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Homology Groups And Persistence Homology

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Outline Introduction Simplicial Complex Boundary Operator Homology Triangulation Persistent Homology 2

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Introduction Why we need homology ? 3 Connected Components =2 Holes=20 Connected Components =1 Tunnels=1059, Cavities=0

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Simplices 0-SimplexPoint∆0∆0 1-SimplexLine Segment∆1∆1 2-SimplexTriangle∆2∆2 3-SimplexTetrahedron∆3∆3 Persistence Homology4

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Simplicial Complex 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.. Persistence Homology5 Simplicial complex Invalid Simplicial complex J. R. Munkres, Elements of Algebraic Topology, p. 7, К={(1,2,3) (1,2),(1,3),(2,3),(2,4),(3,4) (1),(2),(3),(4)}

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Chain Complex 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 Persistence Homology6 C 0 =A+B C 1 =a+b+c

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Boundary Operator The boundary operator ∂, acting on simplices is a following map Boundaries have no boundaries Persistence Homology7 Algebraic Topology, Hatcher

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Cycles and Boundaries 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) ) Persistence Homology8

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Finally Homology!! 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 Persistence Homology9 Intelligent Perception, Computer Vision Primer

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Cycles and Boundaries 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 Persistence Homology10

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Homology of a Circle (S 1 ) Persistence Homology11 Vertices v Edges e Boundary ( ∂ 1 ) ∂e=v-v ∆ 0 (S 1 ) ∆ 1 (S 1 ) H0H0 H1H1

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Computing Homology 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 Persistence Homology12 Computing Homology Group Generators of Images Using Irregular Graph Pyramids, S. Peltier

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Computing Homology 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 Persistence Homology13 Computing Homology Group Generators of Images Using Irregular Graph Pyramids, S. Peltier

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The problem of assigning simplices to point cloud Persistence Homology14

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Delauny Triangulation 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 Persistence Homology15

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Application (Terrain Model) 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? Persistence Homology16 * Delauny Triangulations by Glenn Eguchi

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Alpha Shapes 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 Persistence Homology17

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Alpha Shapes Continued..... For sufficiently large alpha, the alpha shape is identical to the convex hull of S For α=0, it reduces to point cloud Persistence Homology18 Three Dimensional Alpha Shapes, Herbert Edelsbrunner

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Filtration A persistence complex C is a family of chain complexes C * i, together with a chain map Persistence Homology19 Computing Persistent Homology, Afra Zomorodian

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Persistence 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. Persistence Homology20 Persistent Homology of Complex Networks, D. Horak

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Persistence of complex Networks 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 Persistence Homology21 Persistent Homology of Complex Networks, D. Horak

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Persistence continued Cycles which have low persistence can be regarded as topological noise Persistence Homology22 Barcodes: The Persistent Topology Of Data, Robert Ghrist

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Persistent Homology The p-persistent k-th homology groupof K l is Persistence Homology23 Barcodes: The Persistent Topology Of Data, Robert Ghrist Topological persistence and Simplification, Edelsbrunner, Zomorodian

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Protein Structure 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 Persistence Homology24 Applications of Computational Homology, Master thesis, Marshall University

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Protein Structure 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 Persistence Homology25

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Another application!!! Astronomers have measured the location of about 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 Persistence Homology26 Topology for computing, Afra Zomorodian, p-228 M D Dyksterhouse. An alpha-shape view of our universe.

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Summary 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. Persistence Homology27

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References 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 , Feb 2005 H. Edelsbrunner, Ernst Muecke, Three-dimensional alpha shapes, ACM Transactions on Graphics, January, 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. Persistence Homology28

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References 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 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 Persistence Homology29

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