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Quadrilateral and Tetrahedral Mesh Stripification Using the 2-Factor Partitioning of the Dual Graph

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## Quadrilateral and Tetrahedral Mesh Stripification Using the 2-Factor Partitioning of the Dual Graph

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### Quadrilateral and Tetrahedral Mesh Stripification Using the 2-Factor Partitioning of the Dual Graph

Talk outlineTalk outlineTalk outlineTalk outlineTalk outlineTalk outline

Pablo Diaz-Gutierrez

M. Gopi

University of California, Irvine

Problem description

- Input: Quadrilateral or Tetrahedral mesh
- Output: Partition the input mesh into primitive strip(s).
- Approach: Use a graph matching algorithm on the dual graph of the mesh.

http://graphics.ics.uci.edu

Problem descriptionDual graphs

- Every quad/tetrahedron is a node in the dual graph.
- Arcs between these nodes if the corresponding mesh elements are adjacent in the mesh.

From a cube to its dual graph

http://graphics.ics.uci.edu

Problem descriptionDual graphs: properties

- Dual quad-graph: Dual graph of a quadrilateral manifold mesh
- Every node has degree four. (4-regular)
- Dual tetra-graph: Dual graph of a tetrahedral mesh
- Every node has degree four or less.

http://graphics.ics.uci.edu

Problem descriptionGraph factorization

- K-factor of a graph G:
- A spanning, k-regular sub-graph of G

1-factor

(perfect matching)

2-factor

3-factor

http://graphics.ics.uci.edu

Problem descriptionOur Solution

- A 2-factor F of a graph G determines a set of disjoint loops in G
- Finding a 2-factor in the 4-regular dual graph of the mesh partitions the mesh into strip loops of mesh primitives.

http://graphics.ics.uci.edu

Complementary strips

- A 2-factor defines 2 complementary sets of disjoint loops.

http://graphics.ics.uci.edu

Talk outline

- Problem description
- Related work
- Stripification
- 2-pass matching
- Template substitution
- Merging strips
- Nodal simplex processing
- Simplex subdivision
- Results
- Conclusion, Q&A

http://graphics.ics.uci.edu

Relevant related work

- Several papers on triangle stripification
- [Gopi and Eppstein 04], etc
- [Pascucci 04] on GPU for isosurface extraction
- Tetra strips to reduce BW
- 2-factoring of sparse graphs
- [Pandurangan 05]

Images from [Gopi et al. 04] and Pascucci 04]

http://graphics.ics.uci.edu

Talk outline

- Problem description
- Related work
- Stripification
- 2-pass matching
- Template substitution
- Merging strips
- Nodal simplex processing
- Simplex subdivision
- Results
- Conclusion, Q&A

http://graphics.ics.uci.edu

StripificationTwo techniques

- 2-pass matching method
- Straightforward
- Fast
- Not always applicable
- Template substitution method
- More complicated
- Slower (problem size grows 18x)
- Universal

http://graphics.ics.uci.edu

- A 1-factor of a graph is called a perfect matching.

Perfect matching in a 3-regular graph

http://graphics.ics.uci.edu

Talk outline

- Problem description
- Related work
- Stripification
- 2-pass matching
- Template substitution
- Merging strips
- Nodal simplex processing
- Simplex subdivision
- Results
- Conclusion, Q&A

http://graphics.ics.uci.edu

Stripification2-pass matching method

- Repeat twice on a 4-regular graph:
- Find a perfect matching.
- Remove the matched edges

http://graphics.ics.uci.edu

Graph with a 2-factor but without a perfect matching

Stripification2-pass matching method- Advantages:
- Fast
- Existing code for perfect matching
- Simple to implement
- Disadvantages
- Does not work on graphs with odd # of vertices

© Algorithmic Solutions

http://graphics.ics.uci.edu

- Problem description
- Related work
- Stripification
- 2-pass matching
- Template substitution
- Merging strips
- Nodal simplex processing
- Simplex subdivision
- Results
- Conclusion, Q&A

http://graphics.ics.uci.edu

StripificationTemplate substitution method

- Transform (inflate) graph G to lower degree, larger graph G’
- G has degree 4 and less
- G’ has degree 3 and less
- Perfect matching in G’ ↔ 2-factor in G

G

Induce

2-factor

Inflate

G’

Perfect matching

http://graphics.ics.uci.edu

StripificationTemplate substitution method

- Transformation by substituting each vertex V in G by a template V’ (expand V)

G

Induce

2-factor

Inflate

G’

Perfect matching

http://graphics.ics.uci.edu

StripificationTemplate substitution method(2-factor from matching)

- Theorem: G’ has a perfect matching iff G has 2-factor

G

G’

http://graphics.ics.uci.edu

Template substitutionSimple example

http://graphics.ics.uci.edu

- Problem description
- Related work
- Stripification
- 2-pass matching
- Template substitution
- Merging strips
- Nodal simplex processing
- Simplex subdivision
- Results
- Conclusion, Q&A

http://graphics.ics.uci.edu

Merging strips

Disjoint (cyclic) strips in

a 4-regular graph

(i.e. quad-graph)

Disjoint strips in a

degree-4-and-less graph

(i.e. tetra-graph)

http://graphics.ics.uci.edu

Merging stripsStrip compatibility

- Loop + loop = loop
- Loop + linear = linear
- Linear + linear -> 2 linear strips! (pointless)

http://graphics.ics.uci.edu

- Problem description
- Related work
- Stripification
- 2-pass matching
- Template substitution
- Merging strips
- Nodal simplex processing
- Simplex subdivision
- Results
- Conclusion, Q&A

http://graphics.ics.uci.edu

Merging stripsnodal simplex processing: graph

- A nodal simplex:
- An (n-2) dimensional simplex
- A vertex in a quad mesh, or an edge in a tetrahedral mesh
- Around which matches alternate
- Incident cycles are unique
- Toggle matching

Strip loops

A face in the dual graph corresponding to a nodal simplex in the mesh

http://graphics.ics.uci.edu

Merging stripsnodal simplex processing:geometric realization on quads

Nodal

vertex

http://graphics.ics.uci.edu

- Problem description
- Related work
- Stripification
- 2-pass matching
- Template substitution
- Merging strips
- Nodal simplex processing
- Simplex subdivision
- Results
- Conclusion, Q&A

http://graphics.ics.uci.edu

Merging stripsMesh subdivision

- Often there are not enough nodal simplices and we still need to reduce #strips
- Subdivide two adjacent primitives belonging to different cycles
- Reassign dual edge matchings to merge cycles

http://graphics.ics.uci.edu

Merging stripsDual Graph Subdivisions

Dual Tetra Graph

(Non-planar subdivision)

Dual Quad Graph

http://graphics.ics.uci.edu

Merging stripsQuadrilateral subdivision(reassigning matching)

- After subdividing, identify a nodal simplex
- Apply nodal simplex processing to merge strips

http://graphics.ics.uci.edu

Merging stripsQuadrilateral subdivision(geometric realization)

http://graphics.ics.uci.edu

Merging stripsTetrahedral subdivision(dual graph re-matching)

C

C

A

B

A

B

a

c

b

a

c

b

A’

B’

A’

B’

C’

C’

http://graphics.ics.uci.edu

- Problem description
- Related work
- Stripification
- 2-pass matching
- Template substitution
- Merging strips
- Nodal simplex processing
- Simplex subdivision
- Results
- Conclusion, Q&A

http://graphics.ics.uci.edu

ResultsTetrahedral stripification table

http://graphics.ics.uci.edu

ResultsQuadrilateral strips

http://graphics.ics.uci.edu

ResultsQuadrilateral strips

http://graphics.ics.uci.edu

ResultsTetrahedral strips

http://graphics.ics.uci.edu

- Problem description
- Related work
- Stripification
- 2-pass matching
- Template substitution
- Merging strips
- Nodal simplex processing
- Simplex subdivision
- Results
- Conclusion, Q&A

http://graphics.ics.uci.edu

Summary & conclusion

- Two algorithms for 2-factorization of graphs of degree 4 and less
- Unified approach for quadrilateral and tetrahedral stripification
- Subdivision techniques for reduction of number of tetrahedral and quadrilateral strips

http://graphics.ics.uci.edu

Future work

- Tetrahedral mesh compression using strips
- Investigate approximate matching algorithms
- Reduce number of strips without subdivision (maintaining original mesh)

http://graphics.ics.uci.edu

Acknowledgements

- ICS Computer Graphics Lab @ UCI
- http://graphics.ics.uci.edu

http://graphics.ics.uci.edu

The End

- Thanks for listening!!
- Questions?
- Comments?
- Corrections?
- Suggestions?
- Complaints?
- Divagations?

http://graphics.ics.uci.edu

D

BCDE

C

A

B

ABCD

ABED

Dual graphs- Every dual quad-graph is 4-regular
- But not every 4-regular graph is the dual of a valid quadrangulated manifold

?

http://graphics.ics.uci.edu

Merging stripsnodal simplex processing: tetrahedra

- (n-2) dimensional nodal simplex
- Geometric edge
- Dual edges correspond to faces of tetrahedra
- Matching of dual edges (primal faces) is toggled around nodal edge

Nodal

edge

Faces of

tetrahedra

Nodal simplex processing

in a dual tetra-graph

http://graphics.ics.uci.edu

Merging stripsTetrahedral subdivision(subdivision in dual graph)

C

A

B

a

c

b

A’

B’

C’

http://graphics.ics.uci.edu

Merging stripsTetrahedral subdivision(geometric interpretation)

- Two tetrahedra split into 6 (3 each)
- Red line: nodal simplex (an axis)
- A,B,C: Graph edges (faces of tetrahedra)

A

C

B

http://graphics.ics.uci.edu

Merging stripsInitial considerations

- We have a number of primitive strips
- Want to merge into fewer strips
- Issues:
- Can we modify the meshes?
- Different types of strips

http://graphics.ics.uci.edu

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