Hierarchyless simplification stripification and compression of triangulated two manifolds
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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 l.jpg

Hierarchyless Simplification, Stripification and Compression of Triangulated Two-Manifolds

Pablo Diaz-Gutierrez

M. Gopi

Renato Pajarola

University of California, Irvine


Introduction l.jpg
Introduction of Triangulated Two-Manifolds

  • 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 l.jpg
Introduction: Three problems of Triangulated Two-Manifolds

  • 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 l.jpg
Talk outline of Triangulated Two-Manifolds

  • Hierarchyless simplification

  • Simplification and stripification

  • Connectivity compression

  • Results

  • Conclusion

http://graphics.ics.uci.edu


Mesh simplification l.jpg
Mesh simplification of Triangulated Two-Manifolds

  • Popular approach: Edge collapse/vertex split

  • Problem: Dependencies between collapsed edges

    • Hierarchy of collapse/split operations

Edge collapse

Vertex split

http://graphics.ics.uci.edu


Edge collapse dependencies l.jpg
Edge-collapse dependencies of Triangulated Two-Manifolds

Edge-collapse A

can’t be split before

Edge-collapse B

A

B

http://graphics.ics.uci.edu


Definition l.jpg

Multi-edges of Triangulated Two-Manifolds

Definition

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

Edge collapse

Vertex split

http://graphics.ics.uci.edu


Edge collapse dependencies8 l.jpg
Edge-collapse dependencies of Triangulated Two-Manifolds

A

Multi-edges

B

Collapsing

multi-edges

produces

dependencies

http://graphics.ics.uci.edu


Avoiding dependencies l.jpg

Multi-edges of Triangulated Two-Manifolds

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

7

8

4

4

2

2

3

3

http://graphics.ics.uci.edu


Hierarchyless simplification l.jpg
Hierarchyless simplification of Triangulated Two-Manifolds

  • 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 l.jpg
Hierarchyless simplification of Triangulated Two-Manifolds

  • 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 l.jpg
Simplification example of Triangulated Two-Manifolds

  • Genus 0 manifold

  • 3 connected sets

  • Of collapsible edges

http://graphics.ics.uci.edu


Simplification example13 l.jpg
Simplification example of Triangulated Two-Manifolds

Multi-edges

http://graphics.ics.uci.edu


Simplification example14 l.jpg
Simplification example of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Simplification example15 l.jpg
Simplification example of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Simplification example16 l.jpg
Simplification example of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Simplification example17 l.jpg
Simplification example of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Simplification example18 l.jpg
Simplification example of Triangulated Two-Manifolds

Equivalent

vertices

http://graphics.ics.uci.edu


Simplification example19 l.jpg
Simplification example of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Simplification example20 l.jpg
Simplification example of Triangulated Two-Manifolds

Equivalent

vertices

http://graphics.ics.uci.edu


Simplification example21 l.jpg
Simplification example of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Simplification example22 l.jpg
Simplification example of Triangulated Two-Manifolds

  • 3 connected components of collapsible edges

  • 3 vertices after complete simplification

http://graphics.ics.uci.edu


Extremal simplification l.jpg
Extremal simplification of Triangulated Two-Manifolds

  • 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 l.jpg
Extremal simplification of Triangulated Two-Manifolds

  • 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 l.jpg
Extremal simplification of Triangulated Two-ManifoldsConnecting collapsible edges

Initially, 3 connected

components of

collapsible edges

Choose an edge to swap

Only two connected

components now

http://graphics.ics.uci.edu


Extremal simplification26 l.jpg
Extremal simplification of Triangulated Two-Manifolds

  • 2 connected components of collapsible edges

  • 2 vertices after complete simplification

http://graphics.ics.uci.edu


Talk outline27 l.jpg
Talk outline of Triangulated Two-Manifolds

  • Hierarchyless simplification

  • Simplification and stripification

  • Connectivity compression

  • Results

  • Conclusion

http://graphics.ics.uci.edu


Simplification and stripification l.jpg
Simplification and stripification of Triangulated Two-Manifolds

  • 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 l.jpg
Extremal simplification and of Triangulated Two-Manifoldssingle-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 l.jpg
Maintaining strips during simplification of Triangulated Two-Manifolds

  • 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 l.jpg
Maintaining strips during simplification of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Maintaining strips during simplification32 l.jpg
Maintaining strips during simplification of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Maintaining strips during simplification33 l.jpg
Maintaining strips during simplification of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Maintaining strips during simplification34 l.jpg
Maintaining strips during simplification of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Maintaining strips during simplification35 l.jpg
Maintaining strips during simplification of Triangulated Two-Manifolds

http://graphics.ics.uci.edu


Quality considerations l.jpg
Quality considerations of Triangulated Two-Manifolds

  • 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

0

0

0

1

0

3

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

http://graphics.ics.uci.edu


Talk outline37 l.jpg
Talk outline of Triangulated Two-Manifolds

  • Hierarchyless simplification

  • Simplification and stripification

  • Connectivity compression

  • Results

  • Conclusion

http://graphics.ics.uci.edu


Collapsible edges connectivity compression l.jpg
Collapsible edges & connectivity compression of Triangulated Two-Manifolds

  • 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 l.jpg

1 of Triangulated Two-Manifolds

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 l.jpg
“Hand and Glove” compression of Triangulated Two-ManifoldsGenus-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 l.jpg
Talk outline of Triangulated Two-Manifolds

  • Hierarchyless simplification

  • Simplification and stripification

  • Connectivity compression

  • Results

  • Conclusion

http://graphics.ics.uci.edu


Results simplification l.jpg
Results: Simplification of Triangulated Two-Manifolds

Models with 1358, 454, 54 and 4 triangles.

http://graphics.ics.uci.edu


Results simplification43 l.jpg
Results: Simplification of Triangulated Two-Manifolds

Models with 19778, 7238, 1500 and 778 triangles.

Models with 16450, 6450, 2450 and 450triangles.

http://graphics.ics.uci.edu


Results simplification44 l.jpg
Results: Simplification of Triangulated Two-Manifolds

Models with 101924, 33924, 9924 and 1924 triangles.

http://graphics.ics.uci.edu


Results view dependent simplification l.jpg
Results: View-dependent simplification of Triangulated Two-Manifolds

Notice dramatic change

in simplification

http://graphics.ics.uci.edu


Results stripification with asymmetric simplification l.jpg
Results: Stripification of Triangulated Two-Manifoldswith asymmetric simplification

http://graphics.ics.uci.edu


Results compression bit ratios l.jpg
Results: Compression bit ratios of Triangulated Two-Manifolds

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 l.jpg
Talk outline of Triangulated Two-Manifolds

  • Hierarchyless simplification

  • Simplification and stripification

  • Connectivity compression

  • Results

  • Conclusion

http://graphics.ics.uci.edu


Summary and conclusion l.jpg
Summary and conclusion of Triangulated Two-Manifolds

  • 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 l.jpg
Future work of Triangulated Two-Manifolds

  • 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 l.jpg
Acknowledgments of Triangulated Two-Manifolds

  • ICS Computer Graphics Lab @ UC Irvine

    • http://graphics.ics.uci.edu

http://graphics.ics.uci.edu


The end l.jpg
THE END of Triangulated Two-Manifolds

  • Thanks for your time

  • Questions?

  • Comments?

  • Suggestions?

http://graphics.ics.uci.edu