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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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Voronoi Diagrams This set is a convex polyhedra since it is an intersection of half spaces. These polyhedra define a decomposition of R d. 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 -> 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 Courtesy Dey, Giesen and John 04. Edelsbrunner: Union of balls and alpha shapes are homotopy equivalent for all alpha.

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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

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