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# Alpha Shapes - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Alpha Shapes' - paul2

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Presentation Transcript

### Alpha Shapes

• Shape Modelling

• Creates shapes out of point sets

• Gives a hierarchy of shapes.

• Has been used for detecting pockets in proteins.

• For reverse engineering

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.

The smallest convex set that contains the entire point set.

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.

• Post offices for the population in an area

• Subdivision of the plane into cells.

• Always Convex cells

• Curse of Dimension cells.

• Map

• Map Convex Hull back -> Delaunay

• Map

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

• General Position of points in

• k-simplex, Simplicial Complex

• Flipping in 2D and 3D

Delaunay triangulations are simplicial complexes.

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

Alpha Controls the desired level of detail.

• Decide on Speed / Accuracy Trade off

• Exact Arithmetic : Keep Away

• SoS : Keep Away

• Simple Solution: Juggle Juggle and Juggle

(To get to General Position)

Lot of Algorithms available!!!

• Incremental Flipping?

• Divide and Conquer?

• Sweep?

• Randomized or Deterministic?

• Do I calculate Voronoi or Delaunay??

• . . . . . . . . . .

( I got confused  )

• 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 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!”

• Lets get started, Simple things first

• Union of Balls

“If the facts don't fit the theory, change the facts.”

--Albert Einstein

Weighted Voronoi: Seems not so tough yet

Edelsbrunner: Union of balls and alpha shapes are

homotopy equivalent for all alpha.

Courtesy Dey, Giesen and John 04.

The Dual Complex: Assuming General position, at most

3 Voronoi Cells meet at a point.

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

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

It’s a Simplicial Complex all the way

( for a topologist )

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

A Filtration??? (sequence of complexes)

• Filteration = All Alpha Shapes!!!

• Alpha Shapes in 3D!!

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

• 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! 

• 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??

• 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.