homological methods in image analysis

Outline

Homological Methods in Image AnalysisSubproject 3: Constrained Level Sets and Shape

Understanding

Jochen Abhau

University of Innsbruck, Department of Computer Science

1st FSP MeetingNovember 23 – 25, 2005

Jochen Abhau University of Innsbruck

Homological Methods in Image Analysis

Outline

1 Cubical Homology for ImagesTopological PrerequisitesApplicationsComputation via Unions of Balls

2 Meshing SurfacesTopological PrerequisitesApplications

Outline

Cubical Homology for Images Meshing Surfaces

Topological Prerequisites

Outline

Equivalent Computations of Homology

Usual homology computation:

simplicial , using a triangulation of the space

cellular , basic ingredients are cells

singular , for (important) theoretical purposes

rather new idea: cubical , for a space consisting ofcubes

Equivalent Computations of Homology

Usual homology computation:

simplicial , using a triangulation of the space

cellular , basic ingredients are cells

singular , for (important) theoretical purposes

rather new idea: cubical , for a space consisting ofcubes

Cubical Spaces

Consider intervals

I = [a, b] with a ≤ b ∈ Z

If a = b, I is called degenerate, otherwise nondegenrate.

Construct J = I1 × · · · × Id ⊂ Rd ; a n-dimensional cube, ifn(≤ d) intervals are nondegenerate.

If X ⊂ Rd admits a representation

X =l⋃

Ji of cubes,

X is called a cubical space.Jochen Abhau University of Innsbruck

Cubical Spaces

Consider intervals

X =l⋃

Ji of cubes,

Cubical Spaces

Consider intervals

X =l⋃

Ji of cubes,

A cubical space

Cubical Homology, Chain groups

For a cubical space X , and 0 ≤ n ≤ d , define

Fn := {J ⊂ X | J is n − dimensional cube },

the set of n-dimensional faces of X .

Now let

Cn := free abelian group, generated of Fn

the chain group of dimension n.

Cubical Homology, Differentials

Define the differentials

∂n : Cn → Cn−1

∂ in(J) := I1 × · · · × Ii−1 × {bi} × Ii+1 · · · × Id

- I1 × · · · × Ii−1 × {ai} × Ii+1 · · · × Id

∂n(J) :=d∑

(−1)i∂ in(J)

Extend ∂n linearly to linear combinations of cubes J .

∂n : Cn → Cn−1

∂ in(J) := I1 × · · · × Ii−1 × {bi} × Ii+1 · · · × Id

- I1 × · · · × Ii−1 × {ai} × Ii+1 · · · × Id

∂n(J) :=d∑

(−1)i∂ in(J)

∂n : Cn → Cn−1

∂ in(J) := I1 × · · · × Ii−1 × {bi} × Ii+1 · · · × Id

- I1 × · · · × Ii−1 × {ai} × Ii+1 · · · × Id

∂n(J) :=d∑

(−1)i∂ in(J)

Orientations are reflected by the signs

Cubical Homology, Definition

Obtain a chain complex

0 → Cd∂d→ . . . C1

∂1→ C0 → 0

and define for n ∈ Z

Hcubn (X ) := Ker(∂n)/Im(∂n+1)

Cubical Homology

Theorem

For n ∈ Z :

Hcubn (X ) is isomorphic to Hn(X )

Proof:Cubical Homology is a special case of cellular homology

Important Properties of Homology Groups

Some interesting properties of homology groups (forarbitrary spaces):

homology is homotopy invariant

disjoint unions ⇒ direct sum in homology

first homology group is abelianization of thefundamental group

Interpretation of Homology Groups

Interpretation of the homology groups Hi(X )

H0(X ): gives number of connected componentsof X

Hi(X ): gives number of ”i-holes” of X

Hd(X ) = Z, if X is a closed, connected,orientable d -manifold

H0(X ) = Z = H1(X ), otherwise Hi(X ) = 0

Applications

Outline

Applications

Applications of Cubical Homology

Cubical homology can be directly applied to images

X =2D or 3D binary image is a cubical space,homology explains structure

X =2D or 3D greyscale picture, use a ceilingscheme and compute cubical homology severaltimes to detect ”strong” and ”weak” holes

Applications

Computation via Unions of Balls

Outline

Homotopy equivalence(1)

Let Br (P) := {x ∈ Rd | ||x − P|| ≤ r}

Theorem

X ⊂ Rd cubical space, d ≤ 3 and P1, ..., Pk the centers of thecubes. Then

X ∼=k⋃

B√d(Pi)

Proof, Cubes covered by ...

... Balls

Definitions

B = {Br1(P̃1), . . . , Brk (P̃k)}, balls in Rd in generalposition

dpow(x , Bri (P̃i)) := ||x − P̃i ||2 − r 2i , the power distance

pow(Bri (P̃i)) = {x | dpow(x , Bri ) ≤ dpow(Brj ) ∀j} the

power cell of Bri (P̃i)

P = set of intersections of the power cells, the powerdiagram of BQ = intersection of P with the union of balls.

R = dual to P , the regular triangulation

K = dual to QS = the geometric realization of K

Homotopy equivalence(2)

Theorem

S is homotopy equivalent to the union of balls⋃k

i=1 Bri (P̃i).

Proof: Edelsbrunner 1992

Cubical Space Homotopy Equivalent to Alpha

Set P̃i = small perturbation of Pi (→ simulation of simplicity)and let ri =

√d . We get

Corollary

X is homotopy equivalent to S

S is the α-shape to the P̃i , with α = d .

Computation of the Alpha Shape

Due to Edelsbrunner, the complex K can be computed by:

constructing the regular triangulation R by incrementaledge flipping

filtering out the simplices belonging to KExpected running time: If the P̃i are iid, altogether linear withnumber of simplices of the regular triangulation.

Computation of the Alpha Shape

Due to Edelsbrunner, the complex K can be computed by:

constructing the regular triangulation R by incrementaledge flipping

filtering out the simplices belonging to KExpected running time: If the P̃i are iid, altogether linear withnumber of simplices of the regular triangulation.

Cubical Homology via Alpha Shapes

Advantages of computing cubical homology via alpha shapes

reduce the order of the chain groups, SNF is worst part

alpha shape is itself interesting

possible generalization of cubical homology to ”weightedcubical homology”

Outline

Homology of Closed, Orientable Surfaces

Let Fg be a closed, orientable surface of genus g .

Proposition

Hn (Fg) =

0 n ≥ 3Z n = 2

Z2g n = 1Z n = 0

Torus Example

Homology of a Closed Surface minus Points

Proposition

For p1 . . . pk pairwise distinct points in Fg , we have

Hn (Fg\{p1 . . . pk}) =

0 n ≥ 2

Z2g+k−1 n = 1Z n = 0

Homology of a Closed Surface minus Points, Proof

Proof:

Fg\{p1 . . . pk} is noncompact manifold

Mayer-Vietoris for k = 1, D ⊂ Fg small disk aroundp = p1:

. . . Hn+1(Fg ) → Hn(S1)α→ Hn(Fg\D)⊕Hn({p}) → Hn(Fg ) . . .

Notice that α = 0 for n = 1.

further application of Mayer-Vietoris-Sequence giveshomology of Fg\{p1 . . . pk}

Proof:

Interpretation of the homology groups for surfaces:

H0: gives number of connected components

H2: same as H0, if all components are closedsurfaces

H1: gives number of handles

Applications

Outline

Applications

Applications in Surface Meshing

Given a 3D voxel image and a (possibly incorrect)surface triangulation of the image, the homologygoups

can prove a triangulation to be incorrect

can help debug the underlying meshingalgorithm

make the structure of the image computable

Applications

Cyst with Needle

Applications

mesh with 8639 vertices, 12960 edges, 4321 facesH0(X ) = Z = H2(X ), H1(X ) = Z2 (in ≤ 0.1 sec)

Applications

corrupt mesh with 7397 vertices, 11097 edges, 3699 facesH0(X ) = Z, H1(X ) = Z2 ⊕ Z2, H2(X ) = 0 (in ≤ 0.1 sec)

Applications

homological methods in image analysis

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