Cutting a surface into a Disk

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Cutting a surface into a Disk

• Jie Gao
• Nov. 27, 2002

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Papers

• Optimally cutting a surface into a disk, by Jeff
  Erickson and Sariel Har-Peled, in SoCG02.
• Geometry images, by Xianfeng Gu, Steven J.
  Gortler and Hugues Hoppe, in Siggraph02.

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• Given a 3d mesh M, find a set of edges (called
  cut graph) whose removal transforms the surface
  into a topological disk.

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Outline

• Theory work
• Minimize the length of the cut graph
• Algorithms for exact/approximate solutions.
• In practice
• Geometry Images.
• Heuristics

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Theory work

• Optimally cutting a surface into a disk, by Jeff
  Erickson and Sariel Har-Peled, in SoCG02.
• Minimize the total weight of the cut graph.
• e.g., the total length of the cut.

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Definitions

• M compact 2-manifold with boundary.
• Genus g maximum number of disjoint
  non-separating cycles of M.
• k number of boundary components.
• 1-skeleton M1 of M is the graph consisting of all
  the vertices and edges.
• A cut graph G is a subgraph of M1 so that M\G
  (polyhedral schema) is homeomorphic to a disk.
• Goal find a polyhedral schema with minimum
  perimeter.

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• Computing min cut graph of M with fixed g or k
  is NP-hard reduction from rectilinear Steiner
  tree problem.
• Each point puncture
• Cut graph Steiner tree
• To get high genus attach tori or cross-caps to
  punctures.

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• Cut path from one branch point (degreegt2) or
  boundary point to another, without branching
  point in the middle.
• Lemma Any cut path in the minimum cut graph G
  can be decomposed into 2 equal-length shortest
  path in M1 .
• Proof If there is a shorter path, then we can
  cut and re-glue and get a shorter cut graph.

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Min cut graph

• Reduced cut graph
• Remove vertices of degree-1 and their edges.
• Replace maximal path through degree-2 vertices by
  a single edge
• Degree-3, 4g2k-2 vertices, 6g3k-3 edges
  (Eulers formula).
• Min cut graph of M with fixed g and k can be
  computed in time O(nO(gk)).
• Find all-pairs shortest paths
• Enumerating all possible solutions.

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Approximate Min Cut Graph

• Convert M to punctured manifold M without
  boundary.
• Contract every boundary to a puncture point.
• Claim G(M)G(M), M includes all punctures.
• Cut along short essential cycles (doesnt bound a
  disk with lt2 punctures) until we get a set of
  punctured spheres.
• Connect the punctures by cutting along a MST.
• Re-glue some previously cut edges back.

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Approximate Min Cut Graph

• Computing shortest essential cycle is expensive
  O(n2logn).
• Use 2-approximation O(nlogn).
• O(log2g)-approximate min cut graph.
• Running time O(g2nlogn).

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In practice

• Geometry images, by Xianfeng Gu, Steven J.
  Gortler and Hugues Hoppe, in Siggraph02.
• We want not only the cut, but also a geometry
  image D a unit square.
• Find a cut ? and parametrization F.
• F piecewise linear map from D to M\?.

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Why a disk?

• Texture mapping.
• Hardware rendering
• Compression and decompression
• Wavelet-based coder

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Cut parametrize

• Find an initial cut parametrization.
• Improve the cut with the info from the
  parametrization.
• Iterate until no improvement.

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Cut parametrize

• Find an initial cut parametrization.
• Any cut, e.g., the previous one.
• Heuristics find a cut and locally shorten a cut
  path.
• Improve the cut with the info from the
  parametrization.
• Iterate until no improvement.

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Cut parametrize

• Find an initial cut parametrization.
• boundary refinement
• Improve the cut with the info from the
  parametrization.
• Iterate until no improvement.

optimize a few interior pts
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Cut parametrize

• Find an initial cut parametrization.
• Improve the cut with the info from the
  parametrization.
• Search for regions with large geometric stretch.
• Pick a extremal vertex v.
• Add to ? the shortest path from v to the current
  boundary.
• Iterate until no improvement.

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Example
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More example