Voronoi diagrams of

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Voronoi diagrams of nice point sets

• Nina Amenta
• UC Davis
• The World a Jigsaw

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Why we love the Voronoi diagram
Interesting, elegant structure.
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Why we love the Voronoi diagram

• Its useful! for surface reconstruction....

output
input
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Point Set Capture
Point Grey Bumblebee
Cyberware
Boom in surface reconstruction research.
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Why we love the Voronoi diagram

• ...and for mesh generation.

Jonathan Shewchuk, triangle
Triangles with no small angles, respect size of
input features.
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Why we love the Voronoi diagram
Shewchuk, 98

• Three-dimensional tetrahedralization

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Why we hate the Voronoi diagram

• Hard to compute because of numerical instability.

Ten years of research on computing this both
robustly and quickly.
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Why we hate the Voronoi diagram

• Fundamental problem for moving points.

Many unnecessary connectivity adjustments.
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Why we hate the Voronoi diagram

• Worst-case size is exponential in the dimension.

O( nd/2 )
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Why we hate the Voronoi diagram

• Worst-case size is exponential in the dimension.

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Why we hate the Voronoi diagram

• Dual sliver tetrahedra!

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Worst-case approach

• We have done a great job of designing programs
  and algorithms which robustly handle all inputs.

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Consider nice inputs!
Users want to reconstruct surface or mesh object,
they dont care what the point set is.
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Surface reconstruction

• Requires 3D Delaunay triangulation.
• O(n2)?
• Only if you really, really try...these points are
  not on a 1D set, they lie on a 2D surface.
• Dwyer, 91, uniform random points have O(n)-size
  Delaunay triangulation in any fixed dimension.
  But what about points on (near) lower dimensional
  manifolds?

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Practice suggests O(n)
A Choi, 01
Adding samples on surfaces in random order, Del.
tetrahedra grows linearly.
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Linear special cases
Golin Na 00 - Uniform random on surface of
convex polyhedron Golin Na 02 - Uniform random
on polyhedral surface, O(n log4n) (almost
linear!)
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Attali Boissonnat 03
Complexity of Delaunay triangulation is O(n).
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Nicely sampled polygons
Fixed set of polygons in R3. Area, boundary
length, number of polygons constant.
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Sampling model
Every point has at least one and at most k
samples within distance 1/n2. Consider behavior
as n-gtinfinity. Golin Na have similar results
in random case.
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Interiors
x is reflection of x across planar bisector.
Voronoi balls nearly tangent to face at x are
close to x.
x
x
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Interiors
Any ball touching a point far from x on the
opposite plane contains too much area to be
empty.
x
x
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Interiors
All Delaunay edges from x into interior of
another polygon have to end within (1/n2) ball
around x, so only O(1) such.
x
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Edges
Sample near an edge can be connected to more
other samples....but there are only sqrt(n) of
them. The difficult part of the proof!
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Open Questions

• Attali, Boissonnat Lieuter, 03, Generic
  smooth surface S, O(n log n) - best
  possible?
• Erickson, 01, wickedly chosen smooth surface can
  be O(n3/2)
• Noise? Higher dimensions? Argument we just did
  applies to interiors of (d-1)-planes in Rd.

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Mesh Generation
All triangles approximately equilateral, input
features (or given density function) controls
maximum size.
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Harder in 3D
Have to avoid slivers!
Recent approach Alliez, Cohen-Steiner, Yvinec
Desbrun, Siggraph 05. Related to....
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Llyods method, 1982
Compute Voronoi diagram. Move each sample to the
center of mass of its Voronoi cell. Repeat.
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Lloyds method
Apply to all points simultaneously.
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Centroidal Voronoi Diagram
When density is uniform, converges at a minimum
of ?x (x - c(x) )2 c(x) center of Voronoi
cell containing x
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2D, uniform distribution
modulo edge effects
bounded domain
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2D, uniform distribution
Modulo edge effects, only minimum is tightest
possible packing of uniform balls - Gersho,
1979. Round Voronoi cells, all similar size -gt
round Delaunay triangles, all similar size.
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2D, non-uniform distribution
Distribution given by darkness of photo -
stippling patterns! Adrian Secord, NPAR 2002
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In 3D, more difficult
No proof that converges to closest sphere
packing. Hales, 98, showed closest sphere
packing in 3D is regular.
Cannonball packing is not unique - each layer has
two alternate translations.
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If so, no slivers...
Dual of any cannonball packing will be made of
two kinds of tetrahedra, both round.
But does it converge, or does it get stuck in
local minima? Making Voronoi cells round does
not directly make Delaunay tetrahedra round.
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Different iteration
Does not seem to work well in practice. Making
Voronoi cells round does not necessarily make
Delaunay tetrahedra round. Idea (Chen and Xu)
Directly optimize Delaunay tetrahedra.
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Property to dualize?
x -gt ( x, x2 ) Convex Hull projects down to
Delaunay triangulation Intersection of tangent
halfspaces projects down to Voronoi diagram.
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Property to dualize?
local minimum of E ?x (x - c(x) )2
c(x)
E error of Voronoi polytope as approximation of
paraboloid.
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Dualization
xi
Minimize error of Delaunay polytope as
approximation of paraboloid.
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Error minimizing iteration
x
New x 1/area(region) ?T (area(T)
circumcenter(T) )
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Centroidal Delaunay Triangulation
Centroidal Voronoi diagram
Centrodial Delaunay triangulation
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Properties in 2D?
Does this converge to the optimal packing in 2D?
Or can it get stuck in local minima?
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Good in practice
Many open questions in theory...
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Open Questions

• CVD and CDT form sphere packings in 3D?
• Can we compute them without iterating Delaunay
  triangulation?
• Can we compute the DT of a nice distribution
  using only floating-point arithmetic?

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Moving Points

• Show that some iterative process maintains a
  distribution that avoids unnecessary flips (no 5
  points near a sphere).

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Thank you.