1. Sajid Ghuffar 24.June.2010

2.  Introduction  Simplicial Complex  Boundary Operator  Homology  Triangulation  Persistent Homology 6/24/2010 2 Persistence Homology

3.  Why we need homology ? Connected Components =2 Holes=20 Connected Components =1 Tunnels=1059, Cavities=0 CHOMP, Computational Homology Project 6/24/2010 3 Persistence Homology

4. 0-SimplexPoint∆0∆0 1-SimplexLine Segment∆1∆1 2-SimplexTriangle∆2∆2 3-SimplexTetrahedron∆3∆3 6/24/2010 4 Persistence Homology

5.  A simplicial complex К is a finite collection of set of simplices that satisfies the following conditions:  Every face of a simplex of К is also in К.  The intersection of any two simplices of К is a face of each of them.. Simplicial complex Invalid Simplicial complex J. R. Munkres, Elements of Algebraic Topology, p. 7, 1984. 6/24/2010 5 Persistence Homology К={(1,2,3) (1,2),(1,3),(2,3),(2,4),(3,4) (1),(2),(3),(4)}

6.  Let К ={ σ i k } be a simplicial complex with simplices σ i k, where k denotes the simplex dimension. A simplicial k-chain is a formal sum of k-dimensional simplices C 0 =A+B C 1 =a+b+c 6/24/2010 6 Persistence Homology

7.  The boundary operator ∂, acting on simplices is a following map  Boundaries have no boundaries Algebraic Topology, Hatcher 6/24/2010 7 Persistence Homology

8.  A chain is a cycle when its boundary is zero  The cycles form a subgroup Z k (К) of chain group C k (К), which is the kernel of boundary operator (Z is because of german word of cycle) Z k (К) =ker( ∂ k )  The elements in Im(∂ k+1 ) are called boundaries  The k-boundary group of К is the set of boundaries of (k+1)-chains in К, i.e. Its the Image of the (k+1)-chain group B k (К)= ∂(C k+1 (К) ) 6/24/2010 8 Persistence Homology

9.  The homology group is the quotient vector space of cycles over boundaries H k (К)= Z k (К) / B k (К)  Suppose that V= {(x 1,x 2,x 3 )} and W= {(x 1,0,0)}, then quotient vector space V/W (read as " mod") is isomorphic to {(x 2,x 3 )}=R 2 6/24/2010 9 Persistence Homology Intelligent Perception, Computer Vision Primer

10. 0-Simplex = {A,B,C} 1-Simplex = {a,b,c} 2-Simplex = empty 0-Simplex = {A,B,C} 1-Simplex = {a,b,c} 2-Simplex = {f} ∂ 2 f=a+b+c H 1 =Z 1 /B 1 =0 6/24/2010 10 Persistence Homology

11. 6/24/2010 Persistence Homology 11 Vertices v Edges e Boundary ( ∂ 1 ) ∂e=v-v ∆ 0 (S 1 ) ∆ 1 (S 1 ) H0H0 H1H1

12.  Cycles which generate the n dimensional holes are called homology generators  Agoston Algorithm(1976) ◦ Build incidence matrices ◦ Reduce to smith normal form ◦ Compute homology Generators 6/24/2010 Persistence Homology 12 Computing Homology Group Generators of Images Using Irregular Graph Pyramids, S. Peltier

13.  For large no. of vertices Agoston algorithm becomes computationally very expensive  Solution: Build a pyramid  It reduces no. of cells  Apply Agoston Algorithm at top level  Generators fit nicely on borders 6/24/2010 Persistence Homology 13 Computing Homology Group Generators of Images Using Irregular Graph Pyramids, S. Peltier

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15.  For a set P of points in the plane, a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P).  The DT(P) is unique if all the points are in general position in e.g in 2-dimensional space ◦ No three points are on same line ◦ No four points are on same circle 6/24/2010 15 Persistence Homology

16.  Set of data points A  R 2  Height ƒ(p) defined at each point p in A  How can we most naturally approximate height of points not in A? * Delauny Triangulations by Glenn Eguchi 6/24/2010 16 Persistence Homology

17.  Given a finite point set S, and a real parameter alpha: The set of all real numbers alpha leads to a family of shapes capturing the intuitive notion of "crude'' versus "fine'' shape of a point set 6/24/2010 17 Persistence Homology

18.  For sufficiently large alpha, the alpha shape is identical to the convex hull of S  For α=0, it reduces to point cloud Three Dimensional Alpha Shapes, Herbert Edelsbrunner 6/24/2010 18 Persistence Homology

19.  A persistence complex C is a family of chain complexes C * i, together with a chain map 6/24/2010 Persistence Homology 19 Computing Persistent Homology, Afra Zomorodian

20.  Persistence is a measure of importance of an n-cycle defined to be the difference between the  for which the cycle is created, to the  it is filled by adding an (n+1)-simplex. 6/24/2010 Persistence Homology 20 Persistent Homology of Complex Networks, D. Horak

21.  New approach to study highly interconnected dynamic systems such as scale free networks (e.g. Airline traffic routes)  Persistence of the complex gives important information about robustness of the network against addition or removal of nodes 6/24/2010 Persistence Homology 21 Persistent Homology of Complex Networks, D. Horak

22.  Cycles which have low persistence can be regarded as topological noise 6/24/2010 22 Persistence Homology Barcodes: The Persistent Topology Of Data, Robert Ghrist

23. Topological persistence and Simplification, Edelsbrunner, Zomorodian  The p-persistent k-th homology groupof K l is 6/24/2010 23 Persistence Homology

24.  Protein function is in part determined by its shape  This shape allows it to bind to a target molecule  One important and challenging goal of proteomics, the study of proteins, is the identification and characterization of protein binding sites.  Protein data bank website contain 34,303 structures 6/24/2010 Persistence Homology 24 Applications of Computational Homology, Master thesis, Marshall University

25.  Shapes such as letter “C” may nearly be like a circle, but not quite, so we want to capture such structures as well  Hand like structures (TAQ Polymerase) can not be perceived by just looking at the betti numbers of the structure  Amount of time a cycles is created and detroyed can give important features 6/24/2010 Persistence Homology 25

26. 6/24/2010 Persistence Homology 26 Topology for computing, Afra Zomorodian, p-228 M D Dyksterhouse. An alpha-shape view of our universe.  Astronomers have measured the location of about 170000 galaxies, each one represented by a point in three-dimensional space.  It appears that a large number of galaxies are located on or close to sheet-like and to lament-shaped structures.  In other words, large subsets of the points are distributed in a predominantly two- or one- dimensional manner

27.  Homology classifies objects based on their connectivity and n- dimensional holes  Computing homology using pyramids produces nice generators and is computationally inexpensive than previous methods  Alpha shapes provide new tool in analyzing topological properties of the objects  Current research of alpha shapes and persistence homology has mostly focused on molecular biology, but its application in other fields is growing fast. 6/24/2010 Persistence Homology 27

28.  Elements of Algebraic Topology, James R Munkres, MIT, Massacheussets 1984  Algebraic topology, Allen Hatcher, Cambridge University Press, 2002  A. Zomorodian, Gunnar Carlsson, Computing Persistent Homology, Afra Zomorodian, Discrete and Computational Geometry Archive, page 249-274, Feb 2005  H. Edelsbrunner, Ernst Muecke, Three-dimensional alpha shapes, ACM Transactions on Graphics, January, 1994.  Delaunay Triangulations, Presented by Glenn Eguchi, Computational Geometry, October 11, 2001  Computer Vision Primer: beginner's guide to methods of image analysis, data analysis, related mathematics (especially topology), image analysis software, and applications in sciences and engineering. http://inperc.com/wiki/index.php?title=Main_Page 6/24/2010 Persistence Homology 28

29.  Danijela Horak, Slobodan Maletić and Milan Rajković, Persistent Homology of Complex Networks, Institute of Nuclear Sciences Vinča, Belgrade 11001, Serbia Max Planck Institute for Mathematics in the Natural Sciences, D-04103 Leipzig, Germany, Journal of Statistical Mechanics: Theory and Experiment, Volume 2009, March 2009  H. Edelsbrunner, D. Letscher A. Zomorodian,Topological persistence and Simplification, Proceedings of the 41st Annual Symposium on Foundations of Computer Science, 2000  Barcodes: The Persistent Topology Of Data, Robert Ghrist, Department of Mathematics, University of Illinois, Urbana Champaign, 2007  Topology for computing, Afra Zomorodian, Cambridge Monographs on applied and comptational mathematics, 2005  M. D. Dyksterhouse. An alpha-shape view of our universe. Master's thesis, Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, Illinois, 1992  Applications of Computational Homology, Aaron Johnson, Master Thesis, Department of Mathematics, Marshall University, 2006  CHOMP: Computational Homology Project http://chomp.rutgers.edu/ 6/24/2010 Persistence Homology 29