Alpha Shapes

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Alpha Shapes
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Used for

• Shape Modelling
• Creates shapes out of point sets
• Gives a hierarchy of shapes.
• Has been used for detecting pockets in proteins.
• For reverse engineering

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Convexity
A set S in Euclidean space is said to be convex
if every straight line segment having its two
end points in S lies entirely in S.
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Convex Hulls
The smallest convex set that contains the entire
point set.
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Triangulations
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Triangulations
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Voronoi Diagrams
This set is a convex polyhedra since it is an
intersection of half spaces. These polyhedra
define a decomposition of Rd. The voronoi complex
V(P) of P is the collection of all voronoi
objects. Delaunay complex is the dual of the
voronoi complex.
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Delaunay Triangulations
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Voronoi Diagrams

• Post offices for the population in an area
• Subdivision of the plane into cells.
• Always Convex cells
• Curse of Dimension cells.

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Lifting Map Magic

• Map
• Map Convex Hull back -gt Delaunay
• Map
• mapped back to lower dimension is the
  Voronoi diagram!!!

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Other Definitions

• General Position of points in
• k-simplex, Simplicial Complex
• Flipping in 2D and 3D

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k-simplex
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Simplicial Complex
Delaunay triangulations are simplicial complexes.
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Flipping
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Alpha Shapes
The space generated by point pairs that can be
touched by an empty disc of radius alpha.
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Alpha Shapes
Alpha Controls the desired level of detail.
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Sample Outputs
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Sample Output
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Implementing Alpha Shapes

• Decide on Speed / Accuracy Trade off
• Exact Arithmetic Keep Away
• SoS Keep Away
• Simple Solution Juggle Juggle and Juggle
• (To get to General Position)

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Delaunay How???

• Lot of Algorithms available!!!
• Incremental Flipping?
• Divide and Conquer?
• Sweep?
• Randomized or Deterministic?
• Do I calculate Voronoi or Delaunay??
• . . . . . . . . . .
• ( I got confused ? )

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Predicates??

• What are Predicates???
• Why do I bother??
• Which one do I pick?
• When do I use Exact Predicates?
• What else is available?

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What Data Structure!

• What data structure is used to compute Delaunay?
• Which algorithm is easy to code?
• How do I implement the Alpha Shape in my code?
• Any example codes available to cheat?
• Creativity is
  the art of hiding Sources!

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Theory

• Its not so bad)
• Lets get started, Simple things first
• Union of Balls

If the facts don't fit the theory, change the
facts.
--Albert Einstein
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That was simple!
Weighted Voronoi Seems not so tough yet
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An example in the dual
Edelsbrunner Union of balls and alpha shapes are
homotopy equivalent for
all alpha.
Courtesy Dey, Giesen and John 04.
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What Next?
The Dual Complex Assuming General position, at
most 3 Voronoi Cells meet at a point.
For fixed weights, alpha, Its a alpha complex!
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Example of Dynamic Balls!
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Alpha Complex

• The subset of delaunay tesselation in
  d-dimensions that has simplices having
• Circumradius greater than Alpha.
• Its a Simplicial Complex all the way
• ( for a topologist )

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Filter and Filtration
A Filter!!!! (an order on the simplices)
A Filtration??? (sequence of complexes)
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Filteration???

• Filteration All Alpha Shapes!!!
• Alpha Shapes in 3D!!
• Covers, Nerves, Homotopy, Homology?? (Keep Away
  for now) ?

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Alpha Shapes??

• What the hell were Alpha Shapes???
• As the Balls grow(Alpha becomes bigger) on
  the input point set, the dual marches thru the
  Filteration, defining a set of shapes.
• Thats it!! Wasnt it a cute idea for 1983! ?

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So Far So Good!

• How do I calculate Alpha?? ?
• How do I decide the weights for a weighted Alpha
  shape? ?
• Is there an Alpha Shape that is Piecewise Linear
  2-Manifold?
• Isnt the sampling criterion too strict??
• Delaunay is Costly ?, Can we use Point Set
  Distribution information??

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Future Work

• U want to work on Alpha Shapes??
• (And get papers accepted too, Thats tough)
• Alpha shapes is old now, u could try something
  new!
• What else can we try? Try Energy Minimization,
  Optimization! Noise. With provability thrown in,
  That is still open.

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Thats all Folks