ImageAnalStereol2009;28:63-67
OriginalResearchPaper
THE PIVOTALTESSELLATION
LUISM CRUZ-ORIVE
DepartmentofMathematics,StatisticsandComputation,Fa cultyofSciences,UniversityofCantabria,Avda.
LosCastross/n,E-39005Santander,Spain
e-mail: luis.cruz@unican.es
(AcceptedApril24,2009)
ABSTRACT
The tessellation studied here is motivated by some stereolo gical applications of a new expression for the
motion invariant density of straight lines in R3. The term ‘pivotal’ stems from the fact that the tessellatio n
is constructed within a plane which is isotropic through a fix ed, ‘pivotal’ origin. Consider either a stationary
point process, or a stationaryrandomlattice of pointsin th at plane. Througheach point event draw a straight
line which is perpendicular to the axis determined by the ori gin and the point event. The union of all such
lines (called p-lines) constitutes the mentioned tessellation. We concen trate on the pivotal tessellation based
ona stationaryandisotropicplanarPoisson pointprocess; weshowthat thistessellation isnotstationary.
Keywords: geometric probability, p-line, pivotal tessellation, Poisson point process, stere ology, stochastic
geometry.
INTRODUCTION
The purposeof this paperis to explore elementary
properties of a special planar tessellation stemming
from the application of recent stereological results
(Cruz-Orive,2005;2008;Gual-ArnauandCruz-Orive,
2009).
Consider the equatorial disk B2,t=B3∩L3
2[0],
whereB3⊂R3represents a ball of radius Rcentred
at the origin OandL2[0]denotes an isotropic plane
throughOwith normaldirection t∈S2
+.
Within the disk B2,t, generate Nindependent
and identically distributed uniform random points
{z1,z2,...,zN}, (Fig. 1a). For each i=1,2,...,N, draw
a straight line L1(zi)through the point ziand normal
to the axis Ozi. ThusL1(zi)is effectively a “point
sampled” straight line which will be called a p−line.
The union of all p−lines constitutes a tessellation
in the reference disk, (Fig. 1b) which will be called
a pivotal tessellation, inasmuch as the containing
planeL2[0]can only rotate around a fixed ‘pivot’ O.
The practical interest of this construction lies in the
followingfact.Consideranonvoidcompactsubset Y⊂
B3of volume ν3(Y)with piecewise smooth boundary
∂Yofareaν2(∂Y).Then,
/hatwideν2(∂Y) =2aN−1N
∑
i=1ν0{(∂Y∩B2,t)∩L1(zi)},
/hatwideν3(Y) =aN−1N
∑
i=1ν1{(Y∩B2,t)∩L1(zi)},(1)
are unbiased estimators of ν2(∂Y)andν3(Y),
respectively, where a:=πR2, andν0,ν1denotenumberofintersectionsandchordlength,respectively
(Cruz-Orive,2005;2008).
Here we are interested in some properties of
the pivotal tessellation constituted by the p−lines
associatedwith a planarPoissonpointprocess.
PRELIMINARIES
Given a point z∈R2of polar coordinates (ρ,ω),
ρ∈(0,∞),ω∈(0,2π), we define a p-lineL1(z)as a
straightline with normalcoordinates (ρ,ω),namely,
L1(z):=L1(ρ,ω)
=/braceleftbig
x= (x1,x2)∈R2:x1cosω+x2sinω=ρ/bracerightbig
.
(2)
Considereithera stationaryplanarpointprocess
Φ=/uniondisplay
i∈Nzi, (3)
with realizationsin R2, orastationaryrandomlattice
Λz=Λ0+z=/uniondisplay
i∈Nzi, (4)
whereΛ0⊂R2is a fixed regular lattice of points and
zis a uniform random point in a fundamental tile of
Λ0, (Fig. 3a). In either case, the pivotal tessellation
associatedwith either ΦorΛzis
Ψ=/uniondisplay
i∈NL1(zi), (5)
63
CRUZ-ORIVELM:Thepivotaltessellation
O O
(a) (b)
Fig. 1.(a) Disk centred at O containing 50 independent uniform rand om point events. (b) Associated pivotal
tessellationformedbythecorresponding p-lineswith resp ectto O.
namelythecorrespondingprocessof p-lines,(Figs.1b,
3b).
In this paper P(dx)represents the probability
element of a random variable X, namely P(dx):=
P(x<X≤x+dx). IfXadmits a probability density
function f(x), then P(dx) =f(x)dx. This notation
extendstohigherdimensionsina naturalmanner.
THE PIVOTAL POISSON
TESSELLATIONIN R2
When the associatedprocess Φis a stationary and
isotropic planar Poisson point process (Stoyan et al.,
1995), then the corresponding pivotal tessellation Ψ
will be called the pivotal Poisson tessellation. Let τ
denotethe fixedintensityof Φ, namely,
τ=E{ν0(Φ∩B)}
ν2(B),0<ν2(B)<∞,(6)
whereBdenotes any subset from the Borel σ-algebra
inR2, andνqdenotes the q-dimensional Hausdorff
measure in R2, (thusν2represents area, ν1curve
length,and ν0the countingmeasure).
Next we obtain some properties of the tesselation
Ψ.Todo thiswe considertherandomintersection
Ψ∩BR=N/uniondisplay
i=1L1(zi)∩BR, (7)
(Fig. 1b), where BR⊂R2is a closed disk of radius
Rcentred at the origin O, whereas {z1,z2,...,zN}
represent Nindependent and identically distributed
(i.i.d.) uniform random(UR)pointsin BR, and
N∼Poisson (πR2τ), (8)so thatNis a Poisson random variable with mean and
varianceequalto πR2τ.
For ap-lineL1(z):=L1(ρ,ω)such that the point
zof polar coordinates (ρ,ω)is UR inBR, it is easy to
show that ρandωare independent random variables
with
P(dρ) =2R−2ρdρ,0<ρ<R,
P(dω) = (2π)−1dω,0<ω<2π.(9)
Lemma 1. Let z ∈BRdenote a UR point in B R.
Then the mean chord length determined in B Rby the
corresponding p-line is,
E{ν1(L1(z)∩BR)}=4
3R. (10)
Proof.Straightforward bearing in mind that
ν1(L1(z)∩BR) =2/radicalbig
R2−ρ2andusingEq.9. /square
Proposition 1. The mean total length per unit disk
areaof the straightline segmentsdeterminedin B Rby
the p-linesof Ψis
λ2
1(R):= (πR2)−1E{ν1(Ψ∩BR)}
=4
3τR. (11)
Proof.Conditional on the number Nofp-lines
fromΨhittingBR, byEq.10we have,
E{ν1(Ψ∩BR)|N}=4
3RN. (12)
Using the premise (Eq. 8) and dividing by πR2, the
resultfollows. /square
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ImageAnalStereol2009;28:63-67
Consequence. From Eq. 11 we see that λ2
1(R) =
O(R), which implies that the pivotal Poisson
tessellation Ψassociatedwith Φis notstationary.
Remark 1. The mean area of the equatorial disk
B2,tconsidered in the Introduction (see Fig. 2) per
unit volume of the corresponding ball B 3, isλ3
2(R) =
πR2/(4πR3/3)=3/(4R).Ontheotherhand,themean
total chord length of the bounded pivotal Poisson
tessellation in B 2,t, per unit area of B 2,t, is given by
Eq. 11. Therefore,the mean total chordlength of such
planar tessellation per unit volume of the reference
ballB3, is
λ3
1(R) =λ2
1(R)·λ3
2(R) =τ,(13)
namely a constant. This result is consistent with the
factthat p-linesareeffectivelymotioninvariantin R3.
Lemma2. Letz1,z2denotetwoi.i.d.URpointsinB R.
Then,
P{L1(z1)∩L2(z2)∈BR}=3
8,∀R∈(0,∞),(14)
that is, the probability that the corresponding two p-
lines intersect inside B Ris a known constant equal to
3/8 foranyR >0.
ρR
Oz1
BR
Fig. 2.The probability that a p-line L 1(z2)associated
with a UR point z 2∈BRhits a given p-chordL 1(z1)∩
BR(thickstraightlinesegmentinthefigure)isequalto
the probability that z 2falls in the support set (shaded
region)ofthegiven p-chordwith respecttoO.
Proof.Fix one of the two points, e.g.,z1=
(ρ,ω),ρ∈(0,R),ω∈(0,2π), and denote by
p(ρ,ω;R)the required probability conditional on
(ρ,ω). By the definition of support set (Cruz-Orive,
2005; Gual-Arnau and Cruz-Orive, 2009), it follows
thatp(ρ,ω;R)is the probability that z2falls in thesupport set HL1(z1)∩BRof the chord L1(z1)∩BRwith
respect to the disk centre O(see Fig. 2). Bearing in
mind that P(dz2) = (πR2)−1dz2,z2∈BR, wehave
p(ρ,ω;R):=P{L1(z1)∩L2(z2)∈BR|ρ,ω}
=/integraldisplay
HL1(z1)∩BRP(dz2) (15)
=1
2−2
π·gdisk/parenleftbigg/radicalBig
1−ρ2/R2/parenrightbigg
,
where
gdisk(x) =1
2/parenleftBig
cos−1x−x/radicalbig
1−x2/parenrightBig
,(0≤x≤1),
(16)
isthegeometriccovariogramofadiskofunitdiameter.
It isreadily verifiedthat,
P{L1(z1)∩L2(z2)∈BR}=/integraldisplayR
0p(ρ,ω;R)P(dρ)
=3
8, (17)
where P(dρ)is givenbythefirstEq.9. /square
Proposition 2. Letλ(0)
0(R),λ(1)
0(R), andλ(2)
0(R)
denote the mean total numbers per unit disk area of
the vertices, edges and connectedregions constituting
the boundedtessellation Ψ∩BR,respectively.Then,
λ(0)
0(R) =3π
16τ2R2+2τ,
λ(1)
0(R) =3π
8τ2R2+3τ, (18)
λ(2)
0(R) =3π
16τ2R2+τ+1
πR2,
where the terms following the first one in the right
hand side of the preceding identities represent the
contributionsofthe diskboundary ∂BR.
Proof.WeusethemethodofSantal´ o(1940;1976
p.51).Conditionalonthenumber Nofp-linesfrom Ψ
hittingBR,letVBo(N),V∂B(N)denotethemeannumber
of verticesinterior to BRand in∂BR, respectively,and
setVBo(N)+V∂B(N) =V(N).ThenusingLemma2,
E{V(N)|N}=/parenleftbiggN
2/parenrightbigg3
8+2N.(19)
Likewise, let EBo(N),E∂B(N)denote the mean
number of edges interior to BRand in ∂BR,
respectively, and set EBo(N) +E∂B(N) =E(N). At
each interior vertex there meet 4 edges, but they are
counted twice because each edge has two vertices as
endpoints.On the other hand,at eachboundaryvertex
65
CRUZ-ORIVELM:Thepivotaltessellation
(a) (b)O O
Fig. 3.(a) A realization of a stationary random square lattice insi de a disk. The centre O of the disk is uniform
randominatileofthelattice(shadedsquare).(b)Theassoc iatedpivotallatticetessellationinsidethedisk,which
is in factthekindofprobeusedin theapplications(Cruz-Or ive,2005;2008).
theremeet3edges,buttheyarealsocountedtwice for
the samereason.Therefore,
E{E(N)|N}=2/parenleftbiggN
2/parenrightbigg3
8+3N.(20)
Finally, let F(N)denote the total number of
connected regions or “faces”. By Euler’s formula we
haveV(N)+F(N)−E(N) =1, andtherefore,
E{F(N)|N}=/parenleftbiggN
2/parenrightbigg3
8+N+1.(21)
Taking expectations on both sides of each of the
identities (Eqs. 19–21) with respect to N, bearing
Eq. 8 in mind, and dividing by πR2in each case, the
correspondingidentities (Eq. 18)are obtained. /square
Definition. Themeannumberofvertices(orofsides),
the mean boundary length, and the mean area of a
connected region from the bounded tessellation Ψ∩
BR, aredefinedrespectivelyasfollows,
E{N(R)}=2λ(1)
0(R)
λ(2)
0(R),
E{B(R)}=2λ2
1(R)+2/R
λ(2)
0(R), (22)
E{A(R)}=1
λ(2)
0(R).
Proposition 3. The characteristics given in the
preceding definition satisfy the following asymptoticrelations,
E{N(R)}=4+O(R−2),
E{B(R)}=O(R−1), (23)
E{A(R)}=O(R−2).
Proof.Substitute the results Eq. 11 and Eq. 18
into Eq.22. /square
CONCLUSIONSANDCOMMENTS
Concerning the planar pivotal Poisson tessellation
Ψ, the main conclusion is that it is not stationary, as
illustrated by the results (Eqs.11, 18, and 23). The
asymptotic mean number of vertices of a polygon is
4, as in the ordinary Poisson tessellation of straight
lines(Stoyan etal.,1995),buttheremainingproperties
change with the distance from the origin. The non
stationarity is intuitively plausible on seeing Fig. 1b.
A priori one might think that, because p-lines on an
isotropic plane L2[0]are effectively motion invariant
inR3, and because the associated point process is
stationary Poisson, then Ψwould also be stationary in
R2, but this is not the case. As confirmed by Eq. 13,
the length density of the p−lines of the planar pivotal
Poisson tessellation must be constant in R3because
they are motion invariant in R3. Note that the plane
L2[0]islessandless“dense”awayfromtheorigin;this
effectmustbecompensatedbyahigherandhigherline
length density in that plane away from the origin, and
this isindeedwhathappens.
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ImageAnalStereol2009;28:63-67
For estimation purposes via Eq. 1 it is simpler
and more efficient to start with a stationary random
lattice of points (Fig. 3a), instead of a Poisson point
process. The corresponding pivotal lattice tessellation
(Fig. 3b) will enjoy similar properties. An exactstudy
ofthelattermightbeprohibitive,however,becausethe
number of lattice points inside a disk is a complicated
oscillating functionofthediskdiameter.
ACKNOWLEDGMENTS
I am indebted to a referee for pointing out a
mistake in Lemma 2 from the original typescript
which affected the value of the constant in Eq. 14,
as well as some imprecisions. Work supported by
the Spanish Ministry of Education and Science I+D
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