Homology Groups And Persistence Homology

Outline

• Introduction• Simplicial Complex• Boundary Operator• Homology• Triangulation• Persistent Homology

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Introduction• Why we need homology ?

Connected Components =2

Holes=20

Connected Components =1

Tunnels=1059, Cavities=0

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Simplices

0-Simplex Point ∆0

1-Simplex Line Segment ∆1

2-Simplex Triangle ∆2

3-Simplex Tetrahedron ∆3

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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.

.

Simplicial complex Invalid Simplicial complex

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

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 ={σik} be a simplicial complex with simplices σi

k, where k denotes the simplex dimension. A simplicial k-chain is a formal sum of k-dimensional simplices

C0=A+B

C1=a+b+c

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Boundary Operator

• The boundary operator ∂, acting on simplices is a following map

• Boundaries have no boundaries

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 Zk(К) of chain group Ck(К), which is the

kernel of boundary operator (Z is because of german word of cycle) Zk(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

Bk(К)= ∂(Ck+1(К))

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Finally Homology!!

• The homology group is the quotient vector space of cycles over boundaries

Hk (К)= Zk(К) / Bk(К)

• 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

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}∂2f=a+b+c

H1=Z1/B1=0

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Homology of a Circle (S1)

Vertices v

Edges e

Boundary (∂1) ∂e=v-v

∆0(S1)

∆1(S1)

H0

H1

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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

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

Computing Homology Group Generators of Images

Using Irregular Graph Pyramids, S. Peltier

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The problem of assigning simplices to point cloud

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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

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Application (Terrain Model)

• Set of data points A R2

• 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

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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

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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

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

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.

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

Persistent Homology of Complex Networks, D. Horak

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Persistence continued

• Cycles which have low persistence can be regarded as topological noise

Barcodes: The Persistent Topology Of Data, Robert Ghrist

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Persistent Homology• The p-persistent k-th homology groupof Kl is

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

plH,

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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

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

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Another 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

Topology for computing, Afra Zomorodian, p-228M 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.

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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 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

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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-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/