson, 1997; Klein and Gumhold, 1998; El-Sana and Varshney, 1999.

Solid Modeling 2002

Background notionsBackground notions

Wire-edge:Wire-edge: no triangle incident into it

Two-dimensional simplicial complexes in 3D (triangle-segmenttriangle-segment meshes)

Triangle-edge:Triangle-edge: at least one triangle incident into it

Solid Modeling 2002

Non-Manifold Multi-Tessellation (NMT)Non-Manifold Multi-Tessellation (NMT)

Extension of the Multi-TessellationMulti-Tessellation (De Floriani,

Magillo, Puppo, 1997) to the non-manifold domain

Basic ingredients:Basic ingredients: mesh modification + dependency relation

Modification of a meshModification of a mesh : replace a set of elements E of with a new set E’ so that the result ’ is still a mesh

Solid Modeling 2002

Non-Manifold Multi-Tessellation (NMT)Non-Manifold Multi-Tessellation (NMT)

’

A modification:

Solid Modeling 2002

Non-Manifold Multi-Tessellation (NMT)Non-Manifold Multi-Tessellation (NMT)

Modification as a pair M=(M-,M+) where– M+: refinement modificationrefinement modification defined by EE’

– M-: coarsening modificationcoarsening modification defined by E’E

Solid Modeling 2002

Non-Manifold Multi-Tessellation (NMT)Non-Manifold Multi-Tessellation (NMT)

NMT: partially ordered partially ordered set of nodes {M0, ….Mq}:

each node Mi: refinement modification Mi+ and its inverse

coarsening one Mi-

Partial order induced by dependency relationdependency relation:

by Mj depends on Mi iff some element introduced by Mi+ is

deleted Mj+

Minimum element in the order: creation of the coarsest mesh

Solid Modeling 2002

Non-Manifold Multi-Tessellation (NMT)Non-Manifold Multi-Tessellation (NMT)

Solid Modeling 2002

Non-Manifold Multi-Tessellation (NMT)Non-Manifold Multi-Tessellation (NMT)

Solid Modeling 2002

Selective refinement on a NMTSelective refinement on a NMT

Selective refinement:Selective refinement: extract a mesh from a NMT satisfying some application-dependent requirements (LOD criterion + maximal size)

Extracted meshesExtracted meshes correspond to set S of modifications closed closed with respect to the partial order

Solid Modeling 2002

Selective refinement: an incremental Selective refinement: an incremental approachapproach

Modify a previously extracted mesh– by adding necessary nodes– by removing superfluous nodes

Primitives on the current mesh:– mesh refinementmesh refinement (apply M+)– mesh coarseningmesh coarsening (apply M-)

Solid Modeling 2002

A data structure for a 2D instance of a NMTA data structure for a 2D instance of a NMT NMT built through vertex-pair contraction:vertex-pair contraction: (v’,v”) v

Each NMT node M:NMT node M: vertex-pair contraction (M-) and vertex expansion (M+)

NMT nodesNMT nodes : encoding technique by Popovic and Hoppe (1997) Partial orderPartial order : implicit encoding technique by El-Sana and

Varshney (1999)

Solid Modeling 2002

A data structure for a 2D instance of a NMTA data structure for a 2D instance of a NMT

Total cost:Total cost: 32n bytes (where n is the number of vertices in the mesh at full resolution)

Around 90%Around 90% of the cost of storing the mesh at full resolution (by encoding only connectivity information)

Less than halfLess than half than the cost of storing the mesh at full resolution with connectivity and face adjacencies

Solid Modeling 2002

A data structure for 2D simplicial A data structure for 2D simplicial complexescomplexes For each vertex v:For each vertex v:

– all vertices to which v is connected through a wire-edge (Vertex-Vertex Vertex-Vertex relationrelation)

– One triangle incident in v for each connected component (“fan”) of the link of v (partial partial Vertex-Triangle relationVertex-Triangle relation)

Solid Modeling 2002

A data structure for 2D simplicial A data structure for 2D simplicial complexescomplexes

For each triangle t:For each triangle t: – link to its three vertices (Triangle-Vertex relationTriangle-Vertex relation)

– for each edge e of t: two triangles if e is non-manifold,one triangle otherwise (partial Triangle-partial Triangle-Triangle relationTriangle relation)

Solid Modeling 2002

A data structure for 2D simplicial A data structure for 2D simplicial complexescomplexes

It can be traversed through edge adjacencies and around a vertex in optimal timeoptimal time (linear in the output size)

Scalability to manifold meshes:Scalability to manifold meshes: overhead of one byte per one byte per vertexvertex wrt indexed data structure with adjacencies when applied to manifold meshes

More compactMore compact than specialization of existing non-manifold data structures to 2D simplicial complexes

Solid Modeling 2002

Implementation of mesh refinementImplementation of mesh refinement Input:Input:

: current mesh encoded in the previous data structure

– M+: feasible vertex expansion v (v’,v”) as specified in the corresponding NMT node

Output:Output: mesh ’ obtained from by expanding v

Solid Modeling 2002

Basic operations in mesh refinementBasic operations in mesh refinement

Vertex into an edgeVertex into an edge (if v’ and v” were connected by an edge in the mesh at full resolution): v e=(v’,v”)

On wire-edges:On wire-edges:

– move extreme vertex: (w,v) (w,v’)

– duplicate edge: (w,v) (w,v’) + (w,v”)

– edge into triangle: (v,w) (v’,v”,w)

Solid Modeling 2002

Basic operations in mesh refinementBasic operations in mesh refinement

On triangle-edges:On triangle-edges:– duplicate edge: (w,v) (w,v’) + (w,v”)

– edge into triangle: (v,w) (v’,v”,w)

Solid Modeling 2002

Basic operations in mesh refinementBasic operations in mesh refinement

On triangles:On triangles:

– move extreme vertex: (w1,w2,v) (w1,w2,v”)

– duplicate triangle: (w1,w2,v) (w1,w2,v’) + (w1,w2,v”)

Solid Modeling 2002

Some experimental resultsSome experimental results

NMT built through a “simple” simplification strategy based on collapsing the shortest edge

Choice of the simplification strategy affects the shape of the NMT

Results:Results:– Number of wire-edges incident at a vertex < 4– Number of “fans” of triangles incident at a vertex < 4– Average number of “fans” of triangles incident at a vertex < 1.2– Only 2% of the vertices are extremes of wire-edges

Solid Modeling 2002

Selective refinement at workSelective refinement at work

21,648 triangles

0 wire edges

1,909 triangles

9 wire edges

3,311 triangles

6 wire edges

3,509 triangles

7 wire edges

Solid Modeling 2002

SummarySummary

NMT:NMT: a model for multi-resolution simplicial complexes in arbitrary dimension

A compact data structurecompact data structure for a 2D instance of a NMT A compact and scalable topological data structurecompact and scalable topological data structure for

2D simplicial complexes Algorithms for performing vertex-pair contractionvertex-pair contraction and

vertex expansionvertex expansion (basic ingredients for selective refinement) on the previous data structure

Solid Modeling 2002

Current and future workCurrent and future work

Our aim:Our aim: extension of the work presented here to 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 less or equal to d (on-going work).

Solid Modeling 2002

Solid Modeling 2002

Background notionsBackground notions

Manifold edge:Manifold edge: exactly one or two triangles incident into it

manifold edges non-manifold edge

Solid Modeling 2002

Background notionsBackground notions

Manifold vertex:Manifold vertex: – No triangle incident in it and– one or two edges incident into itOR– no wire-edges incident into it and – its incident triangles form a single fan

manifold vertices non-manifold vertices

Solid Modeling 2002

Selective refinement: basic primitivesSelective refinement: basic primitives

On the partial order:– node insertion testnode insertion test– node removal testnode removal test– dependencies retrievaldependencies retrieval

On the current mesh:– mesh refinementmesh refinement (apply M+)

– mesh coarseningmesh coarsening (apply M-)