Topology Preserving Edge Contraction by Vpld86i


									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
 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||.
More Definitions
 d–1 sphere: Sd-1 = {x  Rd | ||x|| = 1}
    1-Sphere – Circle, 2-Sphere-Sphere (hollow)
 d-ball: Bd = {x  Rd | ||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  Rd | x1 = 1}
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
Lots more Definitions 
 k-Simplex is the convex hull of k+1 affinely
 independent point k  0
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.

                Example of faces:
                      {p0}, {p1}, {p0, p0p1},
                      {p0, p1, p2, p0p1, p0p2, p1p2, p0p1p2}
  p1            p2

 Proper faces
(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.
Simplicial Complex
 A collection of simplices, K, such that
    if s  K and t  s, then t  K i.e. for each face in K, all the
    faces of it is there K and all their subfaces are there etc.
    s, s’  K =>
        ss’ = f or

        ss’  s and ss’  s’

    i.e. if two faces intersect, they intersect on their face.
Simplicial Complex
                          Examples of a simplicial complex:
                                {p0}, {p0, p1, p2, p0p1}
                                {p0, p1, p2, p0p1, p0p2, p1p2, p0p1p2}
                    Examples of a non-simplicial complex:
p1               p2       {p0, p0p1}
                             Examples of a non-simplicial complex:
                          p4       {p0, p1, p2, p3, p4,
                p3                  p0p1, p1p2, p2p0, p3p4}

     p1              p2
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| = UsK int s
 Let B  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.

                      p1      p2
 The star of B is the set of all cofaces of simplices
 in B.
 Link of B is the set of all faces of cofaces of
 simplices in B that are disjoint from the simples
 in B
Mathematically Speaking
  B = {τ  K | τ  σ  B}
  St B = {σ  K | σ  τ  B}
  Lk B = St B - St B

Or Simply,
 A subdivision of K is a complex Sd K such that
   |Sd K| = |K| and
   s  K => s  Sd K
 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.
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

            u Vert K
                        bu ( x )  f (u )
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.
 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??
Order (contd.)
 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.
Edge Contraction (Finally!!)
In the Language of Math…
 Contraction is a surjective simplicial map
 jab:|K||L| defined by a surjective vertex map
                  u if u  Vert K – {a, b}
         f(u) =
                  c if u  {a, b}

 Outside |St ab|, the mapping is unity.
 Inside, it is not even injective.
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??!!!
How do I get there?
 Basically, the underlying space should not be
 affected in order to maintain topology.
So, What IS the Condition?!
 If I were to overlay the two stars, the links must
 be the same!
 The condition is: Lk a  Lk b = Lk ab

           Wake up now!!

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