alpha shapes l.
Skip this Video
Loading SlideShow in 5 Seconds..
Alpha Shapes PowerPoint Presentation
Download Presentation
Alpha Shapes

Loading in 2 Seconds...

play fullscreen
1 / 34

Alpha Shapes - PowerPoint PPT Presentation

  • Uploaded on

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

Alpha Shapes

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
used for
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

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
Convex Hulls

The smallest convex set that contains the entire point set.

voronoi diagrams
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
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
Lifting Map: Magic
  • Map
  • Map Convex Hull back -> Delaunay
  • Map
  • mapped back to lower dimension is the Voronoi diagram!!!
other definitions
Other Definitions
  • General Position of points in
  • k-simplex, Simplicial Complex
  • Flipping in 2D and 3D
simplicial complex
Simplicial Complex

Delaunay triangulations are simplicial complexes.

alpha shapes15
Alpha Shapes

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

empty disc of radius alpha.

alpha shapes16
Alpha Shapes

Alpha Controls the desired level of detail.

implementing alpha shapes
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
Delaunay: How???

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
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!”

  • 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
That was simple!

Weighted Voronoi: Seems not so tough yet

an example in the dual
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
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
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
Filter and Filtration

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) 
alpha shapes31
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
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
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.