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Alpha Shapes 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 Convexity A set S in Euclidean space is said to be convex if every straight line segment

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Used for l.jpg
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


Convexity l.jpg
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.


Convex hulls l.jpg
Convex Hulls

The smallest convex set that contains the entire point set.




Voronoi diagrams l.jpg
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.



Voronoi diagrams9 l.jpg
Voronoi Diagrams

  • Post offices for the population in an area

  • Subdivision of the plane into cells.

  • Always Convex cells

  • Curse of Dimension cells.


Lifting map magic l.jpg
Lifting Map: Magic

  • Map

  • Map Convex Hull back -> Delaunay

  • Map

  • mapped back to lower dimension is the Voronoi diagram!!!


Other definitions l.jpg
Other Definitions

  • General Position of points in

  • k-simplex, Simplicial Complex

  • Flipping in 2D and 3D



Simplicial complex l.jpg
Simplicial Complex

Delaunay triangulations are simplicial complexes.



Alpha shapes15 l.jpg
Alpha Shapes

The space generated by point pairs that can be touched by an

empty disc of radius alpha.


Alpha shapes16 l.jpg
Alpha Shapes

Alpha Controls the desired level of detail.




Implementing alpha shapes l.jpg
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)


Delaunay how l.jpg
Delaunay: How???

Lot of Algorithms available!!!

  • Incremental Flipping?

  • Divide and Conquer?

  • Sweep?

  • Randomized or Deterministic?

  • Do I calculate Voronoi or Delaunay??

  • . . . . . . . . . .

    ( I got confused  )


Predicates l.jpg
Predicates??

  • What are Predicates???

  • Why do I bother??

  • Which one do I pick?

  • When do I use Exact Predicates?

  • What else is available?


What data structure l.jpg
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!”


Theory l.jpg
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


That was simple l.jpg
That was simple!

Weighted Voronoi: Seems not so tough yet


An example in the dual l.jpg
An example in the dual

Edelsbrunner: Union of balls and alpha shapes are

homotopy equivalent for all alpha.

Courtesy Dey, Giesen and John 04.


What next l.jpg
What Next?

The Dual Complex: Assuming General position, at most

3 Voronoi Cells meet at a point.

For fixed weights, alpha, It’s a alpha complex!



Alpha complex l.jpg
Alpha Complex

The subset of delaunay tesselation in d-dimensions that has simplices having

Circumradius greater than Alpha.

It’s a Simplicial Complex all the way

( for a topologist )


Filter and filtration l.jpg
Filter and Filtration

A Filter!!!! (an order on the simplices)

A Filtration??? (sequence of complexes)


Filteration l.jpg
Filteration???

  • Filteration = All Alpha Shapes!!!

  • Alpha Shapes in 3D!!

  • Covers, Nerves, Homotopy, Homology?? (Keep Away for now) 


Alpha shapes31 l.jpg
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.

    That’s it!! Wasn’t it a cute idea for 1983! 


So far so good l.jpg
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??


Future work l.jpg
Future Work

  • U want to work on Alpha Shapes??

    (And get papers accepted too, That’s 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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