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Hierarchyless Simplification, Stripification and Compression of Triangulated Two-Manifolds

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### Hierarchyless Simplification, Stripification and Compression of Triangulated Two-Manifolds

Talk outlineTalk outline

Pablo Diaz-Gutierrez

M. Gopi

Renato Pajarola

University of California, Irvine

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

- 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

- Hierarchyless simplification
- Simplification and stripification
- Connectivity compression
- Results
- Conclusion

http://graphics.ics.uci.edu

Mesh simplification

- 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

Edge-collapse A

can’t be split before

Edge-collapse B

A

B

http://graphics.ics.uci.edu

Definition

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

Edge collapse

Vertex split

http://graphics.ics.uci.edu

Edge-collapse dependencies

A

Multi-edges

B

Collapsing

multi-edges

produces

dependencies

http://graphics.ics.uci.edu

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

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Collapsing

edge

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http://graphics.ics.uci.edu

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 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

- Genus 0 manifold
- 3 connected sets
- Of collapsible edges

http://graphics.ics.uci.edu

Simplification example

http://graphics.ics.uci.edu

Simplification example

http://graphics.ics.uci.edu

Simplification example

http://graphics.ics.uci.edu

Simplification example

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Simplification example

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Simplification example

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Simplification example

- 3 connected components of collapsible edges
- 3 vertices after complete simplification

http://graphics.ics.uci.edu

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 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 simplificationConnecting 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 simplification

- 2 connected components of collapsible edges
- 2 vertices after complete simplification

http://graphics.ics.uci.edu

Talk outline

- Hierarchyless simplification
- Simplification and stripification
- Connectivity compression
- Results
- Conclusion

http://graphics.ics.uci.edu

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 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

- 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 simplification

http://graphics.ics.uci.edu

Maintaining strips during simplification

http://graphics.ics.uci.edu

Maintaining strips during simplification

http://graphics.ics.uci.edu

Maintaining strips during simplification

http://graphics.ics.uci.edu

Maintaining strips during simplification

http://graphics.ics.uci.edu

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

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Talk outline

- Hierarchyless simplification
- Simplification and stripification
- Connectivity compression
- Results
- Conclusion

http://graphics.ics.uci.edu

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

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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 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

- Hierarchyless simplification
- Simplification and stripification
- Connectivity compression
- Results
- Conclusion

http://graphics.ics.uci.edu

Results: Simplification

Models with 19778, 7238, 1500 and 778 triangles.

Models with 16450, 6450, 2450 and 450triangles.

http://graphics.ics.uci.edu

Results: Simplification

Models with 101924, 33924, 9924 and 1924 triangles.

http://graphics.ics.uci.edu

Results: View-dependent simplification

Notice dramatic change

in simplification

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Results: Stripificationwith asymmetric simplification

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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

- Hierarchyless simplification
- Simplification and stripification
- Connectivity compression
- Results
- Conclusion

http://graphics.ics.uci.edu

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

- 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

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

http://graphics.ics.uci.edu

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