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 - PowerPoint PPT Presentation

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

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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

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
• 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

Perfect Matching

• 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
• Fast
• Existing code for perfect matching
• Simple to implement
• Does not work on graphs with odd # of vertices

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

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

DOPES

V

V’

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• 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

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

Merging strips

Disjoint (cyclic) strips in

a 4-regular 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

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

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

Nodal

vertex

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

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)

http://graphics.ics.uci.edu

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

http://graphics.ics.uci.edu

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

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

ResultsTetrahedral stripification table

http://graphics.ics.uci.edu

http://graphics.ics.uci.edu

http://graphics.ics.uci.edu

ResultsTetrahedral strips

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

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?
• Corrections?
• Suggestions?
• Complaints?
• Divagations?

http://graphics.ics.uci.edu

E

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

E

D

BCDE

C

A

B

ABCD

ABED

Stripification2-pass matching method

?

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

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