1 / 52

Creating a simplicial complex

MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 25, 2013: Applicable Triangulations. Fall 2013 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics

loren
Download Presentation

Creating a simplicial complex

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 25, 2013: Applicable Triangulations. Fall 2013 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics Applied Mathematical and Computational Sciences, University of Iowa http://www.math.uiowa.edu/~idarcy/AppliedTopology.html

  2. Creating a simplicial complex 1.) Adding 1-dimensional edges (1-simplices) Add an edge between data points that are “close”

  3. Vietoris Rips complex = flag complex = clique complex 2.) Add all possible simplices of dimensional > 1.

  4. Creating the Čech simplicial complex 1.) B1 … Bk+1 ≠ ⁄ , create k-simplex {v1, ... , vk+1}. 0 U U

  5. Consider X an arbitrary topological space. Let V= {Vi | i = 1, …, n } where Vi X , The nerve of V = N(V) where The k -simplices of N(V) = nonempty intersections of k +1 distinct elements of V. For example, Vertices = elements of V Edges = pairs in Vwhich intersect nontrivially. Triangle = triples in V which intersect nontrivially. U http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf

  6. Nerve Lemma:If Vis a finite collection of subsets of X with all non-empty intersections of subcollections of Vcontractible, then N(V) is homotopicto the union of elements of V. http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf

  7. Choose data point v. The Voronoicell associated with v is H(v,w) U w ≠ v The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

  8. Voronoi diagram Suppose your data points live in Rn. Choose data point v. The Voronoicell associated with v is H(v,w) U w ≠ v The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

  9. The delaunay triangulation is the dual to the voronoidiagram If Cv≠ 0, then s is a simplex in the delaunay triangulation. Nerve of {Cv : v in data set} ⁄ U w in s The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

  10. The delaunay triangulation is the dual to the voronoidiagram If Cv≠ 0, then s is a simplex in the delaunay triangulation. Nerve of {Cv : v in data set} ⁄ U w in s The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

  11. voronoi diagram

  12. voronoi diagram

  13. The delaunay triangulation is the dual to the voronoidiagram If Cv≠ 0, then s is a simplex in the delaunay triangulation. Nerve of {Cv : v in data set} ⁄ U w in s The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

  14. The delaunay triangulation is the dual to the voronoidiagram If Cv≠ 0, then s is a simplex in the delaunay triangulation. Nerve of {Cv : v in data set} ⁄ U w in s The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

  15. Delaunay triangulation Čech

  16. Alpha complex Nerve of {CvBv(r): v in data set} U The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

  17. Alpha complex Nerve of {CvBv(r): v in data set} U The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

  18. Alpha complex Čech

  19. Čech

  20. Alpha complex

  21. Let D = set of vertices. v0,v1,...,vkspan a Delaunay k-simplex iff the Voronoi cell associated with vi meet iff there is a point w ∈ Rn, whose k+1 nearest neighbours in D are v0,v1,...,vkand which is equidistant from them.

  22. comptop.stanford.edu/preprints/witness.pdf

  23. Witness complex Let D = set of point cloud data points. Choose L D, L = set of landmark points. U

  24. v0,v1,...,vkspan a k-simplex iff there is a point w ∈ D, whose k+1 nearest neighbours in L are v0,v1,...,vkand which is equidistant from them ?????????????????????? Witness complex Let D = set of point cloud data points. Choose L D, L = set of landmark points. U

  25. v0,v1,...,vkspan a k-simplex iff there is a point w ∈ D, whose k+1 nearest neighbours in L are v0,v1,...,vk and all the faces of {v0,v1,...,vk} belong to the witness complex. w is called a “weak” witness. W∞(D) = Witness complex Let D = set of point cloud data points. Choose L D, L = set of landmark points. U

  26. v0,v1,...,vkspan a k-simplex iff there is a point w ∈ D, whose k+1 nearest neighbours in L are v0,v1,...,vk and all the faces of {v0,v1,...,vk} belong to the witness complex. w is called a “weak” witness. W∞(D) = Witness complex Let D = set of point cloud data points. Choose L D, L = set of landmark points. U

  27. W∞(D) = Witness complex

  28. W∞(D) = Witness complex

  29. W1(D) = Lazy witness complex Let L = set of landmark points. 1-skeletion of W1(D) = 1-skeletion of W∞ (D). Create the flag (or clique) complex: Add all possible simplices of dimensional > 1.

  30. W1(D) = Lazy witness complex Let L = set of landmark points. 1-skeletion of W1(D) = 1-skeletion of W∞ (D). Create the flag (or clique) complex: Add all possible simplices of dimensional > 1.

  31. W1(D) = Lazy witness complex Let L = set of landmark points. 1-skeletion of W1(D) = 1-skeletion of W∞ (D). Create the flag (or clique) complex: Add all possible simplices of dimensional > 1.

  32. Choosing Landmark points: A.) Random B.) Maxmin 1.) choose point l1 randomly 2.) If {l1, …, lk-1} have been chosen, choose lksuch that {l1, …, lk-1} is in D - {l1, …, lk-1} and min {d(lk, l1), …, d(lk, lk-1)} ≥ min {d(v, l1), …, d(v, lk-1)}

  33. Choosing Landmark points

  34. Choosing Landmark points

  35. Choosing Landmark points

  36. Choosing Landmark points

  37. Choosing Landmark points

  38. comptop.stanford.edu/preprints/witness.pdf

  39. Strong witness complex: Let D = set of point cloud data points. Choose L D, L = set of landmark points. Let mv = dist (v, L) = min{ d(v, l ) : l in L } U {l1, …, lk+1} is a k-simplex iff d(v, li) ≤ mv + ε for all i v is the witness

  40. Weak witness complex: Let D = set of point cloud data points. Choose L D, L = set of landmark points. U s = {l1, …, lk+1} is a k-simplex iff d(v, li) ≤ d(v, x) for all i and all x not in s v is the weak witness

  41. Weak witness complex: Let D = set of point cloud data points. Choose L D, L = set of landmark points. U s = {l1, …, lk+1} is a k-simplex iff d(v, li) ≤ d(v, x) + e for all i and all x not in s v is the e-weak witness

  42. The Theory of Multidimensional Persistence, Gunnar Carlsson, AfraZomorodian "Persistence and Point Clouds" Functoriality, diagrams, difficulties in classifying diagrams, multidimensional persistence, Gröbnerbases, Gunnar Carlsson http://www.ima.umn.edu/videos/?id=862

  43. From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www.ima.umn.edu/2008-2009/ND6.15-26.09/activities/Carlsson-Gunnar/imafour-handout4up.pdf

  44. From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www.ima.umn.edu/2008-2009/ND6.15-26.09/activities/Carlsson-Gunnar/imafour-handout4up.pdf

  45. From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www.ima.umn.edu/2008-2009/ND6.15-26.09/activities/Carlsson-Gunnar/imafour-handout4up.pdf

  46. From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www.ima.umn.edu/2008-2009/ND6.15-26.09/activities/Carlsson-Gunnar/imafour-handout4up.pdf

  47. From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www.ima.umn.edu/2008-2009/ND6.15-26.09/activities/Carlsson-Gunnar/imafour-handout4up.pdf

More Related