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**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**The smallest convex set that contains the entire point set.**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 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**• Map • Map Convex Hull back -> Delaunay • Map • mapped back to lower dimension is the Voronoi diagram!!!**Other Definitions**• General Position of points in • k-simplex, Simplicial Complex • Flipping in 2D and 3D**Simplicial Complex**Delaunay triangulations are simplicial complexes.**Alpha Shapes**The space generated by point pairs that can be touched by an empty disc of radius alpha.**Alpha Shapes**Alpha Controls the desired level of detail.**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???**Lot of Algorithms available!!! • Incremental Flipping? • Divide and Conquer? • Sweep? • Randomized or Deterministic? • Do I calculate Voronoi or Delaunay?? • . . . . . . . . . . ( I got confused )**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!**• 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**• 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!**Weighted Voronoi: Seems not so tough yet**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?**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**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**A Filter!!!! (an order on the simplices) A Filtration??? (sequence of complexes)**Filteration???**• Filteration = All Alpha Shapes!!! • Alpha Shapes in 3D!! • Covers, Nerves, Homotopy, Homology?? (Keep Away for now) **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!**• 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**• 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.