Homology Groups And Persistence Homology. Outline Introduction Simplicial Complex Boundary Operator...

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

Persistence Homology 4

Simplices

0-Simplex Point ∆0

1-Simplex Line Segment ∆1

2-Simplex Triangle ∆2

3-Simplex Tetrahedron ∆3


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)}


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


Boundary Operator

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

• Boundaries have no boundaries

Algebraic Topology, Hatcher


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))


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


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


Homology of a Circle (S1)

Vertices v

Edges e

Boundary (∂1) ∂e=v-v

∆0(S1)

∆1(S1)

H0

H1


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


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


The problem of assigning simplices to point cloud


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

http://en.wikipedia.org/wiki/File:Delaunay_circumcircles.png


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


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


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


Filtration

• A persistence complex C is a family of chain complexes C*i , together

with a chain map

Computing Persistent Homology, Afra Zomorodian


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


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


Persistence continued

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

Barcodes: The Persistent Topology Of Data, Robert Ghrist


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,


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


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


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.


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.


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


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/

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