Table Of ContentLecture Notes in Statistics 87
Edited by S. Fienberg, J. Gani, K. Krickeberg,
I. OIkin, and N. Wemmth
Je sper
M~ller
Lectures On Randolll
Voronoi Tessellations
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Jesper M~ller
Department of Theoretical Statistics
Institute of Mathematics
University of Aarhus
DK-8000 Aarhus C
DENMARK
library of Congress Cataloging-in-Publication Data
Mj'jller, Jesper.
Lectures on random Voronoi tessellations 1J esper Mj'jller.
p. em. -- (Lecture notes in statistics; 87)
Includes bibliographical references and index.
1. Voronoi polygons. 2. Spatial analysis (Statistics). I. Title.
II. Series: Lecture notes in statistics (Springer-Verlag); v. 87.
QA278.2.M64 1994
133.9'01'3 --dc20 94-248
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© 1994 Springer-Verlag New York, Inc.
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ISBN-13: 978-0-387-94264-3 e-ISBN-13: 978-1-4612-2652-9
DOl: 10.10071978-1-4612-2652-9
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Preface
These notes were prepared for a series of lectures pre
sented at a summer course 1992 on 'Random Voronoi Tessella
tions' organized by Professor Julian Besag, Department of
Statistics, University of Washington.
Hopefully the notes may provide a helpful supplement to
the existing mathematical literature on random Voronoi tessel
lations, which might be difficult for non-specialists to enter.
The notes were prepared just before the excellent book by Okabe
et al. (1992) appeared. While Okabe et al. (1992) contains a
comprehensive collection of concepts and results of Voronoi
tessellations but often without mathematical details and
proofs, the exposition in the present booklet is mathematically
rigorous and detailed proofs are given making the notes largely
self-contained. Also no background knowledge on either the
subject of this booklet or of spatial statistics and stochastic
geometry is assumed.
Finally, I am indebted to Julian Besag for giving me the
opportunity to visit Seattle and Department of statistics,
University of Washington, and to Oddbj0rg Wethelund for her
eminent secretarial assistance.
December 1992
Jesper M0ller
Contents Page
Preface
1. Introduction and background 1
1.1. Definitions, assumptions, and characteristics 3
1.2. History and applications 9
1.3. Related tessellations 11
2. Geometrical properties and other background material 15
2.1. On the geometric structure of Voronoi and Delaunay
tessellations 15
2.2. Short diversion into integral geometry 27
3. Stationary Voronoi tessellations 43
3.1. Spatial point processes and stationarity 44
3.2. Palm measures and intensities of cells and facets 48
3.3. Mean value relations 60
3.4. Flat sections 69
4. Poisson-Voronoi tessellations 83
4.1. The homogeneous Poisson process 83
4.2. Mean value characteristics of Poisson-Voronoi facets 88
4.3. On the distribution of the typical poisson-Delaunay
cell and related statistics 104
4.4. On the distribution of the typical Poisson-Voronoi
cell and related statistics 112
4.5. Simulation procedures for Poisson-Voronoi
tessellations and other related models 118
References 125
Subject and author index 129
Notation index 132
1
1. Introduction and background
A tessellation or mosaic of the d-dimensional Euclidean
space IRd is a subdivision IRd = U C. into d-dimensional
i J.
non-overlapping sets such arrangements occur in many
natural situations and depending on the situation the sets Ci
might be called cells, crystals, regions, tiles, etc.
Many real-life tessellations are random. Random tessella
tions have been studied for a long time in stochastic geometry
and a general theory has now been established, see e.g. Stoyan
et al. (1987), Zahle (1988), M01ler (1989), and Mecke et al.
(1990). Usually the cells are assumed to be bounded and convex,
and the aggregate of cells is locally bounded in the sense that
the number of cells intersecting any bounded subset of IRd is
finite. Then the cells become d-dimensional convex polytopes,
i.e. bounded intervals (d 1) , convex polygons (d = 2),
convex polyhedra (d = 3), etc. Of course, most interest in
'practice' concerns planar and spatial random tessellations.
Indeed there is a large variety of specific probabilistic
models for random tessellations. Typically, the random mechan
ism is given by some stochastic process of simple geometrical
objects which generate the tessellation in accordance to some
rule. One such example is a process of lines in 1R2, which in
an obvious way determines a random planar tessellation of 1R2,
see Figure 1.1. Line-generated tessellations provide one of the
mathematically most tractable class of models of tessellations,
especially if the line process is Poisson, see e.g. Mecke et
al. (1990) and the references therein. However, their import
ance for 'practical' applications seem to be somewhat limited.
In these lecture notes we shall consider another construc
tion: voronoi tessellations generated by point processes as
described in section 1.1. In the first part, Chapters 1-2, we
discuss essentially non-random properties of Voronoi tessella
tions, i.e. the geometric structure of the cell aggregate when
the realization of the associated point process is given; also
2
Figure 1.1. An example of a simulated Poisson line tessella
tion observed within a disc.
some background material on integral geometry is presented. The
remaining Chapters 3-4, which constitute the major part, are
devoted to the study of random Voronoi tessellations. Chapter 3
treats arbitrary stationary Voronoi tessellations using Palm
measure theory. Chapter 4 concerns Poisson-Voronoi tessella
tions. The homogeneous Poisson process seems to be the only
non-trivial stochastic model for which a reasonable collection
of theoretical results for the associated Voronoi tessellation
is derivable. For specificity and ease of exposition, we shall
often restrict attention to planar and spatial Voronoi tessel
lations. However, most concepts and results hold as well in
IRd, and sometimes it will be more appropriate to state the
definitions and results for arbitrary dimensions d.
The notes in no way attempt to cover all aspects of random
Voronoi tessellations. Much deeper and fascinating results have
had to be omitted, like e.g. David G. Kendall's shape theory of
Poisson-Delaunay triangles, see Kendall (1989). Further, many
results hold as well for more general models of tessellations.
This is in particular the case for the mean value relations
3
stated in Chapter 3 as established by Joseph Mecke and many
others. Also I must acknowledge Roger E. Miles and Kenneth A.
Brakke for most of the material on Poisson-Voronoi and Poisson
Delaunay tessellations as presented in Chapter 4.
Finally, it should be noted that the exposition will main
ly concentrate on the probabilistic aspects of random Voronoi
tessellations. Detailed proofs will be given with the exception
of some easy ~roofs which are stated as exercises.
1.1. Definitions. assumptions. and characteristics
In this section we introduce the Voronoi and Delaunay
tessellations and describe some of their geometric structure.
More details are to be found in section 2.1.
Consider a set of points X. called
1
nuclei. Each nucleus generates a cell
where 11·11 denotes Euclidean distance. Thus C{x. ,~) consists
1
of all points which have xi as nearest nucleus, see Figures
1.1.1-1.1.2. Equivalently,
(1.1.2)
where is the closed halfspace
H{x.,x.) {yElRd , (y-z .. ). {x.-x.»O}
1 ) 1) 1)-
containing and bounded by the bisecting hyperplane
of and that is the hyperplane which con-
tains the midpoint
4
z.. -12 (X,+X,)
1J 1 J
and is perpendicular to the line through x. and x .. Here
1 J
is the usual inner product on IRd. Thus c(xilt) is a closed
convex set. Further, c(xilt) n c(x·lt) ~ G (xi' Xj ) , so the
J
cells are seen to have disjoint topological interiors. Conse
quently, the cells constitute a tessellation of IRd provided
that any point y € IRd has a nearest nucleus and the cells are
of dimension d. This is easily seen to be the case if t is
locally finite in the sense that the number of nuclei within
any bounded subset of IRd is finite (see Exercise 1.1.1).
Figure 1.1.1. Planar Voronoi tessellation generated by a bi
nomial process of 150 independent and uniformly distributed
points on the unit square.
Definition 1.1.1. Suppose is locally
finite. Then c(xilt) is called the Voronoi cell generated by
