Page 1
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
Graz
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 2
Outline
Outline
1 Cubical Homology for ImagesTopological PrerequisitesApplicationsComputation via Unions of Balls
2 Meshing SurfacesTopological PrerequisitesApplications
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 3
Outline
Outline
1 Cubical Homology for ImagesTopological PrerequisitesApplicationsComputation via Unions of Balls
2 Meshing SurfacesTopological PrerequisitesApplications
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 4
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Outline
1 Cubical Homology for ImagesTopological PrerequisitesApplicationsComputation via Unions of Balls
2 Meshing SurfacesTopological PrerequisitesApplications
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 5
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 6
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 7
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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⋃
i=1
Ji of cubes,
X is called a cubical space.Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 8
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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⋃
i=1
Ji of cubes,
X is called a cubical space.Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 9
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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⋃
i=1
Ji of cubes,
X is called a cubical space.Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 10
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
A cubical space
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 11
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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.
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 12
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Cubical Homology, Differentials
Define the differentials
∂n : Cn → Cn−1
by
∂ in(J) := I1 × · · · × Ii−1 × {bi} × Ii+1 · · · × Id
- I1 × · · · × Ii−1 × {ai} × Ii+1 · · · × Id
∂n(J) :=d∑
i=1
(−1)i∂ in(J)
Extend ∂n linearly to linear combinations of cubes J .
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 13
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Cubical Homology, Differentials
Define the differentials
∂n : Cn → Cn−1
by
∂ in(J) := I1 × · · · × Ii−1 × {bi} × Ii+1 · · · × Id
- I1 × · · · × Ii−1 × {ai} × Ii+1 · · · × Id
∂n(J) :=d∑
i=1
(−1)i∂ in(J)
Extend ∂n linearly to linear combinations of cubes J .
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 14
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Cubical Homology, Differentials
Define the differentials
∂n : Cn → Cn−1
by
∂ in(J) := I1 × · · · × Ii−1 × {bi} × Ii+1 · · · × Id
- I1 × · · · × Ii−1 × {ai} × Ii+1 · · · × Id
∂n(J) :=d∑
i=1
(−1)i∂ in(J)
Extend ∂n linearly to linear combinations of cubes J .
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 15
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Orientations are reflected by the signs
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 16
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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)
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 17
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Cubical Homology
Theorem
For n ∈ Z :
Hcubn (X ) is isomorphic to Hn(X )
Proof:Cubical Homology is a special case of cellular homology
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 18
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 19
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 20
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 21
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 22
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 23
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 24
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
H0(X ) = Z = H1(X ), otherwise Hi(X ) = 0
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 25
Cubical Homology for Images Meshing Surfaces
Applications
Outline
1 Cubical Homology for ImagesTopological PrerequisitesApplicationsComputation via Unions of Balls
2 Meshing SurfacesTopological PrerequisitesApplications
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 26
Cubical Homology for Images Meshing Surfaces
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 27
Cubical Homology for Images Meshing Surfaces
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 28
Cubical Homology for Images Meshing Surfaces
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 29
Cubical Homology for Images Meshing Surfaces
Computation via Unions of Balls
Outline
1 Cubical Homology for ImagesTopological PrerequisitesApplicationsComputation via Unions of Balls
2 Meshing SurfacesTopological PrerequisitesApplications
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 30
Cubical Homology for Images Meshing Surfaces
Computation via Unions of Balls
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⋃
i=1
B√d(Pi)
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 31
Cubical Homology for Images Meshing Surfaces
Computation via Unions of Balls
Proof, Cubes covered by ...
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 32
Cubical Homology for Images Meshing Surfaces
Computation via Unions of Balls
... Balls
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 33
Cubical Homology for Images Meshing Surfaces
Computation via Unions of 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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 34
Cubical Homology for Images Meshing Surfaces
Computation via Unions of Balls
Homotopy equivalence(2)
Theorem
S is homotopy equivalent to the union of balls⋃k
i=1 Bri (P̃i).
Proof: Edelsbrunner 1992
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 35
Cubical Homology for Images Meshing Surfaces
Computation via Unions of Balls
Cubical Space Homotopy Equivalent to Alpha
Shape
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 .
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 36
Cubical Homology for Images Meshing Surfaces
Computation via Unions of Balls
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.
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 37
Cubical Homology for Images Meshing Surfaces
Computation via Unions of Balls
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.
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 38
Cubical Homology for Images Meshing Surfaces
Computation via Unions of Balls
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”
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 39
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Outline
1 Cubical Homology for ImagesTopological PrerequisitesApplicationsComputation via Unions of Balls
2 Meshing SurfacesTopological PrerequisitesApplications
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 40
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 41
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Torus Example
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 42
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 43
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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}
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 44
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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}
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 45
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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}
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 46
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
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}
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 47
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Interpretation of Homology Groups
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 48
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Interpretation of Homology Groups
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 49
Cubical Homology for Images Meshing Surfaces
Topological Prerequisites
Interpretation of Homology Groups
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 50
Cubical Homology for Images Meshing Surfaces
Applications
Outline
1 Cubical Homology for ImagesTopological PrerequisitesApplicationsComputation via Unions of Balls
2 Meshing SurfacesTopological PrerequisitesApplications
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 51
Cubical Homology for Images Meshing Surfaces
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 52
Cubical Homology for Images Meshing Surfaces
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 53
Cubical Homology for Images Meshing Surfaces
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
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 54
Cubical Homology for Images Meshing Surfaces
Applications
Cyst with Needle
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 55
Cubical Homology for Images Meshing Surfaces
Applications
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 56
Cubical Homology for Images Meshing Surfaces
Applications
mesh with 8639 vertices, 12960 edges, 4321 facesH0(X ) = Z = H2(X ), H1(X ) = Z2 (in ≤ 0.1 sec)
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 57
Cubical Homology for Images Meshing Surfaces
Applications
corrupt mesh with 7397 vertices, 11097 edges, 3699 facesH0(X ) = Z, H1(X ) = Z2 ⊕ Z2, H2(X ) = 0 (in ≤ 0.1 sec)
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis
Page 58
Cubical Homology for Images Meshing Surfaces
Applications
Jochen Abhau University of Innsbruck
Homological Methods in Image Analysis