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Hierarchyless Simplification, Stripification and Compression of Triangulated Two-Manifolds Pablo Diaz-Gutierrez M. Gopi Renato Pajarola University of California, Irvine Introduction

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hierarchyless simplification stripification and compression of triangulated two manifolds

Hierarchyless Simplification, Stripification and Compression of Triangulated Two-Manifolds

Pablo Diaz-Gutierrez

M. Gopi

Renato Pajarola

University of California, Irvine

introduction
Introduction
  • Intersection of and relationship between three mesh problems: Simplification, stripification and connectivity compression
  • Each problem must be constrained
  • Imposed constraints allow further applicability

http://graphics.ics.uci.edu

introduction three problems
Introduction: Three problems
  • Simplification
  • Decimation
  • Vertex clustering
  • Edge collapsing
  • Compression
  • Valence-driven
  • Strip/edge-graph based
  • Stripification
  • Alternating linear strips
  • Generalized strip loops

http://graphics.ics.uci.edu

talk outline
Talk outline
  • Hierarchyless simplification
  • Simplification and stripification
  • Connectivity compression
  • Results
  • Conclusion

http://graphics.ics.uci.edu

mesh simplification
Mesh simplification
  • Popular approach: Edge collapse/vertex split
  • Problem: Dependencies between collapsed edges
    • Hierarchy of collapse/split operations

Edge collapse

Vertex split

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edge collapse dependencies
Edge-collapse dependencies

Edge-collapse A

can’t be split before

Edge-collapse B

A

B

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definition

Multi-edges

Definition
  • Multi-edge: Edge representing multiple edges from the original mesh, after simplification.

Edge collapse

Vertex split

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edge collapse dependencies8
Edge-collapse dependencies

A

Multi-edges

B

Collapsing

multi-edges

produces

dependencies

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avoiding dependencies

Multi-edges

Avoiding dependencies
  • When one edge of a triangle is collapsed, the other two become multi-edges.
  • If we don’t collapse multi-edges: Only one edge per triangle is collapsed

Edge collapse

6

5

6

5

1

Collapsing

edge

1

9

9

7/8

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hierarchyless simplification
Hierarchyless simplification
  • Each triangle has at most one collapsible edge
    • One collapsible partner across that edge
  • Problem: Choosing one edge prevents others from collapsing
  • Optimize choice of collapsible edges

http://graphics.ics.uci.edu

hierarchyless simplification11
Hierarchyless simplification
  • Pose as graph problem in the dual graph of the triangle mesh
  • Choose collapsible edges
    • Graph matching
  • Maximal set of collapsible edges (triangle pairs)
    • Perfect graph matching
  • No multi-edges collapsed!!
    • (No collapsing dependencies)

http://graphics.ics.uci.edu

simplification example
Simplification example
  • Genus 0 manifold
  • 3 connected sets
  • Of collapsible edges

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simplification example13
Simplification example

Multi-edges

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simplification example14
Simplification example

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simplification example15
Simplification example

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simplification example16
Simplification example

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simplification example17
Simplification example

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simplification example18
Simplification example

Equivalent

vertices

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simplification example19
Simplification example

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simplification example20
Simplification example

Equivalent

vertices

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simplification example21
Simplification example

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simplification example22
Simplification example
  • 3 connected components of collapsible edges
  • 3 vertices after complete simplification

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extremal simplification
Extremal simplification
  • All triangles collapsed
    • But not all vertices collapsed
  • Collapsible edges organized in connected components
    • Each connected component collapses to 1 vertex

http://graphics.ics.uci.edu

extremal simplification24
Extremal simplification
  • Goal: Reduce number of vertices in final model
    • By reducing number of connected components of collapsible edges
    • Apply two operations:
      • Edge swap (next slide)
      • Matching reassignment
  • Minimum 1 or 2 connected components
  • In general, connected components are trees
    • Might have loops

http://graphics.ics.uci.edu

extremal simplification connecting collapsible edges
Extremal simplificationConnecting collapsible edges

Initially, 3 connected

components of

collapsible edges

Choose an edge to swap

Only two connected

components now

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extremal simplification26
Extremal simplification
  • 2 connected components of collapsible edges
  • 2 vertices after complete simplification

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talk outline27
Talk outline
  • Hierarchyless simplification
  • Simplification and stripification
  • Connectivity compression
  • Results
  • Conclusion

http://graphics.ics.uci.edu

simplification and stripification
Simplification and stripification
  • Each triangle has one collapsible edge
    • The other two connect it to a triangle strip loop
  • Removing collapsible edges creates disjoint triangle strip loops

http://graphics.ics.uci.edu

extremal simplification and single stripification
Extremal simplification andsingle-stripification
  • Connected components of collapsible edges are trees
    • Triangles around them form loops
  • Fewer collapsible edge components → fewer loops:
    • Reduce number of connected components of matched edges.
  • In manifolds, collapsible edges can be grouped in 1 or 2 connected components:
    • All triangulated manifolds can be made a single triangle strip loop.

Schematic representation of triangle

strips and medial axes.

http://graphics.ics.uci.edu

maintaining strips during simplification
Maintaining strips during simplification
  • Hierarchyless simplification automatically maintains triangle strips
    • Edge-collapses shorten strips and medial axes
    • But don’t change topology
  • Let’s see an example…

http://graphics.ics.uci.edu

maintaining strips during simplification31
Maintaining strips during simplification

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maintaining strips during simplification32
Maintaining strips during simplification

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maintaining strips during simplification33
Maintaining strips during simplification

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maintaining strips during simplification34
Maintaining strips during simplification

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maintaining strips during simplification35
Maintaining strips during simplification

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quality considerations
Quality considerations
  • Quality of simplification
    • Choose collapsible edges with the least quadric error
  • Quality of stripification
    • Application dependent (i.e. maximize strip locality)
  • Assign edge weights
  • Choose weight minimizingset of collapsible edges
    • Diaz-Gutierrez et al. "Constrained strip generation and management for efficient interactive 3D rendering“, CGI 2005

0

0

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3

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0

0

0

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1

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0

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0

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0

http://graphics.ics.uci.edu

talk outline37
Talk outline
  • Hierarchyless simplification
  • Simplification and stripification
  • Connectivity compression
  • Results
  • Conclusion

http://graphics.ics.uci.edu

collapsible edges connectivity compression
Collapsible edges & connectivity compression
  • Important: topological, not geometric compression.
  • Some existing techniques produce triangle strips as byproduct of compression.
  • A few compress connectivity along with strips.
  • We exploit duality of strips and medial axes.

Images from http://www.gvu.gatech.edu/~jarek/

http://graphics.ics.uci.edu

hand and glove compression genus 0 triangulated manifolds

1

0

1

0

1

0

1

0

Encoding of a strip as a zipping of two trees

“Hand and Glove” compression Genus-0 triangulated manifolds
  • Encode genus 0 mesh as:
    • Two vertex spanning trees of collapsible edges (“hand and glove” trees)
    • A bit string zips the trees together along the single strip loop
  • Guaranteed upper bound: 2 bits/face (i.e. 4 bits/vertex)

http://graphics.ics.uci.edu

hand and glove compression genus 0 triangulated manifolds40
“Hand and Glove” compression Genus-0 triangulated manifolds
  • Predict direction of strip to improve compression
  • Slight modifications to handle:
    • Higher genus
    • Boundaries
    • Quadrilateral manifolds
    • Etc.
  • Very simple to code
    • One day for prototype program

http://graphics.ics.uci.edu

talk outline41
Talk outline
  • Hierarchyless simplification
  • Simplification and stripification
  • Connectivity compression
  • Results
  • Conclusion

http://graphics.ics.uci.edu

results simplification
Results: Simplification

Models with 1358, 454, 54 and 4 triangles.

http://graphics.ics.uci.edu

results simplification43
Results: Simplification

Models with 19778, 7238, 1500 and 778 triangles.

Models with 16450, 6450, 2450 and 450triangles.

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results simplification44
Results: Simplification

Models with 101924, 33924, 9924 and 1924 triangles.

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results view dependent simplification
Results: View-dependent simplification

Notice dramatic change

in simplification

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results compression bit ratios
Results: Compression bit ratios

Bits per vertex obtained with Hand & Glove method.

Comparison with Edgebreaker. The output of both

methods is compressed with an arithmetic encoder.

http://graphics.ics.uci.edu

talk outline48
Talk outline
  • Hierarchyless simplification
  • Simplification and stripification
  • Connectivity compression
  • Results
  • Conclusion

http://graphics.ics.uci.edu

summary and conclusion
Summary and conclusion
  • This paper lays a theoretical foundation for combining three important areas of geometric computing.
  • By computing and appropriately managing sets of collapsible edges, we achieved:
    • Hierarchyless mesh simplification
    • Dynamic management of triangle strip loops
    • Efficient connectivity compression

http://graphics.ics.uci.edu

future work
Future work
  • Explore and improve Hand & Glove mesh compression.
  • Design a lighter data structure for computing errors in view-dependent simplification.
  • Extend current results on stripification (partially completed).

http://graphics.ics.uci.edu

acknowledgments
Acknowledgments
  • ICS Computer Graphics Lab @ UC Irvine
    • http://graphics.ics.uci.edu

http://graphics.ics.uci.edu

the end
THE END
  • Thanks for your time
  • Questions?
  • Comments?
  • Suggestions?

http://graphics.ics.uci.edu