Topology Preserving Edge Contraction

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# Topology Preserving Edge Contraction - PowerPoint PPT Presentation

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

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.
• 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 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
• 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 ³ 0

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)
• 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 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 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
• 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

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.
Star
• 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
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 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
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
• 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
• 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…
• 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…
• 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?!
• 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,
• THANKS!!!
• Wake up now!!