VIEWS: 7 PAGES: 27 POSTED ON: 6/12/2012
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 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. p0 Example of faces: {p0}, {p1}, {p0, p0p1}, p3 {p0, p1, p2, p0p1, p0p2, p1p2, p0p1p2} p1 p2 Proper faces 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 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. and s, s’ K => ss’ = f or ss’ s and ss’ 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: p1 p2 {p0, p0p1} p0 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| = UsK int s Closure 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. p0 p1 p2 Star The star of B is the set of all cofaces of simplices in B. Link 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 L Or Simply, Subdivision A subdivision of K is a complex Sd K such that |Sd K| = |K| and s K => s Sd K Homeomorphism Homeomorphism is topological equivalence An intuitive definition? Technical definition: Homeomorphism between two spaces X and Y is a bijection h:XY 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 KVert 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. 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?? Order (contd.) 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. 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?! Simple. If I were to overlay the two stars, the links must be the same! The condition is: Lk a Lk b = Lk ab Finally, THANKS!!! Wake up now!!