Delaunay Meshing of Isosurfaces

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Delaunay Meshing ofIsosurfaces

• Tamal K. Dey
• Joshua A. Levine

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Motivations

• Delaunay refinement a simple strategy for meshing
  a large variety of domains.
• What about DR meshing for isosurfaces?
• Want high quality geometry and topological
  guarantees.
• Efficiency?

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Motivations
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Isosurfacing Background

• Soft Objects, Wyvill et al. 86.
• Marching Cubes, Lorensen/Cline 87.
• Topological Concerns with MC
• Nielson/Hamann 91 Chernyaev 95 Varadhan et
  al. 04
• Bhaniramka/Wenger/Crawfis 04.
• Schreiner/Scheidegger/Silva 06.

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Delaunay Refinement Background

• Furthest Point insertions of Chew 93.
• Smooth surfaces
• Boissonnat/Oudot 03 Cheng et al. 04
• Dey et al. 05 Oudot et al. 05.
• Polygonal (PLC) surfaces/volumes
• Shewchuk 98 Cheng et al. 04.
• More general classes (PSC)
• Cheng/Dey/Ramos 07.

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Formalities

• Assume fR3 ? R is a C2-smooth function and let P
  ? R3 be a discrete point set
• A volume dataset is a set,
• V (p, f(p)) p ? P.
• Assume ? f -1(?) is compact,
• ? is the isosurface at isovalue ?.
• Isosurface Problem
• Given V ?, find a suitable approximation
  (polygonal mesh) for ? by interpolating a
  function g from V such that f -1(?) g -1(?).

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Formalities

• For a finite point set S ? R3 with p ? S
• Voronoi cell
• Vp set of all points in R3 closer to p than any
  other point q ? S.
• Voronoi k-face
• intersection of 4-k Voronoi cells.
• Voronoi Diagram
• Vor S collection of Voronoi faces.
• Delaunay (j-1) simplex
• convex hull of j points which define a Voronoi
  (4-j)-face
• Delaunay Triangulation
• Del S collection of Delaunay simplices.
• Restricted Delaunay Tri. If S is on ?,
• Del S? subcomplex of Del S which consists of
  duals of Voronoi faces with nonempty intersection
  with ?.

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Delaunay Refinement Paradigm

• DelaunayRefine(S)
• Build Del S.
• while (!condition(Del S))
• Insert a new point into S.

• For surface meshing, criteria to refine guided by
  Del S? a working approximation for ?.
• New points to insert are taken as intersections
  of Voronoi edges with ?.
• So we require a primitive that does an
  intersection between a ray/segment and a volume
  dataset.
• Proof burden Does it terminate?

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Topological Ball Property (TBP)

• S has the TBP for ? if each k-face in Vor S
  either does not intersect ? or intersects it in a
  (k-1)-ball.
• Thm by Edelsbrunner and Shah 94 says that if S
  satisfies the TBP then Del S? is homeomorphic to
  ?.

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Improving Delaunay Refinement

• Expensive to maintain Del S Del S?.
• Early in the refinement, Del S? has the correct
  topology for all further insertions.
• Optimization
• Split algorithm into two stages.
• Maintain Del S? without Del S during the second.
• This technique works provided S is dense enough
  so that for all insertions p, both Del S? and
  Del Sp? are homeomorphic to ?.

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Del S? without Del S

• Key Observation(s)
• When inserting new p (an intersection between a
  Voronoi edge and the surface)
• A set of triangles, E, will be encroached by p
  (they will no longer have an empty Delaunay ball)
• A new set of triangles D, incident to p in Del (S
  ? p)?, will fill in this gap.
• Del S? Del (S ? p)? are manifolds, therefore
  E D must both be topological disks.
• Del S? - E Del (S ? p)? - D.
• Thus we can insert a point by computing E,
  removing it, and replacing the hole with D.

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Del S? without Del S

• Notice E can be computed by a local walk
• For each triangle, we maintain its surface
  intersection point.
• This defines a Delaunay ball for each triangle.
• Check if p is within this ball, if so the
  triangle will be encroached.

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Worthwhile?

• 3D
• Single Stage Delaunay Refinement
• DelIso
• Our two stage version
• Times are in seconds.

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Two Stage Algorithm

• DelIso(V,?)
• S ? Initialize()
• Recover(Del S)
• TS ? Refine(Del S)
• return TS.

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Stage 1 Recover

• Based on the algorithm of CDRR04
• First triggers refinement of Del S? based on TBP
  violations to capture topology.
• Followed by a small amount of geometric
  refinement
• Refinement guided by radius/pole height criteria
  AB99
• After this point (we assume) no additional
  topological changes will occur with future
  insertions.

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Stage 2 Refine

• First extracts Del S? into a mesh TS.
• Continues geometric refinement based on pole
  heights and aspect ratios.
• Point insertions done without Del S using the
  technique previously described.

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Intersection Computation

• Points to insert are intersections of Voronoi
  edges with ?.
• Standard technique use a kd-tree search
  WFMSS50 to find intersecting voxels, then solve
  trilinear equation within each voxel PSLHS98.

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Intersection Computation

• During second stage
• We dont have a Voronoi edge, but we do have
  (geometric) dual edges for triangles.
• Moreover, these triangles are already a good
  approximation of the surface, so the intersection
  points are close.
• So we start at circumcenter of triangle and grow
  a line segment outward until it intersects the
  surface.

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Examples - Engine
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Examples - Cadaver
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Conclusions

• High geometric quality isosurfaces with correct
  topology.
• Improved Delaunay refinement.
• Do we need Stage 1?
• We only require a primitive that does
  intersection between a ray/segment and a volume
  dataset.

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Software

• DelIso and other surface meshing algorithms
  freely available on web
• http//www.cse.ohio-state.edu/tamaldey

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Boundary Considerations

• Supporting theory assumes ? is a smooth manifold
  w/ out boundary.
• Most isosurfaces have boundaries though!
• We precompute these boundaries, sample them, and
  allow Manifold() check to pass with a
  half-manifold at boundary.

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Conclusions