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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).

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quadrilateral and tetrahedral mesh stripification using the 2 factor partitioning of the dual graph

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
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 description dual graphs
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 description dual graphs properties
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 description graph factorization
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 description our solution
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
Complementary strips
  • A 2-factor defines 2 complementary sets of disjoint loops.

http://graphics.ics.uci.edu

talk outline
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
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 outline10
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

stripification two techniques
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

slide12

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 outline13
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

stripification 2 pass matching method
Stripification2-pass matching method
  • Repeat twice on a 4-regular graph:
    • Find a perfect matching.
    • Remove the matched edges

http://graphics.ics.uci.edu

stripification 2 pass matching method15

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

talk outline16
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

stripification template substitution method
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

stripification template substitution method18
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

stripification template substitution method19
StripificationTemplate substitution method

DOPES

V

V’

http://graphics.ics.uci.edu

stripification template substitution method 2 factor from matching
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 substitution simple example
Template substitutionSimple example

http://graphics.ics.uci.edu

talk outline22
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
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 strips strip compatibility
Merging stripsStrip compatibility
  • Loop + loop = loop
  • Loop + linear = linear
  • Linear + linear -> 2 linear strips! (pointless)

http://graphics.ics.uci.edu

talk outline25
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 nodal simplex processing graph
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 strips nodal simplex processing geometric realization on quads
Merging stripsnodal simplex processing:geometric realization on quads

Nodal

vertex

http://graphics.ics.uci.edu

talk outline28
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 mesh subdivision
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 strips dual graph subdivisions
Merging stripsDual Graph Subdivisions

Dual Tetra Graph

(Non-planar subdivision)

Dual Quad Graph

http://graphics.ics.uci.edu

merging strips quadrilateral subdivision reassigning matching
Merging stripsQuadrilateral subdivision(reassigning matching)
  • After subdividing, identify a nodal simplex
  • Apply nodal simplex processing to merge strips

http://graphics.ics.uci.edu

merging strips quadrilateral subdivision geometric realization
Merging stripsQuadrilateral subdivision(geometric realization)

http://graphics.ics.uci.edu

merging strips tetrahedral subdivision dual graph re matching
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 outline34
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

results tetrahedral stripification table
ResultsTetrahedral stripification table

http://graphics.ics.uci.edu

results quadrilateral strips
ResultsQuadrilateral strips

http://graphics.ics.uci.edu

results quadrilateral strips37
ResultsQuadrilateral strips

http://graphics.ics.uci.edu

results tetrahedral strips
ResultsTetrahedral strips

http://graphics.ics.uci.edu

talk outline39
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
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
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
Acknowledgements
  • ICS Computer Graphics Lab @ UCI
    • http://graphics.ics.uci.edu

http://graphics.ics.uci.edu

the end
The End
  • Thanks for listening!!
  • Questions?
  • Comments?
  • Corrections?
  • Suggestions?
  • Complaints?
  • Divagations?

http://graphics.ics.uci.edu

dual graphs

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

stripification 2 pass matching method46

E

D

BCDE

C

A

B

ABCD

ABED

Stripification2-pass matching method

?

http://graphics.ics.uci.edu

merging strips nodal simplex processing tetrahedra
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 strips tetrahedral subdivision subdivision in dual graph
Merging stripsTetrahedral subdivision(subdivision in dual graph)

C

A

B

a

c

b

A’

B’

C’

http://graphics.ics.uci.edu

merging strips tetrahedral subdivision geometric interpretation
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 strips initial considerations
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