Union of Balls and α-Complexes

Union of Balls and α-Complexes

Jean-Daniel Boissonnat Geometrica, INRIA

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

Winter School, University of Nice Sophia Antipolis

January 26-30, 2015

Laguerre geometry

Power distance of two balls or of two weighted points.

ballb1(p1, r1), center p1 radiusr1←→ weigthed point(p1, r₁²)∈R^d ballb2(p2, r2), center p2 radiusr2←→ weigthed point(p2, r₂²)∈R^d

π(b1, b2) = (p1−p2)²−r₁²−r₂²

Orthogonal balls

b1, b2 closer ⇐⇒ π(b1, b2)<0⇐⇒(p1−p2)²≤r₁²+r₂² b1, b2 orthogonal ⇐⇒ π(b1, b2) = 0⇐⇒(p1−p2)²=r₁²+r₂² b1, b2further ⇐⇒ π(b1, b2)>0⇐⇒(p1−p2)²≤r₁²+r₂²

Laguerre geometry

Power distance of two balls or of two weighted points.

ballb1(p1, r1), center p1 radiusr1←→ weigthed point(p1, r₁²)∈R^d ballb2(p2, r2), center p2 radiusr2←→ weigthed point(p2, r₂²)∈R^d

π(b1, b2) = (p1−p2)²−r₁²−r₂²

Orthogonal balls

b1, b2 closer ⇐⇒ π(b1, b2)<0⇐⇒(p1−p2)²≤r₁²+r₂² b1, b2 orthogonal ⇐⇒ π(b1, b2) = 0⇐⇒(p1−p2)²=r₁²+r₂² b1, b2further ⇐⇒ π(b1, b2)>0⇐⇒(p1−p2)²≤r₁²+r₂²

Power distance of a point wrt a ball

Ifb1 is reduced to a pointp: π(p, b2) = (p−p2)²−r₂² Normalized equation of bounding sphere :

p∈∂b2⇐⇒π(p, b2) = 0

p∈intb2 ⇐⇒ π(p, b)<0 p∈∂b2 ⇐⇒ π(p, b) = 0 p6∈b2 ⇐⇒ π(p, b)>0 Tangents and secants through p π(p, b) =pt²=pm·pm⁰=pn·pn⁰

p

m

m⁰ t

p2

n

n⁰

Radical Hyperplane

The locus of point ∈R^d with same power distance to ballsb1(p1, r1)andb2(p2, r2)is a hyperplane ofR^d

π(x, b1) =π(x, b2) ⇐⇒ (x−p1)²−r²₁= (x−p2)²−r₂²

⇐⇒ −2p1x+p²₁−r²₁=−2p2x+p²₂−r₂²

⇐⇒ 2(p2−p1)x+ (p²₁−r₁²)−(p²₂−r₂²) = 0

Radical Hyperplane

The locus of point ∈R^d with same power distance to ballsb1(p1, r1)andb2(p2, r2)is a hyperplane ofR^d

π(x, b1) =π(x, b2) ⇐⇒ (x−p1)²−r²₁= (x−p2)²−r₂²

⇐⇒ −2p1x+p²₁−r²₁=−2p2x+p²₂−r₂²

⇐⇒ 2(p2−p1)x+ (p²₁−r₁²)−(p²₂−r₂²) = 0

Power Diagrams

also named Laguerre diagrams or weighted Voronoi diagrams

Sites: n ballsB ={bi(pi, ri), i= 1, . . . n} Power distance: π(x, bi) = (x−pi)²−r_i² Power Diagram: Vor(B)

One cellV(bi)for each site

V(bi) ={x:π(x, bi)≤π(x, bj).∀j6=i} Each cell is a polytope

V(bi)may be empty pi may not belong toV(bi)

Weighted Delaunay triangulations

B =

b

(p

, r

)

a set of balls

B

=

b

(p

, r

), i = 0, . . . k

B B

Del(B )

bi∈Bτ

V (b

)

=

To be proved (next slides):

under a

condition on B ,

B

τ = conv(

p

, i = 0 . . . k

)

embeds Del(B) as a triangulation in

,

called the

Characteristic property of weighted Delaunay complexes

τ

Del(B)

bi∈B_τ

V (b

)

=

⇐⇒ ∃

x

s.t.

b

, b

B

, b

B

B

π(x, b

) = π(x, b

) < π(x, b

)

The space of spheres

b(p, r)ball ofR^d

→point φ(b)∈R^d+1 φ(b) = (p, s=p²−r²)

→polar hyperplane hb=φ(b)^∗∈R^d+1 P ={xˆ∈R^d+1:xd+1=x²}

hb={xˆ∈R^d+1:xd+1= 2p·x−s} ^σ

h(σ) P

Balls will null radius are mapped ontoP hp is tangent toP atφ(p).

The vertical projection ofhb∩ P ontoxd+1= 0is∂b

The space of spheres

b(p, r)ball ofR^d

→point φ(b)∈R^d+1 φ(b) = (p, s=p²−r²)

→polar hyperplane hb=φ(b)^∗∈R^d+1 P ={xˆ∈R^d+1:xd+1=x²}

hb={xˆ∈R^d+1:xd+1= 2p·x−s} ^σ

h(σ) P

Balls will null radius are mapped ontoP hp is tangent toP atφ(p).

The vertical projection ofhb∩ P ontoxd+1= 0is∂b

The space of spheres

b(p, r)ball ofR^d

→point φ(b)∈R^d+1 φ(b) = (p, s=p²−r²)

→polar hyperplane hb=φ(b)^∗∈R^d+1 P ={xˆ∈R^d+1:xd+1=x²}

hb={xˆ∈R^d+1:xd+1= 2p·x−s} ^σ

h(σ) P

Balls will null radius are mapped ontoP hp is tangent toP atφ(p).

The vertical projection ofhb∩ P ontoxd+1= 0is∂b

The space of spheres

b(p, r)ball ofR^d

→point φ(b)∈R^d+1

φ(b) = (p, s=p²−r²)

→polar hyperplane hb=φ(b)^∗∈R^d+1 hb={xˆ∈R^d+1:xd+1= 2p·x−s}

b x

φ^∗(b)

The vertical distance betweenxˆ= (x, x²)andhb is equal to x²−2p·x+s=π(x, b)

The faces of the power diagram ofB are the vertical projections onto xd+1= 0of the faces of the polytopeV(B) =T

ih⁺_b ofR^d+1

The space of spheres

b(p, r)ball ofR^d

→point φ(b)∈R^d+1

φ(b) = (p, s=p²−r²)

→polar hyperplane hb=φ(b)^∗∈R^d+1 hb={xˆ∈R^d+1:xd+1= 2p·x−s}

b x

φ^∗(b)

The vertical distance betweenxˆ= (x, x²)andhb is equal to x²−2p·x+s=π(x, b)

The faces of the power diagram ofB are the vertical projections onto xd+1= 0of the faces of the polytopeV(B) =T

ih⁺_b ofR^d+1

Power diagrams, weighted Delaunay triangulations and polytopes

V

(B ) =

φ(b

)

(B ) = conv

( ˆ P )

Proof of the theorem

B

B,

B

= d + 1, τ = conv(

p

, b

(p

, r

)

B

), φ(τ ) = conv(

φ(b

), b

B

)

∃

b(p, r) s.t. h

= φ(b)

= aff(

φ(b

), b

B

)

φ(τ )

(B) = conv

(

φ(b

)

)

⇐⇒ ∀

b

B

, φ(b

)

h

b

B

, φ(b

)

h

⇐⇒ ∀

b

B

, π(b, b

) = 0

b

B

, π(b, b

) > 0

⇐⇒

p

bi∈B_τ

V (b

)

⇐⇒

τ

Del(B )

Delaunay’s theorem extended

B =

b

, b

. . . b

is said to be in general position wrt spheres if

x

with equal power to d + 2 balls of B

P =

p

, ..., p

: set of centers of the balls of B

Theorem

If

is in general position wrt spheres, the simplicial map

f : vert(Del(B))→P

provides a realization of

Del(B)

is a triangulation of

called the

Delaunay’s theorem extended

B =

b

, b

. . . b

is said to be in general position wrt spheres if

x

with equal power to d + 2 balls of B

P =

p

, ..., p

: set of centers of the balls of B

Theorem

If B is in general position wrt spheres, the simplicial map f : vert(Del(B ))

P

provides a realization of Del(B)

Del(B) is a triangulation of P

P called the

Power diagrams, Delaunay triangulations and polytopes

If B is a set of balls in general position wrt spheres :

V

(B ) = h

1∩

. . .

h

n

duality

−→ D

(B) = conv

(

φ(b

), . . . , φ(b

)

)

↑ ↓

Voronoi Diagram

of B

of B

Complexity and algorithm for weighted VD and DT

Number of faces= Θ

n^{bd+12 c}

(Upper Bound Th.)

Construction can be done in timeΘ

nlogn+n^{bd+12 c}

(Convex hull)

Main predicate

power test(b0, . . . , bd+1) = sign

1 . . . 1

p0 . . . pd+1

p²₀−r₀² . . . p²_d+1−r_d+1²

Complexity and algorithm for weighted VD and DT

Number of faces= Θ

n^{bd+12 c}

(Upper Bound Th.)

Construction can be done in timeΘ

nlogn+n^{bd+12 c}

(Convex hull)

Main predicate

power test(b0, . . . , bd+1) = sign

1 . . . 1

p0 . . . pd+1

p²₀−r₀² . . . p²_d+1−r_d+1²

Complexity and algorithm for weighted VD and DT

Number of faces= Θ

n^{bd+12 c}

(Upper Bound Th.)

Construction can be done in timeΘ

nlogn+n^{bd+12 c}

(Convex hull)

Main predicate

power test(b0, . . . , bd+1) = sign

1 . . . 1

p0 . . . pd+1

p²₀−r₀² . . . p²_d+1−r_d+1²

Power diagrams are maximization diagrams

Cell of bi in the power diagram Vor(B)

V(bi) = {x∈R^d:π(x, bi)≤π(x, bj).∀j6=i}

= {x∈R^d: 2pix−si= maxj∈[1,...n]{2pjx−sj}}

Vor(B)is themaximization diagramof the set of affine functions {fi(x) = 2pix−si, i= 1, . . . , n}

Affine diagrams (regular subdivisions)

Affine diagrams are defined as the maximization diagrams of a finite set of affine functions

They are equivalently defined as the vertical projections of polyhedra intersection of a finite number of upper half-spaces ofR^d+1

Voronoi diagrams and power diagrams are affine diagrams.

Any affine diagram ofR^d is the power diagram of a set of balls.

Delaunay and weighted Delaunay triangulations are regular triangulations Any regular triangulation is a weighted Delaunay triangulation

Affine diagrams (regular subdivisions)

Affine diagrams are defined as the maximization diagrams of a finite set of affine functions

They are equivalently defined as the vertical projections of polyhedra intersection of a finite number of upper half-spaces ofR^d+1

Voronoi diagrams and power diagrams are affine diagrams.

Any affine diagram ofR^d is the power diagram of a set of balls.

Delaunay and weighted Delaunay triangulations are regular triangulations Any regular triangulation is a weighted Delaunay triangulation

Affine diagrams (regular subdivisions)

Affine diagrams are defined as the maximization diagrams of a finite set of affine functions

They are equivalently defined as the vertical projections of polyhedra intersection of a finite number of upper half-spaces ofR^d+1

Voronoi diagrams and power diagrams are affine diagrams.

Any affine diagram ofR^d is the power diagram of a set of balls.

Delaunay and weighted Delaunay triangulations are regular triangulations Any regular triangulation is a weighted Delaunay triangulation

Examples of affine diagrams

1 The intersection of a power diagram with an affine subspace (Exercise)

2 A Voronoi diagram defined with a quadratic distance function kx−ak^Q= (x−a)^tQ(x−a) Q=Q^t

3 korder Voronoi diagrams

k -order Voronoi Diagrams

LetP be a set of sites.

k -order Voronoi diagrams are power diagrams

LetS1, S2, . . .denote the subsets ofkpoints ofP.

Thek-order Voronoi diagram is the minimization diagram ofδ(x, Si): δ(x, Si) = 1

k X

p∈S_i

(x−p)²

= x²−2 k

X

p∈S_i

p·x+1 k

X

p∈S_i

p²

= π(bi, x) where bi is the ball

1 centered atci=_k¹P

p∈Sip

2 withsi=π(o, bi) =c²_i −r²_i = ¹_k P

p∈S_ip²

3 and radiusri²=c²i −¹k

P

p∈S p² .

Combinatorial complexity of k -order Voronoi diagrams

Theorem

IfP be a set ofnpoints inR^d, the number of vertices and faces in all the Voronoi diagrams Vorj(P)

of ordersj ≤k is:

O

k^{dd+12 e}n^{bd+12 c}

Proof

uses :

I bijection betweenk-sets and cells in k-order Voronoi diagrams

I the sampling theorem (from randomization theory)

Combinatorial complexity of k -order Voronoi diagrams

Theorem

IfP be a set ofnpoints inR^d, the number of vertices and faces in all the Voronoi diagrams Vorj(P)

of ordersj ≤k is:

O

k^{dd+12 e}n^{bd+12 c}

Proof

uses :

I bijection betweenk-sets and cells in k-order Voronoi diagrams

I the sampling theorem (from randomization theory)

k -sets and k -order Voronoi diagrams

P a set ofnpoints inR^d

k-sets

Ak-set ofP is a subset P⁰ ofP with sizek that can be separated fromP\P⁰ by a hyperplane

k-order Voronoi diagrams

kpoints ofP have a cell in Vork(P)iff there exists a ball that contains those points and only those

⇒each cell of Vork(P)corresponds to a k-set of

φ(P) ^σ

h(σ) P

k -sets and k -order Voronoi diagrams

P a set ofnpoints inR^d

k-sets

Ak-set ofP is a subset P⁰ ofP with sizek that can be separated fromP\P⁰ by a hyperplane

k-order Voronoi diagrams

kpoints ofP have a cell in Vork(P)iff there exists a ball that contains those points and only those

⇒each cell of Vork(P)corresponds to a k-set of

φ(P) ^σ

h(σ) P

k -sets and k -levels in arrangements of hyperplanes

For a set of pointsP ∈R^d, we consider the arrangement of thedual hyperplanesA(P^∗)

hdefines ak setP⁰ ⇒ hseparatesP⁰ (belowh) fromP\P⁰ (above h)

⇒ h^∗ is below thek hyperplanes ofP^0∗ and above those ofP^∗\P^0∗

k-sets of P are in 1-1 correspondance with the cells ofA(P^∗)of levelk, i.e.

k -sets and k -levels in arrangements of hyperplanes

For a set of pointsP ∈R^d, we consider the arrangement of thedual hyperplanesA(P^∗)

hdefines ak setP⁰ ⇒ hseparatesP⁰ (belowh) fromP\P⁰ (above h)

⇒ h^∗ is below thek hyperplanes ofP^0∗ and above those ofP^∗\P^0∗

k-sets of P are in 1-1 correspondance with the cells ofA(P^∗)of levelk, i.e.

k -sets and k -levels in arrangements of hyperplanes

For a set of pointsP ∈R^d, we consider the arrangement of thedual hyperplanesA(P^∗)

hdefines ak setP⁰ ⇒ hseparatesP⁰ (belowh) fromP\P⁰ (above h)

⇒ h^∗ is below thek hyperplanes ofP^0∗ and above those ofP^∗\P^0∗

k-sets of P are in 1-1 correspondance with the cells ofA(P^∗)of levelk, i.e.

Bounding the number of k -sets

ck(P) : Number ofk-sets ofP = Number ofcellsof levelk inA(P^∗) c_≤k(P) =P

l≤kcl(P)

c⁰_≤k(P): Number ofverticesofA(P^∗)with level at mostk c_≤k(n) = max_|P|=nc_≤k(P)c⁰_≤k(n) = max_|P|=nc⁰_≤k(P)

Hyp. in general position: each vertex∈dhyperplanes incident to2^d cells Vertices of levelkare incident to cells with level∈[k, k+d]

Cells of level khave incident vertices with level∈[k−d, k]

Regions, conflicts and the sampling theorem

O a set ofnobjects.

F(O)set ofconfigurationsdefined byO

each configuration is defined by a subset ofbobjects each configuration is in conflictwith a subset ofO F^j(O)set of configurations in conflict withj objects

|F≤k(O)| number of configurations defined byO in conflict with at mostkobjects ofO f0(r) =Exp(|F⁰(R|)expected number of configurations

defined and without conflict on a randomr-sample ofO.

The sampling theorem

For 2≤k≤ b+1ⁿ , |F≤k(O)| ≤4 (b+ 1)^bk^bf0(n k

)

Regions, conflicts and the sampling theorem

O a set ofnobjects.

F(O)set ofconfigurationsdefined byO

each configuration is defined by a subset ofbobjects each configuration is in conflictwith a subset ofO F^j(O)set of configurations in conflict withj objects

|F≤k(O)| number of configurations defined byO in conflict with at mostkobjects ofO f0(r) =Exp(|F⁰(R|)expected number of configurations

defined and without conflict on a randomr-sample ofO.

The sampling theorem

For 2≤k≤ b+1ⁿ , |F≤k(O)| ≤4 (b+ 1)^bk^bf0(n k

)

Proof of the sampling theorem

f0(r) =X

j

|F^j(O)|

n−b−j r−b

n

r

≥ |F≤k(O)|

n−b−k r−b

n

r

then,we prove that forr= ⁿ_k

n−b−k r−b

n

r

≥ 1 4(b+ 1)^bk^b

n−b−k r−b

n

r

= r!

(r−b)!

(n−b)!

n!

| {z }

≥_{(b+1)b kb}¹

(n−r)!

(n−r−k)!

(n−b−k)!

(n−b)!

| {z }

≥¹₄

Proof of the sampling theorem

end

(n−r)!

(n−r−k)!

(n−b−k)!

(n−b)! =

k

Y

j=1

n−r−k+j n−b−k+j ≥

n−r−k+ 1 n−b−k+ 1

^k

≥

n−n/k−k+ 1 n−k

k

≥ (1−1/k)^k≥1/4 pour(2≤k), r!

(r−b)!

(n−b)!

n! =

b−1

Y

l=0

r−l n−l ≥

b

Y

l=1

r+ 1−b n

≥

b

Y

l=1

n/k−b n

bk 1 n

Bounding the number of k -sets

ck(P) : Number ofk-sets ofP = Number of cells of levelk inA(P^∗).

c_≤k(P) =P

l≤kcl(P)

c⁰_≤k(P): Number of vertices ofA(P^∗)with level at mostk.

ObjectsO: n hyperplanes ofR^d

Configurations: vertices inA(O),b=d Conflictbetweenv andh: v∈h⁺

Sampling th: c⁰_≤k(P)≤4(d+ 1)^dk^df0 n k

Upper bound th: f0(n

k

) =O

nb^d2c b^dc











⇒c⁰_≤k(n) =O

kd^d2enb^d2c

Combinatorial complexities

Number of vertices of level≤kin an arrangement ofnhyperplanes inR^d Number of cells of level≤k in an arrangement ofnhyperplanes in R^d Total number ofj≤ksets for a set ofnpoints inR^d

kd^d2enb^d2c

Total number of faces in the Voronoi diagrams of orderj≤kfor a set ofn points inR^d

kd^d+12 enb^d+12 c

Combinatorial complexities

Number of vertices of level≤kin an arrangement ofnhyperplanes inR^d Number of cells of level≤k in an arrangement ofnhyperplanes in R^d Total number ofj≤ksets for a set ofnpoints inR^d

kd^d2enb^d2c

Total number of faces in the Voronoi diagrams of orderj≤kfor a set ofn points inR^d

kd^d+12 enb^d+12 c

Restriction of Delaunay triangulation

Let Ω

and P

a finite set of points.

Vor(E)

Ω is a cover of Ω. Its

is called the Delaunay triangulation

of E restricted to Ω, noted Del

(P)

Union of balls

What is the combinatorial complexity of the boundary of the union U of n balls of

?

Compare with the complexity of the arrangement of the bounding hyperspheres

How can we compute U ?

What is the image of U in the space of spheres ?

Restriction of Del(B) to U = S

b∈B

b

U =

b∈B

b

V (b) and ∂U

∂b = V (b)

∂b.

The nerve of

is the restriction of

to

, i.e. the subcomplex

of

whose faces have a circumcenter in

∀b, b∩V(b)

is convex and thus contractible

is a good cover of

The nerve of

is a

of

Restriction of Del(B) to U = S

b∈B

b

U =

b∈B

b

V (b) and ∂U

∂b = V (b)

∂b.

The nerve of

is the restriction of Del(B ) to U , i.e. the subcomplex

of Del(B ) whose faces have a circumcenter in U

∀b, b∩V(b)

is convex and thus contractible

is a good cover of

The nerve of

is a

of

Restriction of Del(B) to U = S

b∈B

b

U =

b∈B

b

V (b) and ∂U

∂b = V (b)

∂b.

The nerve of

is the restriction of Del(B ) to U , i.e. the subcomplex

of Del(B ) whose faces have a circumcenter in U

∀

b, b

V (b) is convex and thus contractible

is a good cover of

The nerve of

is a

of

Restriction of Del(B) to U = S

b∈B

b

U =

b∈B

b

V (b) and ∂U

∂b = V (b)

∂b.

The nerve of

is the restriction of Del(B ) to U , i.e. the subcomplex

of Del(B ) whose faces have a circumcenter in U

∀

b, b

V (b) is convex and thus contractible

=

b

V (b), b

B

is a good cover of U

The nerve of

is a

of

Restriction of Del(B) to U = S

b∈B

b

U =

b∈B

b

V (b) and ∂U

∂b = V (b)

∂b.

The nerve of

is the restriction of Del(B ) to U , i.e. the subcomplex

of Del(B ) whose faces have a circumcenter in U

∀

b, b

V (b) is convex and thus contractible

=

b

V (b), b

B

is a good cover of U

The nerve of

is a

of U

Cech complex versus Del

(B)

Both complexes are homotopy equivalent to U

The size of Cech(B) is Θ(n

)

The size of Del

(B) is Θ(n

)

Filtration of a simplicial complex

1

A filtration of K is a sequence of subcomplexes of K

∅

= K

K

K

= K

such that: K

= K

σ

, where σ

is a simplex of K

2

Alternatively a filtration of K can be seen as an ordering σ

, . . . σ

of the simplices of K such that the set K

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

K

K

= K

such that: K

= K

σ

, where σ

is a simplex of K

2

Alternatively a filtration of K can be seen as an ordering σ

, . . . σ

of the simplices of K such that the set K

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

α -filtration of Delaunay complexes

P a finite set of points of

U (α) =

p∈P

B (p, α)

= Del

(P )

The filtration

Del

(P ), α

is called the

of Del(P)

Shape reconstruction using α -complexes (2d)

Shape reconstruction using α -complexes (3d)

Constructing the α -filtration of Del(P )

σ

Del(P ) is said to be Gabriel iff σ

σ

=

a b

a b

A Gabriel edge A non Gabriel edge

Algorithm

for eachd-simplexσ∈Del(P): αmin(σ) =r(σ) fork=d−1, ...,0,

for eachk-faceσ∈Del(P)

αmed(σ) = minσ∈coface(σ)αmin(σ) ifσis Gabriel thenαmin(σ) = r(σ)

Constructing the α -filtration of Del(P )

σ

Del(P ) is said to be Gabriel iff σ

σ

=

a b

a b

A Gabriel edge A non Gabriel edge

Algorithm

for eachd-simplexσ∈Del(P): αmin(σ) =r(σ) fork=d−1, ...,0,

for eachk-faceσ∈Del(P)

αmed(σ) = minσ∈coface(σ)αmin(σ) ifσis Gabriel thenα (σ) = r(σ)

α -filtration of weighted Delaunay complexes

B =

b

= (p

, r

)

W (α) =

B

p

,

r

+ α

r

√r²+α²

p x

π(x, bblue) = (x−p)²−r²−α² π(x, bred) = (x−p)²−r²

α-complex

= Del

(B)

:

Del

(B), α