SingleStrip Triangulation of Manifolds with Arbitrary Topology

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Single-Strip Triangulation of Manifolds with
Arbitrary Topology

• M. Gopi
• David Eppstein
• University of California, Irvine

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Triangle Stripification

• Triangle Strip A sequence of edge-connected
  triangles.
• Stripification Representing a triangulated model
  in terms of one or more triangle strips.

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Applications

• Rendering (fewer vertices through the graphics
  bus)
• Mesh compression
• Mesh streaming
• Space filling curves on surfaces
• Mesh unfolding on planes
• Mesh parameterization

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Stripification Problem

• Finding a single strip passing through all the
  triangles is a Hamiltonian path problem.
• Hamiltonian path problem is NP complete Garey,
  Johnson, Tarjan - 1976.
• Hence change the definition of the problem to
  find a reasonable solution.
• Multiple strips instead of single strip.
• Change triangulation to force a Hamiltonian path.

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Related Work

• Retain the input triangulation - Multiple
  triangle strips. typically all the related work
  in computer graphics
• STRIPE Evans, Skiena, and Varshney 96
• Velho, de Figueiredo, and Gomes 99.
• Chow 97.
• Triangulated Irregular Networks Snoeyink and
  Speckmann 97.
• Xiang, Held, and Mitchell 99.
• Tunnelling for levels of detail Stewart 01,
  Shafae and Pajarola 03
• Vertex caching Hoppe 99, Bogomjakov and Gotsman
  02
• Mesh compression Rossignac 99, Tauma and Gotsman
  98, Isenberg 00

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Related Work

• Change the input triangulation - Single Strip
  most of the related work in computational
  geometry
• Impose triangulation on planar point set Arkin,
  Held, Mitchell, and Skiena 96
• QuadTIN for irregular terrain point data using
  Steiner vertices Pajarola et al. 02
• Hamiltonian path triangulation of quadrilateral
  meshes of manifolds Taubin 02.
• Change the definition of triangle strip -- Strips
  through vertices instead of edges.
• Hamiltonian path exists for any edge connected
  mesh Demaine, et al. 03.

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Our Problem Statement

• INPUT
• Triangulated mesh of a manifold of arbitrary
  genus.
• OUTPUT
• Single Triangle Strip (Loop)
• CONSTRAINT
• Retain the input triangulation as much as
  possible.
• IDEA
• Use Perfect Graph Matching algorithm.
• Split triangles when required.

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Graph Matching

• Matching of a vertex of a graph to a only one of
  its adjacent vertices.

• Perfect matching if every vertex has a matching.

Perfect Matching
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Perfect Matching

• Any 3-regular 3-connected graph has a perfect
  matching Peterson 1891 (Polynomial time
  computation Edmonds 1965)

• Dual graph of any triangulated 2-manifold is a
  3-regular 3-connected graph.

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Stripfication Algorithm
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Stripfication Algorithm
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Stripfication Algorithm
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Stripfication Algorithm
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Stripfication Algorithm
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Stripification Algorithm
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Stripification Algorithm
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Stripification Algorithm
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Questions

• Number of splits depends on a particular perfect
  matching. Can we get a matching that yields
  minimum number of cycles (splits)?
• NP hard
• For a given perfect matching can we change it to
  get a matching that yields a minimal number of
  cycles (splits)?
• Yes!
• IDEA
• Avoid small cycles (three edge cycles)
• Merge cycles without triangle splits if possible.

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Avoid small cycles (Pre-processing)
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Merge cycles without splitting
Nodal vertex k unique incident cycles and 2k
incident triangles.
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Stripfication Algorithm
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Stripfication Algorithm
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Nodal Vertex in Triangle Splitting
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Nodal Vertex in Triangle Splitting
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Algorithm Summary

• Remove ? configuration in input triangulation.
• Perfect graph matching of the dual graph G.
• Re-insert removed triangles.
• Nodal vertex processing till no more nodal
  vertex.
• Construct a graph T whose nodes are disjoint
  cycles of G and edges are matched edges of G.
• Find the spanning tree of T ST(T)
• Split the primal triangle pair corresponding to
  the edges of ST(T), and merge cycles.

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Results
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Results
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Results
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Results
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Analysis of the algorithm

• Theoretical run-time complexity O(nlog3nloglogn)

3 mts.
7 mts.
(Python implementation)
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Current and Future work

• Manifolds with boundaries (done)
• Single strip between any two arbitrary triangles
  (done)
• Coarser control over strips (current work)
• Use weighted perfect matching
• Applications
• Preserve spatial locality for better caching
• Preserve geometric properties for use in culling
  and dynamic strip management
• Major limitation
• Finer and local control over strip growth is
  difficult.
• Apply to mesh compression and transmission
  algorithms (future work)
• Look for more applications.

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Implementation

• LEDA implementation of cardinality and weighted
  graph matching are available.
• Python implementation of graph matching is
  available from David Eppsteins web page.
  (http//www.ics.uci.edu/eppstein)
• C implementation of single loop and single
  strip (between any two arbitrary triangles)
  representation of manifolds with or without
  boundaries will soon be released for public.

Thank you!
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Stripification Problem

• Parameters
• Number of triangle strips
• Quality of the strip
• How many swaps are there in the strip?
• Does the strip makes use of vertex caching?
• Does the triangles in a strip satisfy any
  specific property?
• Variants of the problem statement
• Retain the input triangulation of the model.
• Allowed to change the input triangulation.