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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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topology preserving edge contraction

Topology Preserving Edge Contraction

Paper By

Dr. Tamal Dey et al

Presented by

Ramakrishnan Kazhiyur-Mannar

some definitions lots actually
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||.
more definitions
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}
even more definitions
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
Lots more Definitions 
  • k-Simplex is the convex hull of k+1 affinely independent point k ³ 0
still more definitions

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}

definitions i have given up trying to get unique titles
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.
simplicial complex
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.

simplicial complex9
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

subcomplex closure
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
closure

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.
slide12
Star
  • The star of B is the set of all cofaces of simplices in B.
slide13
Link
  • Link of B is the set of all faces of cofaces of simplices in B that are disjoint from the simples in B
subdivision
Subdivision
  • A subdivision of K is a complex Sd K such that
    • |Sd K| = |K| and
    • s Î K => s Î Sd K
homeomorphism
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.
combinatorial version
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
combinatorial version contd
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.
order
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??
boundary
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.
in the language of math
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) =

one last term
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
How do I get there?
  • Basically, the underlying space should not be affected in order to maintain topology.
so what is the condition
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
finally
Finally,
  • THANKS!!!
  • Wake up now!!