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Homology Groups And Persistence Homology1OutlineIntroductionSimplicial ComplexBoundary OperatorHomologyTriangulationPersistent Homology22IntroductionWhy we need homology ?

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Connected Components =2Holes=20Connected Components =1Tunnels=1059, Cavities=03Simplices0-Simplex Point

01-Simplex Line Segment

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2-SimplexTriangle

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3-SimplexTetrahedron

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4Simplicial ComplexA simplicial complex is a finite collection of set of simplices that satisfies the following conditions: Every face of a simplex of is also in . The intersection of any two simplices of is a face of each of them.

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Simplicial complex Invalid Simplicial complex J. R. Munkres, Elements of Algebraic Topology, p. 7, 1984 .

={(1,2,3) (1,2),(1,3),(2,3),(2,4),(3,4) (1),(2),(3),(4)}Vertices are 0 faces, edge are 1 face etcAdd equations5Chain ComplexLet ={ik} be a simplicial complex with simplices ik, where k denotes the simplex dimension. A simplicial k-chain is a formal sum of k-dimensional simplicesPersistence Homology 6

C0=A+BC1=a+b+c

6Boundary OperatorThe boundary operator , acting on simplices is a following map

Boundaries have no boundariesPersistence Homology 7

Algebraic Topology, Hatcher

7Cycles and BoundariesA chain is a cycle when its boundary is zeroThe cycles form a subgroup Zk() of chain group Ck(), which is the kernel of boundary operator (Z is because of german word of cycle) Zk() =ker(k)

The elements in Im(k+1) are called boundariesThe k-boundary group of is the set of boundaries of (k+1)-chains in , i.e. Its the Image of the (k+1)-chain groupBk()= (Ck+1())

Persistence Homology 88Finally Homology!!The homology group is the quotient vector space of cycles over boundaries

Hk ()= Zk() / Bk()

Suppose that V= {(x1,x2,x3)} and W= {(x1,0,0)}, then quotient vector space V/W (read as " mod") is isomorphic to {(x2,x3)}=R2

Persistence Homology 9Intelligent Perception, Computer Vision Primer9Cycles and Boundaries0-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}2f=a+b+cH1=Z1/B1=0

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10Homology of a Circle (S1)Persistence Homology 11

VerticesvEdgeseBoundary (1)e=v-v0(S1)1(S1)H0H1

11Computing HomologyCycles which generate the n dimensional holes are called homology generators

Agoston Algorithm(1976)Build incidence matricesReduce to smith normal formCompute homology GeneratorsPersistence Homology 12Computing Homology Group Generators of Images Using Irregular Graph Pyramids, S. Peltier

12Computing HomologyFor large no. of vertices Agoston algorithm becomes computationally very expensive

Solution: Build a pyramid It reduces no. of cellsApply Agoston Algorithm at top levelGenerators fit nicely on borders

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Computing Homology Group Generators of Images Using Irregular Graph Pyramids, S. Peltier13The problem of assigning simplices to point cloudPersistence Homology 1414Delauny TriangulationFor 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 spaceNo three points are on same line No four points are on same circlePersistence Homology 15

15Application (Terrain Model)Set of data points A R2Height (p) defined at each point p in AHow can we most naturally approximate height of points not in A?Persistence Homology 16

* Delauny Triangulations by Glenn Eguchi16Alpha ShapesGiven 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 setPersistence Homology 17

17Alpha Shapes Continued.....For sufficiently large alpha, the alpha shape is identical to the convex hull of SFor =0, it reduces to point cloudPersistence Homology 18

Three Dimensional Alpha Shapes, Herbert Edelsbrunner18FiltrationA persistence complex C is a family of chain complexes C*i , together with a chain map Persistence Homology 19

Computing Persistent Homology, Afra Zomorodian

19PersistencePersistence 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.Persistence Homology 20

Persistent Homology of Complex Networks, D. Horak20Persistence of complex NetworksNew 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

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Persistent Homology of Complex Networks, D. Horak21Persistence continuedCycles which have low persistence can be regarded as topological noisePersistence Homology 22

Barcodes: The Persistent Topology Of Data, Robert Ghrist

22Persistent HomologyThe p-persistent k-th homology groupof Kl is

Persistence Homology 23Barcodes: The Persistent Topology Of Data, Robert GhristTopological persistence and Simplification, Edelsbrunner, Zomorodian

23Protein StructureProtein function is in part determined by its shapeThis shape allows it to bind to a target moleculeOne 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 structuresPersistence Homology 24

Applications of Computational Homology, Master thesis, Marshall University24Protein StructureShapes such as letter C may nearly be like a circle, but not quite, so we want to capture such structures as wellHand like structures (TAQ Polymerase) can not be perceived by just looking at the betti numbers of the structureAmount of time a cycles is created and detroyed can give important features

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25Another application!!!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

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Topology for computing, Afra Zomorodian, p-228M D Dyksterhouse. An alpha-shape view of our universe.

26SummaryHomology classifies objects based on their connectivity and n-dimensional holesComputing homology using pyramids produces nice generators and is computationally inexpensive than previous methodsAlpha shapes provide new tool in analyzing topological properties of the objectsCurrent research of alpha shapes and persistence homology has mostly focused on molecular biology, but its application in other fields is growing fast.Persistence Homology 2727ReferencesElements of Algebraic Topology, James R Munkres, MIT, Massacheussets 1984Algebraic topology, Allen Hatcher, Cambridge University Press, 2002A. Zomorodian, Gunnar Carlsson, Computing Persistent Homology, Afra Zomorodian, Discrete and Computational Geometry Archive, page 249-274, Feb 2005H. Edelsbrunner, Ernst Muecke, Three-dimensional alpha shapes, ACM Transactions on Graphics , January, 1994.Delaunay Triangulations, Presented by Glenn Eguchi, Computational Geometry, October 11, 2001Computer 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

Persistence Homology 2828ReferencesDanijela Horak , Slobodan Maleti and Milan Rajkovi, Persistent Homology of Complex Networks, Institute of Nuclear Sciences Vina, 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, 2000Barcodes: The Persistent Topology Of Data, Robert Ghrist, Department of Mathematics, University of Illinois, Urbana Champaign, 2007Topology for computing, Afra Zomorodian, Cambridge Monographs on applied and comptational mathematics, 2005M. D. Dyksterhouse. An alpha-shape view of our universe. Master's thesis, Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, Illinois, 1992Applications of Computational Homology, Aaron Johnson, Master Thesis, Department of Mathematics, Marshall University, 2006CHOMP: Computational Homology Project http://chomp.rutgers.edu/

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