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Topology Preserving Edge Contraction

Topology Preserving Edge Contraction. Paper By Dr. Tamal Dey et al Presented by Ramakrishnan Kazhiyur-Mannar. Some Definitions (Lots actually). Point – a d-dimensional point is a d-tuple of real numbers.

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Topology Preserving Edge Contraction

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  1. Topology Preserving Edge Contraction Paper By Dr. Tamal Dey et al Presented by Ramakrishnan Kazhiyur-Mannar

  2. Some Definitions (Lots actually) • Point – a d-dimensional point is a d-tuple of real numbers. • Norm of a Point – If the point x = (x1, x2, x3…xd), the norm ||x|| = (Sxi2)1/2 • Euclidean Space – A d-dimensional Euclidean space Rd is the set of d-dimensional points together with the euclidean distance function mapping each set of points (x,y) to ||x-y||.

  3. More Definitions • d–1 sphere: Sd-1 = {x ÎRd | ||x|| = 1} • 1-Sphere – Circle, 2-Sphere-Sphere (hollow) • d-ball: Bd = {x ÎRd | ||x|| £ 1} • 2-ball - Disk (curve+interior), 3-ball – Sphere (Solid) • The surface of a d-ball is a d-1 sphere. • d-halfspace: Hd = {x ÎRd | x1 = 1}

  4. Even More Definitions • Manifold: A d-manifold is a non-empty topological space where at each point, the neighborhood is either a Rd or a Hd. • With Boundary/ Without Boundary

  5. Lots more Definitions  • k-Simplex is the convex hull of k+1 affinely independent point k ³ 0

  6. p0 p3 p1 p2 Still more Definitions • Face: If s is a simplex a face of s, t is defined by a non-empty subset of the k+1 points. • Proper faces Example of faces: {p0}, {p1}, {p0, p0p1}, {p0, p1, p2, p0p1, p0p2, p1p2, p0p1p2}

  7. Definitions (I have given up trying to get unique titles) • Coface: If t is a face of s, then s is a coface of t, written as t £ s. • The interior of the simplex is the set of points contained in s but not on any proper face of s.

  8. Simplicial Complex • A collection of simplices, K, such that • if s Î K and t £ s, then t Î K i.e. for each face in K, all the faces of it is there K and all their subfaces are there etc. and • s,s’ Î K => • sÇs’ = f or • sÇs’ £ s and sÇs’ £ s’ i.e. if two faces intersect, they intersect on their face.

  9. Simplicial Complex p0 Examples of a simplicial complex: {p0}, {p0, p1, p2, p0p1} {p0, p1, p2, p0p1, p0p2, p1p2, p0p1p2} p3 Examples of a non-simplicial complex: {p0, p0p1} p1 p2 p0 Examples of a non-simplicial complex: {p0, p1, p2, p3, p4, p0p1, p1p2, p2p0, p3p4} p4 p3 p1 p2

  10. Subcomplex, Closure • A subcomplex of a simplicial complex one of its subsets that is a simplicial complex in itself. • {p0, p1, p0p1} is a subcomplex of {p0, p1, p2, p0p1, p1p2, p2p0, p0p1p2} • The Underlying space is the union of simplex interiors. |K| = UsÎK int s

  11. p0 p2 p1 Closure • Let B Í K (B need not be a subcomplex). • Closure of B is the set of all faces of simplices of B. • The Closure is the smallest subcomplex that contains B.

  12. Star • The star of B is the set of all cofaces of simplices in B.

  13. Link • Link of B is the set of all faces of cofaces of simplices in B that are disjoint from the simples in B

  14. Mathematically Speaking Or Simply, L

  15. Subdivision • A subdivision of K is a complex Sd K such that • |Sd K| = |K| and • s Î K => s Î Sd K

  16. Homeomorphism • Homeomorphism is topological equivalence • An intuitive definition? • Technical definition: Homeomorphism between two spaces X and Y is a bijection h:XY such that both h and h’ are continuous. • If $ a Homeomorphism between two spaces then they are homeomorphic X» Y and are said to be of the same topological type or genus.

  17. Combinatorial Version • Complexes stand for topological spaces in combinatorial domain. • A vertex map for two complexes K and L is a function f: Vert KVert L. • A Simplicial Map f: |K||L| is defined by

  18. Combinatorial Version (contd.) • f need not be injective or surjective. • It is a homeomorphism iff f is bijective and f -1 is a vertex map. • Here, we call it isomorphism denoted by K ~ L. • There is a slight difference between isomorphism and homeomorphism.

  19. Order • Remember manifolds? • What if the neighborhood of a point is not a ball? • For s, a simplex in K, if dim St s = k, the order is the smallest interger I for which there is a (k-i) simplex h such that St s ~ St h • What is that mumbo-jumbo??

  20. Order (contd.)

  21. Boundary • The Jth boundary of a simplicial complex K is the set of simplices with order no less than j. • Order Bound: Jth boundary can contain only simplices of dimensions not more than dim K-j • Jth boundary contains (j+1)st Boundary. • This is used to have a hierarchy of complexes.

  22. Edge Contraction (Finally!!)

  23. In the Language of Math… • Contraction is a surjective simplicial map jab:|K||L| defined by a surjective vertex map • Outside |St ab|, the mapping is unity. Inside, it is not even injective. u if u Î Vert K – {a, b} c if u Î {a, b} f(u) =

  24. One Last Term… • An unfolding i of jab is a simplicial homeomorphism |K|  |L|. • It is local if it differs from jab only inside |E| and it is relaxed if it differs from jab only inside |St E| • Now, WHAT IS THAT??!!!

  25. How do I get there? • Basically, the underlying space should not be affected in order to maintain topology.

  26. So, What IS the Condition?! • Simple. • If I were to overlay the two stars, the links must be the same! • The condition is: Lk a Ç Lk b = Lk ab

  27. Finally, • THANKS!!! • Wake up now!!

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