Piecewise Linear Interpolation between Polygonal Slices

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Piecewise Linear Interpolation between Polygonal
Slices

• Gill Barequet
• Micha Sharir

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Reconstruction from Polygonal Slices

• Input series of parallel planar slices not
  crossing but might be nested.
• Output Polyhedral solid model whose
  cross-sections along the given planes coincide
  with the input slices.

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Wish List

• No input limitations.
• Automatic procedure.
• intuitive surface.

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The Algorithm in Brief

• Match similarities
• Stitch
• Triangulate the rest

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Orientation

• Orient all contours in each slice in alternating
  directions, from the outside to the inside,
  Starting clockwise.

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Discretize

• Discretize each contour polygon into a cyclic
  sequence of vertices with a constant and small
  (enough) length.

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Finding similarities Voting process (1)

• The voting process is done for every pair of
  contour polygons which belong to a different
  slices.
• Indexing The two cyclic contours are break into
  two linear indexed vertices list.
• For each vertex C2(j) with distance less then e
  from vertex C1(i) we place a vote for (j-i)(mod
  lc2).
• Large number of votes reflects long portions of
  contour matching.
• Identify the start and end of each match.

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Finding similarities Voting process (2)

• Score each match according to the number of votes
  and additional parameters (Distance, max.
  consecutive mismatch).
• Notes
• e is an a-priory estimation of the physical
  differences between two slices.
• The distance function considers orientation. This
  guarantees that the material will be on the same
  side of the contours.
• Its most important that every intersected pair of
  contours will be matched.

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Stitching

• We stitch all the pairs vertices chains which
  were discovered.
• Stitching algorithm
• Inverse the orientation of lower chain.
• Choose arbitrary one end to start the stitching
  from.
• Always add the triangle with the shorter
  perimeter.

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Stitching and summing (1)

• Inversing orientation
• Stitching
• Summing (e (-e) 0)

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Stitching and summing (2)

• After summing we remain with 3D polygon (Clefts)
• When projected to xy-plane Clefts are
  non-intersecting. This is due to the matching in
  the interesting neighborhood.
• The Clefts form an hierarchy of nested curves
  when projected to xy-plane.

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Connecting the Clefts

• Build a complete graph with vertices represent
  Clefts and edges represent the minimum distance
  between the two Clefts.
• Calculate Minimum Spanning Tree.
• Build Bridges According to the MST ? simple 3D
  cycle (also a cycle when projected to xy-plane).
• Triangulate Using dynamic programming function
  which minimize a given function (e.g area or
  perimiter).

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Example (1)
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Example (2)
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Conclusions (1)

• The algorithm overcomes the classic problems
  nesting and branching.
• The algorithm doesnt take any assumption on
  input.
• The procedure is automatic except for few
  parameters (discretize length, e distance hard
  decision parameter and cost function of the
  Clefts triangulation).
• The algorithm enforces that planer intersecting
  contours will be connected, and for a reasonable
  e non nested or intersecting contours wont be
  connected.

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Conclusions (2)

• The algorithm divide the solution to a set of
  simple known algorithms and therefore
  implementation should be simple as well.
• The authors claim to test the algorithm on
  several examples including synthetic contours,
  real contours (CT,MRI) and cases taken from
  previous works (boissonat and unsolvable
  problems of gitlin, ORouke and subramanian). The
  Algorithms gave good results with all of those
  examples. However, only few examples are given in
  the paper.

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Conclusions (3)

• Its not proven in the article that bridges wont
  cross contours and Cleft projection is a simple
  circle. The authors does refer to this problem as
  unlikely to happen if using sufficiently good
  parameters.
• The paper is written clearly and is
  self-contained. More and better examples would
  make it simpler to understand.

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Bonus example (1)
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Bonus example (2)
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Bonus example (3)