Spectral sequences for computing persistent homology of ...Spectral sequences for computing...

Spectral sequences for computing persistent homology ofdigital images

Ana Romero1, Jonathan Heras2, Gadea Mata1, Julio Rubio1, andFrancis Sergeraert3

1Universidad de La Rioja (Spain), 2University of Dundee (UK), 3Institut Fourier (France)

Malaga, 3th July 2013

Ana Romero et al. Spectral sequences for computing persistent homology of digital images Malaga, 3th July 2013 1 / 20

Introduction Persistence in digital images

Persistence in digital images

Introduction Preliminary definitions

Persistent homology

Definition

Let K be a simplicial complex. A filtration of K is a sequence ofsubcomplexes: ∅ = K 0 ⊆ K 1 ⊆ K 2 ⊆ · · · ⊆ K m = K .

The filtration produces a sequence of groups and homomorphisms:

0 = Hn(K 0)f 0,1n→ Hn(K 1)

f 1,2n→ · · · f m−1,m

n→ Hn(K m) = Hn(K )

Definition

The n-th persistent homology groups of K are defined asH i ,j

n = Im f i ,jn , for 0 ≤ i ≤ j ≤ m.

If γ is born at K i and dies entering K j , we denote pers(γ) = j − i .

Persistent homology

Definition

0 = Hn(K 0)f 0,1n→ Hn(K 1)

f 1,2n→ · · · f m−1,m

Definition

n = Im f i ,jn , for 0 ≤ i ≤ j ≤ m.

Persistent homology

Definition

0 = Hn(K 0)f 0,1n→ Hn(K 1)

f 1,2n→ · · · f m−1,m

Definition

n = Im f i ,jn , for 0 ≤ i ≤ j ≤ m.

Persistent homology

Definition

0 = Hn(K 0)f 0,1n→ Hn(K 1)

f 1,2n→ · · · f m−1,m

Definition

n = Im f i ,jn , for 0 ≤ i ≤ j ≤ m.

Persistent homology

Definition

0 = Hn(K 0)f 0,1n→ Hn(K 1)

f 1,2n→ · · · f m−1,m

Definition

n = Im f i ,jn , for 0 ≤ i ≤ j ≤ m.

Persistent homology

Example:

1 2 3 4 5 6 7

Roughly speaking:

Homology detects topological features (connected components, holes,and so on).Persistent homology describes the evolution of topological featureslooking at consecutive thresholds.

Persistent homology

Example:

1 2 3 4 5 6 7

Roughly speaking:

Persistent homology

Example:

1 2 3 4 5 6 7

Roughly speaking:

Persistent homology

Example:

1 2 3 4 5 6 7

Roughly speaking:

Persistent homology

Example:

1 2 3 4 5 6 7

Roughly speaking:

Spectral sequences

Definition

A spectral sequence E = (E r , d r )r≥1 is a family of bigraded Z-modulesE r = {E r

p,q}, each provided with a differentiald r = {d r

p,q : E rp,q → E r

p−r ,q+r−1} of bidegree (−r , r − 1) and with

isomorphisms H(E r , d r ) ∼= E r+1 for every r ≥ 1.

Spectral sequences

Definition

Spectral sequences

Definition

• • • • •

dd d34,1

Spectral sequences

Definition

• • • • •

dd d34,1

Spectral sequences

Definition

• • • • •

dd d34,1

Spectral sequences

Definition

• • • • •

dd d34,1

Spectral sequences

Definition

• • • • •

dd d34,1

Since E r+1p,q is a subquotient of E r

p,q for each r ≥ 1, one can define the finalgroups of the spectral sequence as E∞p,q =

⋂r≥1 E r

Spectral sequences

Theorem

Let C be a chain complex with a filtration. There exists a spectralsequence with

E rp,q =

Z rp,q + C p−1

dp+q+1(Z r−1p+r−1,q−r+2) + C p−1

where Z rp,q is Z r

p,q = {a ∈ C pp+q| dp+q(a) ∈ C p−r

p+q−1} ⊆ C pp+q, and

d rp,q : E r

p,q → E rp−r ,q+r−1 is the morphism induced by

dp+q : Cp+q → Cp+q−1. This spectral sequence converges to H∗(C ), thatis, there are natural isomorphisms

E∞p,q∼=

Hpp+q(C )

Hp−1p+q (C )

Spectral sequences

Theorem

Let C be a chain complex with a filtration. There exists a spectralsequence with

E rp,q =

Z rp,q + C p−1

dp+q+1(Z r−1p+r−1,q−r+2) + C p−1

where Z rp,q is Z r

p,q = {a ∈ C pp+q| dp+q(a) ∈ C p−r

p+q−1} ⊆ C pp+q, and

d rp,q : E r

p,q → E rp−r ,q+r−1 is the morphism induced by

dp+q : Cp+q → Cp+q−1. This spectral sequence converges to H∗(C ), thatis, there are natural isomorphisms

E∞p,q∼=

Hpp+q(C )

Hp−1p+q (C )

Computing persistent homology by means of spectral sequences Relation between persistent homology and spectral sequences

“Erroneous” relation

The so called “Spectral sequence theorem” in the book Computationaltopology: an introduction by H. Edelsbrunner and J. Harer claims that:

“Erroneous” theorem

The total rank of the groups of dimension p + q in the level r ≥ 1 of theassociated spectral sequence equals the number of points in the (p + q)-thpersistence diagram whose persistence is r or larger, that is,

m∑p=1

rank E rp,q = card{a ∈ Dgmp+q(f )| pers(a) ≥ r}

where in the left side q decreases as p increases so that the dimensionp + q remains constant.

However, the formula is erroneous because in the left side there can bemore elements than in the right side; the formula should be therefore aninequality.

m∑p=1

The correct relation

Theorem

The total rank of the images of the differential maps in the level r ≥ 1 ofthe spectral sequence equals the number of points in the (p + q)-thpersistence diagram whose persistence is r :

m∑p=1

rank Arp,q = card{a ∈ Dgmp+q(f )| pers(a) = r}

where Arp,q = Im(d r

p+r ,q−r+1 : E rp+r ,q−r+1 → E r

p,q) ⊆ E rp,q.

This theorem gives us an algorithm for computing the groups H i ,jn when

one works with coefficients over a field F . However the information aboutranks is not sufficient if we work with coefficients over Z.

A. Romero, J. Heras, J. Rubio and F. Sergeraert. Defining and computing persistentZ-homology in the general case. Preprint, 2013.

Theorem

m∑p=1

p+r ,q−r+1 : E rp+r ,q−r+1 → E r

p,q) ⊆ E rp,q.

Theorem

m∑p=1

p+r ,q−r+1 : E rp+r ,q−r+1 → E r

p,q) ⊆ E rp,q.

one works with coefficients over a field F .

However the information aboutranks is not sufficient if we work with coefficients over Z.

Theorem

m∑p=1

p+r ,q−r+1 : E rp+r ,q−r+1 → E r

p,q) ⊆ E rp,q.

Theorem

m∑p=1

p+r ,q−r+1 : E rp+r ,q−r+1 → E r

p,q) ⊆ E rp,q.

Computing persistent homology by means of spectral sequences Computing spectral sequences of filtered complexes

A Kenzo module for spectral sequences

In a previous work, we developed a set of programs computingspectral sequences associated with filtered chain complexes.

These algorithms were implemented in Common Lisp as a newmodule for the Kenzo system. Kenzo implements the effectivehomology method and the notion of reduction from one (big) chaincomplex to a smaller one, such that the homology groups of the twocomplexes are explicitly isomorphic and one can compute homology ofinfinite spaces.

The new programs use the effective homology technique and allowthe Kenzo user to determine the different components of spectralsequences of filtered complexes even in some cases where the chaincomplex has infinite type.

These algorithms were implemented in Common Lisp as a newmodule for the Kenzo system.

Kenzo implements the effectivehomology method and the notion of reduction from one (big) chaincomplex to a smaller one, such that the homology groups of the twocomplexes are explicitly isomorphic and one can compute homology ofinfinite spaces.

Computing persistent homology by means of spectral sequences Computing persistent homology by means of spectral sequences

Computing persistent homology

Using our programs for spectral sequences, and thanks to our theoremexpressing the relation between spectral sequences and persistenthomology, one can determine the ranks of the groups H i ,j

In fact a small modification of our programs for spectral sequences issufficient to determine the groups H i ,j

n , which can be expressed as aquotient:

H i ,jn =

Ker dn ∩ C in

dn+1(Z j−ij ,n−j+1)

i ,n−i

The programs are also valid in the integer case and this makes itpossible to solve the possible extension problems.

Our programs can also be applied in the infinite case, where theeffective homology method can be used to determine the groups H i ,j

by means of a reduction of the initial chain complex C to an auxiliarychain complex of finite type.

H i ,jn =

Ker dn ∩ C in

i ,n−i

H i ,jn =

Ker dn ∩ C in

i ,n−i

H i ,jn =

Ker dn ∩ C in

i ,n−i

H i ,jn =

Ker dn ∩ C in

i ,n−i

Persistent homology of digital images Digital algebraic topology

Digital Algebraic Topology

Digital Image

Simplicial Complex Chain Complex

Homology Groups

C0 = verticesC1 = edgesC2 = triangles

H1 = Z⊕ Z⊕ ZH0 = Z⊕ Z

Digital Image

Homology Groups

H1 = Z⊕ Z⊕ ZH0 = Z⊕ Z

Digital Image

Homology Groups

H1 = Z⊕ Z⊕ ZH0 = Z⊕ Z

Digital Image

Homology Groups

H1 = Z⊕ Z⊕ ZH0 = Z⊕ Z

Digital Image

Homology Groups

H1 = Z⊕ Z⊕ ZH0 = Z⊕ Z

Persistent homology of digital images Persistent homology of digital images

Persistent homology

If we have a filtered digital image, we can compute persistent homologygroups.

“Small” example:

> (prst-hmlg-group K 1 4 0) > (prst-hmlg-group K 2 4 1)

Persistent Homology H^{1,4}_0 Persistent Homology H^{2,4}_1

Component Z Component Z

Component Z

Persistent homology

Component Z

Persistent homology

Component Z

Persistent homology

Component Z

Reducing images Discrete Vector Fields and effective homology

Discrete Vector Fields

Definition

Let C∗ = (Cp , dp)p∈Z be a free chain complex with distinguished Z−basis βp ⊂ Cp . A discretevector field on C∗ is a collection of pairs V = {(σi , τi )}i∈β satisfying the conditions:

1 Every σi is some element of βp , in which case the other corresponding componentτi ∈ βp+1. The degree p depends on i and in general is not constant.

2 Every component σi is a regular face of the corresponding component τi .

3 A generator of C∗ appears at most one time in V .

Definition

A V -path of degree p is a sequence π = ((σik , τik ))0≤k<m satisfying:

1 Every pair ((σik , τik )) is a component of V and the cell τik is a p-cell

2 For every 0 < k < m, the component σik is a face of τik−1, non necessarily regular, but

different from σik−1

Definition

A discrete vector field V is admissible if for every p ∈ Z, a function λp : βp → Z is providedsatisfying the property: every V -path starting from σ ∈ βp has a length bounded by λp(σ).

Definition

Example: an admissible discrete vector field

Dvf x Dvf X

Admissible x

Dvf x Dvf X

Admissible x

Admissible X

Discrete Vector Fields and effective homology

Definition

A cell χ which does not appear in a discrete vector field V = {(σi , τi )}i∈β is called a critical cell.

Theorem

Let C∗ = (Cp , dp , βp)p be a free chain complex and V = {(σi , βi )}i∈β be an admissible discretevector field on C∗. Then the vector field V defines a canonical reductionρ : (Cp , dp) =⇒ (C c

p , d′p) where C c

p = Z[βc

]is the free Z−module generated by the critical

p-cells.

• • • • • •

• • •

A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology,2010. http://arxiv.org/abs/1005.5685v1.

Definition

Theorem

p = Z[βc

p-cells.

• • • • • •

• • •

Definition

Theorem

p = Z[βc

p-cells.

• • • • • •

• • •

Definition

Theorem

p = Z[βc

p-cells.

• • • • • •

• • •

Definition

Theorem

p = Z[βc

p-cells.

• • • • • •

• • •

Vector fields and matrices

Differential maps of a chain complex of finite type can be represented as matrices

. . .← Zm2

M←− Zn2 ← . . .

Definition

An admissible discrete vector field V for M is nothing but a set of integer pairs {(ai , bi )}satisfying these conditions:

1 1 ≤ ai ≤ m and 1 ≤ bi ≤ n

2 The entry M[ai , bi ] of the matrix is 1 or −1

3 The indices ai (resp. bi ) are pairwise different

4 Non existence of loops

Algorithm

Input: A matrix MOutput: An admissible discrete vector field for M

. . .← Zm2

M←− Zn2 ← . . .

Definition

1 1 ≤ ai ≤ m and 1 ≤ bi ≤ n

Algorithm

. . .← Zm2

M←− Zn2 ← . . .

Definition

1 1 ≤ ai ≤ m and 1 ≤ bi ≤ n

Algorithm

. . .← Zm2

M←− Zn2 ← . . .

Definition

1 1 ≤ ai ≤ m and 1 ≤ bi ≤ n

Algorithm

. . .← Zm2

M←− Zn2 ← . . .

Definition

1 1 ≤ ai ≤ m and 1 ≤ bi ≤ n

Algorithm

{0, 1} {0, 2} {1, 2} {1, 3} {2, 3} {3, 4} {4, 5} {4, 6} {5, 6}{0} 1 1 0 0 0 0 0 0 0{1} 1 0 1 1 0 0 0 0 0{2} 0 1 1 0 1 0 0 0 0{3} 0 0 0 1 1 1 0 0 0{4} 0 0 0 0 0 1 1 1 0{5} 0 0 0 0 0 0 1 0 1{6} 0 0 0 0 0 0 0 1 1

Discrete vector fields and persistence

Given an admissible discrete vector field on a digital image, areduction is obtained from the big chain complex to the critical(small) chain complex, such that the homology groups of both chaincomplexes are isomorphic.

If we want to determine persistent homology groups, we need adiscrete vector field which is compatible with the filtration (such thatthe obtained reduction is compatible with the filtration).

To obtain such a filtered discrete vector field, we apply our algorithmseparately to the differential submatrices corresponding to eachfiltration index.

This allows us to compute persistent homology groups of big digitalimages by means of a reduced chain complex.

Reducing images Fingerprint example

Fingerprints

Big chain complex Reduced chain complex

Vertices 9082 150

Edges 20364 86

Triangles 11352 6

H1,10 57sec 0, 5sec

H1,20 33min 10sec

H1,30 - 0, 4sec

Fingerprints

Vertices 9082 150

Edges 20364 86

Triangles 11352 6

H1,10 57sec 0, 5sec

H1,20 33min 10sec

H1,30 - 0, 4sec

Fingerprints

Vertices 9082 150

Edges 20364 86

Triangles 11352 6

H1,10 57sec 0, 5sec

H1,20 33min 10sec

H1,30 - 0, 4sec

Fingerprints

Vertices 9082 150

Edges 20364 86

Triangles 11352 6

H1,10 57sec 0, 5sec

H1,20 33min 10sec

H1,30 - 0, 4sec

Conclusions

Persistent homology can be useful to give information of the evolutionof a digital image at different stacks.

The relation between persistent homology and spectral sequencesallows us to reuse our previous programs for spectral sequences tocompute persistent homology groups. The programs are valid for theinteger case and for infinite simplicial sets with effective homology. Inparticular they can be used for digital images.

Discrete vector fields and effective homology can be used to reducethe size of digital images before computing persistent homology.

Conclusions

The relation between persistent homology and spectral sequencesallows us to reuse our previous programs for spectral sequences tocompute persistent homology groups. The programs are valid for theinteger case and for infinite simplicial sets with effective homology.

Inparticular they can be used for digital images.

Conclusions

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