Union of Balls, Alpha-Complexes and Delaunay filtrations

Union of Balls, Alpha-Complexes and Delaunay filtrations

Jean-Daniel Boissonnat DataShape, INRIA

http://www-sop.inria.fr/datashape

Restriction of Delaunay triangulation

Let Ω ⊆ R ^d and P ∈ R ^d a finite set of points.

Vor(P) ∩ Ω is a cover of Ω. Its nerve is called the Delaunay triangulation of P restricted to Ω, noted Del _|Ω (P )

If Vor(P ) ∩ Ω is a good cover of Ω, Del (P ) is homotopy equivalent to Ω

Cech complex versus Del _{| U} (B)

Computational Topology (Jeff Erickson) Examples of Cell Complexes

Corollary 15.1.For any points setPand radius�, the Aleksandrov-ˇCech complexACˇ_�(P)is homotopy- equivalent to the union of balls of radius�centered at points inP.

Aleksandrov-ˇCech complexes and unions of balls for two different radii. 2-simplices are yellow; 3-simplices are green.

15.1.2 Vietoris-Rips Complexes: Flags and Shadows

Theproximity graph N_�(P)is the geometric graph whose vertices are the pointsPand whose edges join all pairs of points at distance at most 2�; in other words,N_�(P)is the 1-skeleton of the Aleksandrov-ˇCech complex. TheVietoris-Rips complex VR�(P)is theflag complexorclique complexof the proximity graphN_�(P). A set ofk+1 points inPdefines ak-simplex inV R_�(P)if and only if every pair defines an edge inN�(P), or equivalently, if the set has diameter at most 2�. Again, the Vietoris-Rips complex is an abstractsimplicial complex.

The Vietoris-Rips complex was used by Leopold Vietoris[57]in the early days of homology theory as a means of creating finite simplicial models of metric spaces.²The complex was rediscovered by Eliayu Rips in the 1980s and popularized by Mikhail Gromov[35]as a means of building simplicial models for group actions. ‘Rips complexes’ are now a standard tool in geometric and combinatorial group theory.

The triangle inequality immediately implies the nesting relationshipACˇ_�(P)⊆V R_�(P)⊆ACˇ2�(P) for any�, where⊆indicates containmentas abstract simplicial complexes. The upper radius 2�can be reduced to�3�/2 if the underlying metric space is Euclidean[21], but for arbitrary metric spaces, these bounds cannot be improved.

One big advantage of Vietoris-Rips complexes is that they determined entirely by their underlying proximity graphs; thus, they can be applied in contexts like sensor-network modeling where the underlying metric is unknown. In contrast, the Aleksandrov-ˇCech complex also depends on the metric of the ambient space that containsP; even if we assume that the underlying space is Euclidean, we need the lengths of the edges of the proximity complex to reconstruct the Aleksandrov-ˇCech complex.

On the other hand, there is no result like the Nerve Lemma for flag complexes. Indeed, it is easy to construct Vietoris-Rips complexes for pointsin the Euclidean planethat contain topological features of arbitrarily high dimension.

2Vietoris actually defined a slightly different complex. LetU={U1,U2, . . .}be a set of open sets that cover some topological spaceX. TheVietoris complexofUis the abstract simplicial complex whose vertices are points inX, and whose simplices

Both complexes are homotopy equivalent to U The size of Cech(B) is Θ(n ^d )

The size of Del _|U (B) is Θ(n ^d

^e )

Del _|U (B) naturally embeds in R ^d but not Cech(B) (general position)

Algorithmic Geometry of Triangulations Union of Balls and Delaunay filtrations J-D. Boissonnat 3 / 12

Filtration of a simplicial complex

1

A filtration of K is a sequence of subcomplexes of K

∅ = K ⁰ ⊂ K ¹ ⊂ · · · ⊂ K ^m = K

such that: K ⁱ⁺¹ = K ⁱ ∪ σ ⁱ⁺¹ , where σ ⁱ⁺¹ is a simplex of K

2

Alternatively a filtration of K can be seen as an ordering σ ₁ , . . . σ _m of the simplices of K such that the set K ⁱ of the first i simplices is a subcomplex of K

The ordering should be consistent with the dimension of the simplices

Filtration plays a central role in topological persistence

Filtration of a simplicial complex

1

A filtration of K is a sequence of subcomplexes of K

∅ = K ⁰ ⊂ K ¹ ⊂ · · · ⊂ K ^m = K

such that: K ⁱ⁺¹ = K ⁱ ∪ σ ⁱ⁺¹ , where σ ⁱ⁺¹ is a simplex of K

2

Alternatively a filtration of K can be seen as an ordering σ ₁ , . . . σ _m of the simplices of K such that the set K ⁱ of the first i simplices is a subcomplex of K

The ordering should be consistent with the dimension of the simplices

Filtration plays a central role in topological persistence

Delaunay filtration

P a finite set of points of R ^d U (α) = S

p∈P B (p, α) α-complex = Del _|U(α) (P )

{ Del _|U(α) (P ), α ∈ R ⁺ } is called the Delaunay filtration of P

Shape reconstruction using α -complexes (2d)

Shape reconstruction using α -complexes (3d)

Gabriel simplices

σ ∈ Del(P ) is said to be Gabriel iff σ ∩ σ ^∗ 6 = ∅

a b

a b

A Gabriel edge A non Gabriel edge

Computing Delaunay filtrations

Input: a finite set of points P in R

Compute Del( P )

for each d-simplex τ ∈ Del( P ) do

α(τ) := r(τ)

(the squared circumradius of τ)

for k = d − 1, ..., 0 do

for each d-simplex τ ∈ Del(P) do if τ is a Gabriel simplex then

α(τ ) := r(τ )

(the squared smallest circumradius of τ ) else

α(τ ) := min

α(σ)

Output: The filtration value of each simplex in Del(P) has been computed

Union of balls

What is the combinatorial complexity of the boundary of the union U of n balls of R ^d ?

Compare with the complexity of the arrangement of the bounding hyperspheres

How can we compute U ?

Restriction of Del(B) to U = S

b ∈ B b

U = S

b∈B b ∩ V (b) and ∂U ∩ ∂b = V (b) ∩ ∂b.

The nerve of C is the restriction of Del(B ) to U , i.e. the subcomplex Del _|U (B) of Del(B ) whose faces have a circumcenter in U

∀ b, b ∩ V (b) is convex and thus contractible

C = { b ∩ V (b), b ∈ B } is a good cover of U

The nerve of C is a deformation retract of U

Restriction of Del(B) to U = S

b ∈ B b

U = S

b∈B b ∩ V (b) and ∂U ∩ ∂b = V (b) ∩ ∂b.

The nerve of C is the restriction of Del(B ) to U , i.e. the subcomplex Del _|U (B) of Del(B ) whose faces have a circumcenter in U

∀ b, b ∩ V (b) is convex and thus contractible

C = { b ∩ V (b), b ∈ B } is a good cover of U

The nerve of C is a deformation retract of U

Restriction of Del(B) to U = S

b ∈ B b

U = S

b∈B b ∩ V (b) and ∂U ∩ ∂b = V (b) ∩ ∂b.

The nerve of C is the restriction of Del(B ) to U , i.e. the subcomplex Del _|U (B) of Del(B ) whose faces have a circumcenter in U

∀ b, b ∩ V (b) is convex and thus contractible

C = { b ∩ V (b), b ∈ B } is a good cover of U

The nerve of C is a deformation retract of U

Restriction of Del(B) to U = S

b ∈ B b

U = S

b∈B b ∩ V (b) and ∂U ∩ ∂b = V (b) ∩ ∂b.

The nerve of C is the restriction of Del(B ) to U , i.e. the subcomplex Del _|U (B) of Del(B ) whose faces have a circumcenter in U

∀ b, b ∩ V (b) is convex and thus contractible

C = { b ∩ V (b), b ∈ B } is a good cover of U

The nerve of C is a deformation retract of U

Restriction of Del(B) to U = S

b ∈ B b

U = S

b∈B b ∩ V (b) and ∂U ∩ ∂b = V (b) ∩ ∂b.

The nerve of C is the restriction of Del(B ) to U , i.e. the subcomplex Del _|U (B) of Del(B ) whose faces have a circumcenter in U

∀ b, b ∩ V (b) is convex and thus contractible

C = { b ∩ V (b), b ∈ B } is a good cover of U

The nerve of C is a deformation retract of U

Filtration of weighted Delaunay complexes

B = { b _i = (p _i , r _i ) } i=1,...,n

W (α) = S n i=1 B

p i , q

r ² _i + α ²

= { x ∈ R ^d , min _B

power(x) ≤ α }

r

√ r

+ α

p x

π(x, b

) = (x − p)

− r

− α

π(x, b

) = (x − p)

− r

α-complex = Del _{W (α)} (B) Filtration : { Del _{W (α)} (B), α ∈ R ⁺ }