Image Anal Stereol 2016;35:181-193 doi:10.5566/ias.1490
Original Research Paper
ASSESSINGDISSIMILARITYOFRANDOMSETSTHROUGHCONVEX
COMPACTAPPROXIMATIONS,SUPPORTFUNCTIONSAND
ENVELOPETESTS
VESNA GOTOVAC
1
, KATE
ˇ
RINA HELISOV
´
A
B
;2
AND IVO UGRINA
3
1
Department of Mathematics, Faculty of Science, University of Split, 21000 Split, Croatia;
2
Department of
Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, 162 27 Prague 6 –
Dejvice, Czech Republic;
3
Faculty of Pharmacy and Biochemistry, University of Zagreb, 10000 Zagreb, Croatia
e-mail: vgotovac@pmfst.hr, helisova@math.feld.cvut.cz, ivougrina@gmail.com
(Received February 24, 2016; revised November 16, 2016; accepted November 17, 2016)
ABSTRACT
In recent years random sets were recognized as a valuable tool in modelling different processes from fields like
biology, biomedicine or material sciences. Nevertheless, the full potential of applications has not still been
reached and one of the main problems in advancement is the usual inability to correctly differentiate between
underlying processes generating real world realisations. This paper presents a measure of dissimilarity of
stationary and isotropic random sets through a heuristic based on convex compact approximations, support
functions and envelope tests. The choice is justified through simulation studies of common random models
like Boolean and Quermass-interaction processes.
Keywords: approximations, dissimilarity, envelope tests, random sets, stochastic geometry, support functions
.
INTRODUCTION
The need for continuous improvement in fields
like biology, biomedicine or material sciences urges
us to develop and explore new techniques and
models in applications. Over few years, one of these
models received a lot of attention due to its strong
mathematical foundations and general nature.
As a model, random sets can describe and explain
many events. Some examples are dynamics of cells in
organisms (Mrkviˇ cka and Mattfeldt, 2011; Hermann
et al., 2015), particles in materials (Helisov´ a, 2014;
Neumann et al., 2016) or the presence of different
plants in ecosystems (Diggle, 1981; Møller and
Helisov´ a, 2010).
Although three dimensional applications are of
interest, applications of random sets are mainly done
through two-dimensional modelling. We are often
interested only in the projection of objects to the plane
(e.g., ground area of plants or trees) or we study
only cross-sections of a mass which create planar
formations and suppose that the behaviour of the
studied object is stationary in the third dimension (e.g.,
organic cells or material particles).
Random sets are represented by different models
ranging from simple and intuitive ones to those that
are highly complex and specialized. One of the basic
models is the Boolean model (Chiu et al., 2013).
Due to its simplicity many theoretical results can
be derived for it but, unfortunately, the model is
not sufficient in many applications. For example, in
Diggle (1981) researchers tried to model a data set
concerning heather plants and the model proved to
be unsatisfactory due to some interactions between
bushes.
An example of a sophisticated model is
Quermass-interaction process. This model introduces
dependencies on geometrical characteristics of sets
like area or boundary. It was first studied by Kendall
et al. (1999) and later extended by Møller and Helisov´ a
(2008).
As with classical statistics, being able to specify
a particular model (e.g., distribution in hypothesis
testing) is of great benefit. However, it is not
necessarily feasible. Mrkviˇ cka and Mattfeldt (2011)
explored tumour cells data with the Boolean model
with the aim to use estimated parameters for
detecting the differences between mammary cancer
and masthopatic tissue. This model proved to be
unsatisfactory since only 7% of images of mammary
cancer and 1% of images of mastopathy were
compatible with the model on the significance level of
0.05. Later in Hermann et al. (2015), the Quermass-
interaction process was fitted to the same data, but the
compatibility was again rejected in many cases (only
27.5% cases of mammary cancer and 6% cases of
mastopathic tissue were accepted with a significance
level of 0.05).
181

GOTOVAC V ET AL: Dissimilarity of random sets
There are some instances where knowledge about
the underlying model is not necessary, although
it could be beneficial. For example, in classical
hypothesis testing we could use rank based statistics
without specifying strict probability distributions.
Translated to the language of random sets, one is
interested only in distinguishing between two sets seen
as realisations of the same or different models. Exactly
this problem, restricted only to stationary and isotropic
random sets, will be the main topic explored in the rest
of the paper.
In its most general form the problem can be
described by the following diagram:
RealisationA RealisationB
Characteristics ofA Characteristics ofB
Dissimilarity measure
Given two realisations of stationary and isotropic
random sets one would like to find characteristics
useful when making a decision about strength of
dissimilarity of the underlying processes that have
generated those sets. Strength of dissimilarity will be
represented as a number a2 [0;1]; where the lower
value of a means lower belief that the realisations
come from the same underlying processes.
Notice that we have posed the problem starting
with realisations of random sets. Contrary to the
classical point measurements, where points can be
described purely by numbers (up to an error),
describing realisations of random sets is much harder.
Especially in practice where one usually obtains an
image in raster format (like JPG or PNG), i.e., a grid
of pixels (usually black and white) representing an
observed realisation of a random set. Also, realisations
of random sets are often of rugged shape and it is hard
to work with them since even the description of the
realisations is complicated.
Due to these reasons, it seems plausible to try to
approximate realisations by sets of more appealing
characteristics. One of the approaches, and the one
taken in this article, is to use convex compact sets due
to some useful properties describing them. Therefore,
we shall assess dissimilarity of the underlying process
generating realisations through assessing dissimilarity
of convex compact sets approximating our realisations.
The approach is graphically presented in Fig. 1 and
can be described in the following steps: starting with
realisations of random sets (images in practice) find a
suitable covering with simple (well described) sets like
balls (step 1), break the covering into convex compact
sets (step 2), derive some characteristics of this
approximation (step 3) and use these characteristics
to assess the dissimilarity of the underlying processes.
To the best of our knowledge, this approach had, until
now, not been taken into consideration in the known
literature.
It is really important to stress that we are
not interested in assessing the similarity of images
representing realisations of random sets like in the
literature on analysis of binary images (e.g., Hall,
1985; Ohser and M¨ ucklich, 2000, or Pratt, 2001) but
on assessing the dissimilarity of underlying processes.
The paper is organised as follows. In the Material
and Methods section we provide basic definitions,
methods and procedures. Roughly, we define support
functions, introduce a procedure for covering a set
by discs, define V oronoi tessellation on the union of
discs, introduce the theory behind envelope tests and
give a few remarks on free parameters. In the Results
section we present a simulation study and numerical
results based on the methodology from the Material
and Methods section. In the last section we give an
overview of the presented results with comments and
remarks.
MATERIAL AND METHODS
BASICS OF RANDOM SETS
In this section, we recall some terms related
to the basic theory of random sets. Definitions 2-5
concerning basic terms can be found in (Chiu et al.,
2013) while Definition 6 of Quermass-interaction
process in this form is taken from (Møller and
Helisov´ a, 2008). Readers familiar with this topic can
skip to the following section.
Definition 1. The set K R
d
is said to be convex if
cx+(1 c)y2 K for all x;y2 K and all 0 < c < 1.
The set K R
d
is said to be compact if it is closed and
bounded.
Definition 2. Let (W;F;P) be a probability space,
C the system of closed sets in R
d
and C = sfC
K
:
K is a compact subset ofR
d
g, whereC
K
=fC2C :
C\ K6= / 0g. Then a random closed set X in R
d
is a
measurable mapping from(W;F) to(C;C).
Definition 3. The distribution P
X
of a random
set X is given by the relation P
X
(F) =
P(fw2W : X(w)2 Fg) for F2C.
182

Image Anal Stereol 2016;35:181-193
P. HERMANN ETAL.
(a) Mastopathic tissue (b) Mammary cancer tissue
Figure17. Mastopathic(a)andmammarycancer(b)tissuepictureofwhichtheHausdorffmeasureisestimated.
Table VII. Hausdorffmeasureofmastopathictissuescalculatedbyusing(10).
No. #ofholes diam(U) !(U) Haus.dimension Haus.Measure
11 20 . 3 7 5 00 . 0 6 5 81 . 5 6 3 63 . 2 7 7 0
23 30 . 7 0 2 90 . 2 3 3 71 . 7 6 6 92 . 2 9 4 7
31 60 . 4 4 4 80 . 0 8 2 61 . 6 0 0 03 . 3 1 1 9
41 10 . 7 1 0 00 . 1 2 9 61 . 6 7 2 24 . 3 5 1 7
5–––– –
61 10 . 3 0 9 80 . 0 6 9 21 . 5 7 1 62 . 2 9 0 9
71 90 . 3 8 7 70 . 1 4 7 81 . 6 9 3 41 . 3 5 9 3
81 30 . 9 9 6 50 . 2 0 3 31 . 7 4 4 44 . 8 8 9 6
921 . 0 0 1 70 . 0 5 0 31 . 5 2 0 41 9 . 9 4 5 2
10 29 1.0063 0.2633 1.7860 3.8404
Table VIII. Hausdorffmeasureofmammarycancertissuescalculatedbyusing(10).
No. #ofholes diam(U) !(U) Haus.dimension Haus.Measure
11 0 30 . 4 2 5 70 . 3 6 0 91 . 8 3 6 50 . 5 7 7 4
29 40 . 3 6 6 40 . 3 3 5 61 . 8 2 4 90 . 4 7 6 9
36 30 . 4 3 0 80 . 2 6 7 21 . 7 8 8 30 . 8 3 0 1
46 20 . 5 1 7 60 . 2 1 0 91 . 7 5 0 41 . 4 9 7 1
57 80 . 2 8 3 20 . 2 6 6 11 . 7 8 7 60 . 3 9 4 0
66 10 . 3 4 0 20 . 2 6 5 51 . 7 8 7 30 . 5 4 8 4
75 90 . 2 5 9 00 . 2 7 5 91 . 7 9 3 50 . 3 2 1 4
86 50 . 1 9 7 70 . 2 5 6 11 . 7 8 1 50 . 2 1 7 6
93 00 . 2 8 0 60 . 1 7 9 51 . 7 2 4 50 . 6 2 2 8
10 58 0.4530 0.3821 1.8457 0.6068
and Table VIII). Tables VII and VIII provide quantitative information about mastopathic (Table VII)
and mammary cancer (Table VIII) tissue dissimilation within the monitored 2D section pattern. The
measurable entries are performed for the sake of rating the layout of the tissue arising (R) within the
parentone:numberofcoherentcircumscribed‘holes’inparenttissue!lledbythemastopathic/mammary
cancertissueequalsthenumberofelementsofR,diameter(diam(R))infactperformsthediameterofthe
greatest hole in the picture (herein, the diameter stands for the maximal distance of two arbitrary points
withinone‘hole’),density !(R)standsfortheratiobetweenmastopathic(ormammarycancer)tissueto
the entire image area, and HM of the pattern depicted on particular picture represents the ratio between
diam(R)and!(R).Theremarkablevalue19.9452inTableVIIforHMresultsfromthecombinationofa
smallamountoflittleareas(twoholes;see!rsttheplotofthethirdrowinFigure17(a)),alargeenough
diameter(morethan1)anddensitywhichhastinyvaluesbecauseofalongandthintissue.
Copyright©2015JohnWiley&Sons,Ltd. Statist. Med. 2015,342636–2661
2653
P. HERMANN ETAL.
(a) Mastopathic tissue (b) Mammary cancer tissue
Figure17. Mastopathic(a)andmammarycancer(b)tissuepictureofwhichtheHausdorffmeasureisestimated.
Table VII. Hausdorffmeasureofmastopathictissuescalculatedbyusing(10).
No. #ofholes diam(U) !(U) Haus.dimension Haus.Measure
11 20 . 3 7 5 00 . 0 6 5 81 . 5 6 3 63 . 2 7 7 0
23 30 . 7 0 2 90 . 2 3 3 71 . 7 6 6 92 . 2 9 4 7
31 60 . 4 4 4 80 . 0 8 2 61 . 6 0 0 03 . 3 1 1 9
41 10 . 7 1 0 00 . 1 2 9 61 . 6 7 2 24 . 3 5 1 7
5–––– –
61 10 . 3 0 9 80 . 0 6 9 21 . 5 7 1 62 . 2 9 0 9
71 90 . 3 8 7 70 . 1 4 7 81 . 6 9 3 41 . 3 5 9 3
81 30 . 9 9 6 50 . 2 0 3 31 . 7 4 4 44 . 8 8 9 6
921 . 0 0 1 70 . 0 5 0 31 . 5 2 0 41 9 . 9 4 5 2
10 29 1.0063 0.2633 1.7860 3.8404
Table VIII. Hausdorffmeasureofmammarycancertissuescalculatedbyusing(10).
No. #ofholes diam(U) !(U) Haus.dimension Haus.Measure
11 0 30 . 4 2 5 70 . 3 6 0 91 . 8 3 6 50 . 5 7 7 4
29 40 . 3 6 6 40 . 3 3 5 61 . 8 2 4 90 . 4 7 6 9
36 30 . 4 3 0 80 . 2 6 7 21 . 7 8 8 30 . 8 3 0 1
46 20 . 5 1 7 60 . 2 1 0 91 . 7 5 0 41 . 4 9 7 1
57 80 . 2 8 3 20 . 2 6 6 11 . 7 8 7 60 . 3 9 4 0
66 10 . 3 4 0 20 . 2 6 5 51 . 7 8 7 30 . 5 4 8 4
75 90 . 2 5 9 00 . 2 7 5 91 . 7 9 3 50 . 3 2 1 4
86 50 . 1 9 7 70 . 2 5 6 11 . 7 8 1 50 . 2 1 7 6
93 00 . 2 8 0 60 . 1 7 9 51 . 7 2 4 50 . 6 2 2 8
10 58 0.4530 0.3821 1.8457 0.6068
and Table VIII). Tables VII and VIII provide quantitative information about mastopathic (Table VII)
and mammary cancer (Table VIII) tissue dissimilation within the monitored 2D section pattern. The
measurable entries are performed for the sake of rating the layout of the tissue arising (R) within the
parentone:numberofcoherentcircumscribed‘holes’inparenttissue!lledbythemastopathic/mammary
cancertissueequalsthenumberofelementsofR,diameter(diam(R))infactperformsthediameterofthe
greatest hole in the picture (herein, the diameter stands for the maximal distance of two arbitrary points
withinone‘hole’),density !(R)standsfortheratiobetweenmastopathic(ormammarycancer)tissueto
the entire image area, and HM of the pattern depicted on particular picture represents the ratio between
diam(R)and!(R).Theremarkablevalue19.9452inTableVIIforHMresultsfromthecombinationofa
smallamountoflittleareas(twoholes;see!rsttheplotofthethirdrowinFigure17(a)),alargeenough
diameter(morethan1)anddensitywhichhastinyvaluesbecauseofalongandthintissue.
Copyright©2015JohnWiley&Sons,Ltd. Statist. Med. 2015,342636–2661
2653
0123456
3.0 3.5 4.0 4.5 5.0 5.5 6.0
0123456
3.0 3.5 4.0 4.5 5.0 5.5 6.0
Dissimilarity
measure
1
1
2
2
3
3
Fig. 1. Steps in assessing dissimilarity of the underlying processes generating realizations through dissimilarity
convex compact sets approximating realizations. The original images (left) were kindly provided by Mrkviˇ cka and
Mattfeldt (2011).
Definition 4. A random set X is stationary if its
distribution is invariant under translation, i.e., for all
v2R
d
, the distribution of X+ v=fu+ v;v2 Xg is
the same as that of X. A random set X is isotropic if
its distribution is invariant under rotation. If a random
set is both stationary and isotropic, it is called motion
invariant.
For A;B R
d
let us denote by A B :=fx+ y :
x2 A;y2 Bg and byjAj the d-dimensional Lebesgue
measure of the set A.
Definition 5. Let Y =fy
1
;y
2
;:::g be a stationary
Poisson point process in R
d
and fB
1
;B
2
;:::g be
a sequence of independent identically distributed
random compact sets in R
d
that are independent of
Y . IfEjB
1
 Kj <¥ for all compact sets K, then the
random set
B=[
¥
n=1
(y
n
+ B
n
) (1)
is called Boolean model.
Definition 6. Consider a planar random disc Boolean
model, i.e., the Boolean model with B
1
being a
disc inR
2
with random radius. Quermass-interaction
process is a random set whose probability measure is
absolutely continuous with respect to the probability
measure of the given Boolean model and the density of
its probability measure with respect to the probability
measure of the given Boolean model is of the form
f
q
(b)=
1
c
q
expfq
1
A(U
b
)+q
2
L(U
b
)+q
3
c(U
b
)g
for each finite disc configuration b =fb
1
:::;b
n
g,
where A = A(U
b
) is the area, L = L(U
b
) is the
perimeter,c =c(U
b
) is Euler-Poincar´ e characteristic
(i.e., the number of connected components minus the
number of holes) of the union U
b
=[
n
i=1
b
i
, q =
(q
1
;q
2
;q
3
) is 3-dimensional vector of parameters and
c
q
is the normalising constant.
SUPPORT FUNCTION OF A CONVEX
COMPACT SET
In the following text, convex compact sets have a
central role in our approach and we are restricted only
to the planar case, i.e., the convex compact sets inR
2
.
One of the most important basic concepts related to
convex compact sets is the support function.
Definition 7. For a (random) convex compact set X
define its support function as
h
X
(u)= sup
x2X
hu;xi; u2¶b(0;1);
where ¶b(0;1) is the unit sphere inR
2
, i.e., the circle
with radius 1 and the centre in the origin.
Fig. 2 illustrates the support function of two sets, a
disc and a square centred in the origin.
The importance of the support function in our
approach is derived from the following result on
the equality of distributions of two random convex
compact sets via their support functions (for the proof
please take a look at Lavie (2000)).
Theorem 8. For two random convex compact sets X
1
and X
2
it holds
X
1
=
(D)
X
2
,(h
X
1
(u
i
))
i2I
=
(D)
(h
X
2
(u
i
))
i2I
for all finite index sets I and (u
n
)
n2N
dense subset of
¶b(0;1):
183

GOTOVAC V ET AL: Dissimilarity of random sets
 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 0123456
0.6 0.8 1.0 1.2 1.4
angles
Support function
 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 0.5 0.0 0.5 1.0 1.5 0123456
1.0 1.1 1.2 1.3 1.4
angles
Support function
Fig. 2. Examples of the support function for a disc and
a square, respectively.
COVERING A SET BY DISCS WITH
IDENTICAL RADII
As mentioned in the Introduction, our aim is to
divide a planar data set S (e.g., Fig. 3a) into a family of
convex compact sets.
In practice we usually start with a digital
binary image I containing the observed set, i.e., a
picture consisting of black and white pixels, as an
approximation of the initial set and its complement.
This approximation can be achieved by pixelisation
through a simple structure, where pixels are squares
with side length D and the binary image I containing
an arbitrary planar set S is a matrix M=[m
i; j
] of black
and white pixels (or the matrix of 1’s and 0’s) such that
the pixel m
i; j
is black if and only if its centre lies in the
set S, i.e., if and only if (i D D=2; j D D=2)2 S,
and it is white otherwise (Fig. 3b). Let us denote by D
the approximation of S in the digital image I.
Having a digital approximation D of our initial set
S, it can be useful to approximate the shape of the
given set S by a geometrical object A with suitable
mathematical properties. One of the approaches, and
the one taken in this article, would be to cover the
pixelised version with discs of identical radii. This
can be achieved by utilising a customized version
of maximal Poisson-disc sampling algorithm (Ebeida
et al., 2011).
In the maximal Poisson-disc sampling algorithm,
set D is covered by discs b(x
i
;r) with centres x
i
and radii r so that the discs centres come from the
Poisson-disc sampling process, i.e., the point process
X =fx
i
; i = 1;:::;ng is constructed point by point
satisfying the following conditions:
1.8x
i
;x
j
2 X;x
i
6= x
j
:jjx
i
 x
j
jj r,
2.If D
i 1
= Dn[
j=1;:::;i 1
b(x
j
;r), then 8x
i
2
X;8B D
i 1
:
P(x
i
2 B)=
jBj
jD
i 1
j
;
3.8x2 D9x
i
2 X :jjx x
i
jj< r.
It is obvious that in this approach the area of
approximating set is larger than the area of the original
set D. Depending on the choice of radius r this can lead
to many white pixels of the digital image I becoming
black, therefore leading to an increase in geometrical
properties like area or boundary length. To account for
this we introduce a customized version of the maximal
Poisson-disc sampling (called CMPD in future) such
that we first reduce the set D by a radius r, i.e., we
construct
D
 r
=fu : b(u;r) Dg;
(Fig. 3c). Now, we cover the newly obtained set D
 r
by the maximal Poisson-disc sampling obtaining a set
ˆ
D
 r
. In order to keep the original shape as precise as
possible, we start covering on the border of D
 r
, i.e.,
the centres are first sampled from the boundary pixels
(Fig. 3d). After covering all the boundary pixels we
choose the centres from the remaining black pixels of
D
 r
(Fig. 3e).
However, since set
ˆ
D
 r
will be a subset of the
starting set D we can try to cover what has not yet been
covered, i.e.,
˜
D= Dn
ˆ
D
 r
, using the conditions 1 and 2
from the maximal Poisson-disc sampling and adjusted
condition 3:
1.8x
i
;x
j
2 X;x
i
6= x
j
:jjx
i
 x
j
jj r,
184

Image Anal Stereol 2016;35:181-193
(a) Basic planar set (b) Digital approximation (c) Reduced set D
 r
(d) Covering on boundary pixels (e) Covering of inner pixels (f) Additional covering based on
PD(r)<d
Fig. 3. Stepwise covering of a planar set by discs of identical radii
2.If D
i 1
=
˜
Dn[
j=1;:::;i 1
b(x
j
;r), then 8x
i
2
X;8B D
i 1
:
P(x
i
2 B)=
jBj
jD
i 1
j
;
3.PD(r)<d,
where PD(r) is a measure of pixel difference defined
below by (2) and d is some threshold. In words, this
can be described as following the same principal of
covering as with the maximal Poisson-disc sampling
except that we do not want to completely cover the
original set but only up to a predefined similarity
(measured by PD). Therefore, we add more discs as
long as PD(r)<d is valid (Fig. 3f).
It should be noted that depending on the choice
of d covering with a predefined radius r might not be
possible at all. Also, adding a disc to Dn
ˆ
D
 r
might fail
the PD(r) < d condition for the first random centre
(disc) while for some other centres it could work.
In the rest of this paper, we have decided to stop
with the first disc failing, therefore not trying with
other/additional (random) centres. This choice was
based on our computational restrictions (feasibility).
Simulation study presented in Results section
justifies (in the sense of pixel difference (Eq. 2), see
below) the choice of covering starting on the boundary.
The choice of the pixel difference measure is based
on the idea of precision from the field of binary
classification and is given by the following formula
PD(r)=
BW(r)+WB(r)
B
(orig)
; (2)
where BW denotes the number of pixels which were
black in the digital image I and are white after the
covering, WB denotes the number of pixels which were
white in the digital image I and are black after the
covering, and B
(orig)
denotes the number of all black
pixels in the original digital image I, i.e., the area of D
(expressed in pixels).
It could be tempting to conclude that instead of
the CMPD we could use the maximal Poisson-disc
sampling and just remove pixels that are outside of the
original set D. Although this approach would be much
simpler it would suffer from a major drawback since
some covering sets would not be convex any more and
convexity is one of our important assumptions to use
support functions.
185

GOTOVAC V ET AL: Dissimilarity of random sets
One problem that still remains open is the
choice of the optimal radius for covering. The
optimal radius would be the one where discs overlap
suitably in the sense that they are not too dense
(due to consequent construction of the corresponding
V oronoi tessellation), the union covers the whole
set satisfactorily (in the sense of condition 3 for
CMPD), the radius should not be too small because
it would not provide satisfactory information about
inner structure of the set D (support functions) and
the radius should not be too large since we would
like to obtain sufficient number of convex compact
objects (cells of the corresponding V oronoi tessellation
in our case). This is needed for testing whether the
cells of V oronoi tessellations corresponding to two
different realisations of random sets come from the
same distribution. More on the choice of radius and
d is given later in the text.
VORONOI TESSELLATION ON A DISC
UNION
The final step for obtaining a family of convex
compact sets from the data set D is to construct a
tessellation over the union of discs A approximating
the set D. We took the inspiration from the study of
power tessellation of a union of discs (for construction,
properties etc., see Møller and Helisov´ a, 2008) which
is the intersection of the union of discs and Laquerre
tessellation (e.g., Imai et al., 1985) built over discs
centres, where the weights are given by the radii of
the discs. Here we construct its simpler version for
discs having the same radii, i.e., for the case when the
Laquerre tessellation corresponds to its special case,
V oronoi tessellation (e.g., Chiu et al., 2013). Thus
the definition of the tessellation we will use is the
following:
Definition 9. Consider a finite configuration of discs
fb
1
:::;b
n
g with centres fc
1
:::;c
n
g and identical
radii, and denote by
B
i
=fy2 b
i
:ky c
i
kk y c
j
k for all j6= ig:
The system B of all sets B
i
is called the V oronoi
tessellation on a union of discs.
It is easy to see that all cells of a V oronoi
tessellation are convex. Moreover, since we intersect
them with discs they are compact. Thus, applying
this procedure to the approximation given by discs,
we obtain a family of convex compact sets from
the data set D. This approximation by a union of
convex compact sets can now serve as a basis for a
dissimilarity measure.
ENVELOPE TESTS
In this section, we introduce a tool (test) used for
testing equality of distributions of random geometrical
objects recently developed by Myllym¨ aki et al. (2016)
and Mrkviˇ cka et al. (2015). Also, we describe the
customization of the test to fit our needs.
Consider a group of s+ 1 geometrical objects,
where k-th object, k = 1;:::;s+ 1, is described by
characteristics T
k
(j) forj2 I, where I is a finite index
set, so T
k
(j) may be considered as a vector of finite
dimension. We suppose that T
k
(j), k = 2;:::;s+ 1;
come from the same distribution, and we would like
to test the hypothesis that T
1
(j) also comes from the
same distribution as T
k
(j), k= 2;:::;s+ 1.
Suppose that T
k
(j), k= 1;:::;s+ 1, are evaluated
on some a priori chosen mesh of values j
j
, j =
1;:::;n. Let r
1; j
;:::;r
s+1; j
denote raw ranks of
T
1
(j
j
);:::;T
s+1
(j
j
) such that the smallest T
k
(j
j
) for
a fixed j has rank 1, the second smallest T
k
(j
j
) has
rank 2, etc., and the largest T
k
(j
j
) has rank s+ 1. In
case of ties, the raw ranks are averaged within ties. The
point-wise ranks are calculated as
R
k; j
= min
 r
k; j
;s+ 1 r
k; j
 :
The extreme rank measure R
k
of the vector T
k
is the
minimum of the point-wise ranks R
k; j
; i.e.,
R
k
= min
j2f1;:::;ng
R
k; j
:
Now, for a given significance level a, we can
define a test as follows. First, we determine an
appropriate low rank R
(a)
which is the smallest value
infR
1
;:::;R
s+1
g for which
s+1
 k=1
1(R
k
 R
(a)
) a(s+ 1):
Secondly, for
I
a
=
 k2f1;:::;s+ 1g : R
k
 R
(a)
 we define values
T
(a)
low
(j
j
)= min
k2I
a
T
k
(j
j
); T
(a)
upp
(j
j
)= max
k2I
a
T
k
(j
j
):
The following results from Myllym¨ aki et al. (2016)
now lead to the definition of the rank envelope test and
present its statistical justification:
P
 T
1
(j
j
) = 2
h
T
a
low
(j
j
);T
(a)
upp
(j
j
)
i
for any jjH
0
  a
and
P
 T
1
(j
j
) = 2
 T
a
low
(j
j
);T
(a)
upp
(j
j
)
 for any jjH
0
 >a;
186

Image Anal Stereol 2016;35:181-193
where H
0
is a simple null hypothesis.
Therefore, if the observed vector T
1
leaves
envelope at some point, i.e., R
1
< R
(a)
the null
hypothesis is rejected at significance level a. If the
observed vector lies completely inside this envelope,
i.e., R
1
> R
(a)
, the null hypothesis is not rejected at
significance level a. If the observed vector coincides
in some point with the border of the envelope, i.e.,
R
1
= R
(a)
the rejection of the null hypothesis remains
undecided.
Testing can also be conducted through a
customized version of p-values which are defined by
assigning an extreme rank measure R
k
to each of the
vectors T
k
, such that the lowest rank corresponds to the
most extreme values of statistic:
p
 =
1
s+ 1
s+1
 k=1
1(R
k
< R
1
);
p
+
=
1
s+ 1
s+1
 k=1
1(R
k
 R
1
):
From the definition of p-values it is easy to see that
R
1
< R
(a)
is equivalent to p
+
 a (null hypothesis
rejected), R
1
> R
(a)
is equivalent to p
 > a (null
hypothesis not rejected) and R
1
= R
(a)
is equivalent
to p
  a < p
+
(the rejection of the null hypothesis
remains undecided).
The width of the interval between p
+
and p
 is
p
+
 p
 =
1
s+ 1
s+1
 k=1
1(R
k
= R
1
) 2n
s+ 1
:
and in the case of lots of ties it can be large.
This problem can sometimes be solved by additional
ordering as described by Myllym¨ aki et al. (2016).
It consists in replacing ranks R
k
by vector ranks R
k
which allow finer ordering among ranks so that in
some cases it eliminates ties. Consider the vectors
of point-wise ordered ranks R
k
= (R
k;(1)
;:::;R
k;(n)
),
wherefR
k;(1)
;:::;R
k;(n)
g =fR
k
(j
1
);:::;R
k
(j
n
)g and
R
k;( j
1
)
 R
k;( j
2
)
whenever j
1
 j
2
. Then
R
k
1
 R
k
2
,9n
0
 n :
R
k
1
;( j)
= R
k
2
;( j)
8 j< n
0
& R
k
1
;(n
0
)
< R
k
2
;(n
0
)
: (3)
Using R
k
instead of R
k
, k= 1;:::;s+1; for calculation
of the width p
+
 p
 of the p-value interval, it may
theoretically happen that the value is the same as for
R
k
, but in the most cases it is significantly narrower.
Now lets describe the application of this test to our
data.
In our case, we compare two realisations of
random sets in the way that first, each of the sets
is covered by discs with identical radii, then the
corresponding V oronoi tessellation is constructed, and
finally, m non-neighbouring cells from each of the
V oronoi tessellations are sampled. The sampling is
carried out so that the first cell B
1
is chosen randomly
uniformly from all the cells in the corresponding
tessellation and having k sampled cells, the (k+ 1)-th
cell B
k+1
is chosen randomly uniformly from the set
of cellsfB
j
: B
j
\ B
k+1
= / 0g for k = 1;:::;m 1. It
means that our data consist of 2 m characteristics of
convex compact sets, more precisely of 2 m support
functions related to the centres of the corresponding
covering discs and evaluated in some (equidistant)
partition of(0;2p]. Thus we do not explicitly have only
one characteristic T
1
(j) to be compared to T
k
(j), k=
2;:::;s+ 1. Therefore we decided to use a permutation
version of the above-mentioned test to attain functional
ANOV A procedure (Mrkviˇ cka et al., 2015). It works as
follows.
We form matrices H
A
and H
B
by putting in
their rows discretized support functions from the first
set and the second set, respectively. We denote by
H
A
(j
j
) and H
B
(j
j
), respectively, the j th column
of matrices H
A
and H
B
while the column represents
values of support functions of sample cells evaluated
at angles j
j
= j
2p
n
, j = 1;2;:::;n: Further, denote
by H
A
(j
j
), H
B
(j
j
), Var(H
B
(j
j
)) and Var(H
A
(j
j
))
the mean values and variances, respectively, of each
column in each matrix.
Assume that there exist non random functions
m
A
(j) andm
B
(j) such that
H
A;i
(j)=m
A
(j)+ e
A;i
(j);
H
B;i
(j)=m
B
(j)+ e
B;i
(j)
for i = 1;:::;m, where e
A;i
(j) and e
B;i
(j) are i.i.d.
samples from distribution G(j) for every j; where
G(j) has zero mean and finite variance. We want to
test the hypothesis
H
0
:m
A
(j) m
B
(j) 0:
The characteristic T
1
(j
j
) is the mean difference of
support functions H
A
(j
j
) and H
B
(j
j
) normalised to
unit variance, i.e.,
T
1
(j
j
)=
H
A
(j
j
) H
B
(j
j
)
p
Var(H
A
(j
j
))+ Var(H
B
(j
j
))
for j = 1;2;:::;n. Characteristics T
k
(j
j
) for k =
2;:::;s+ 1 are given by
T
k
(j
j
)=
H
0
1
(j
j
) H
0
2
(j
j
)
p
Var(H
0
1
(j
j
))+ Var(H
0
2
(j
j
))
;
187

GOTOVAC V ET AL: Dissimilarity of random sets
where H
0
1
; and H
0
2
are obtained by randomly permuting
rows of matrix
 H
A
H
B
 and then splitting back into two
matrices with equal number of rows.
An example of graphical output of the test is
shown in Fig. 4. The introduced plots come from the
simulation study described in details below in Results
section.
0 1 2 3 4 5 6
-0.4 -0.2 0.0 0.2 0.4
angles
Envelopes
0 1 2 3 4 5 6
-0.6 -0.4 -0.2 0.0 0.2 0.4
angles
Envelopes
Fig. 4. Graphical representation of envelopes for
number of permutations s= 4999 when comparing two
Boolean models, where [p
 ; p
+
) = [0:1726;0:1774)
(left), and Cluster model with Repulsive model,
where [p
 ; p
+
)=[0:0054;0:0102) (right), for details
concerning the mentioned models Simulated data
section. Dot-dashed black line represents envelope for
a = 0:05:
ALIGNMENTOFSUPPORTFUNCTIONS
Since we assume that the distribution of original
random set is isotropic, the same should be true for
the distribution of a randomly chosen cell in the
V oronoi tessellation of its coverage. However, the
mean value of support functions of two cells having
identical shape but being rotated by different angles
could significantly differ from their original support
function. In order to avoid this “loss of information”,
we cluster and align support functions of cells that
are “almost equal” if transformed under rotation.
Alignment is achieved through an adjusted version
of agglomerative hierarchical clustering (James et al.,
2013) with average linkage and correlation as the
dissimilarity measure.
In more details, let us denote by H = [h
i; j
] =
[h
i
(j
j
)] a matrix, where h
i
(j
j
) stands for the value
of support function for the i-th sampled cell of the
approximating tessellation at the j-th angle on a mesh
of angles (e.g., j
j
= j
2p
n
; j = 1;:::;n). Also, let us
denote with h
i;1:
the i-th row of the matrix H and
with h
i;l:
=(h
i;l
;h
i;l+1
;:::;h
i;n
;h
i;1
;:::;h
i;l 1
) a shifted
version of h
i;1:
by l places. Now, for rows i
1
and i
2
we
can define the value
f cor(i
1
;i
2
)= max
 cor
 h
i
1
;1:
;h
i
2
;l:
 : l= 1;2;:::;n:
 :
Using this newly defined measure of similarity
between rows of H we can build a dendrogram (or
clustering tree) for rows thus obtaining the clustering
between rows based on the similarity between support
functions regardless of rotation. As with almost all
applications of hierarchical clustering a threshold C
for the correlation should be defined by a researcher
to cut the dendrogram into clusters. Alignment is now
performed within the obtained clusters by shifting
appropriate rows to achieve maximal f cor.
Since the main problem in this paper is comparing
two random sets, matrix H is defined as H =
 H
A
H
B
 where H
A
and H
B
are matrices representing
support functions for the first and the second set,
respectively. Therefore, alignment is done together for
both datasets. This approach removes a problem with
the choice of the starting row within a cluster to align
with since all similar tessellations will be aligned in
the same way regardless of affiliation to the first or the
second random set.
The effects of such alignment when comparing
realisations from different underlying processes
conducted in our later simulation study can be seen
in Fig. 5.
CHOOSING “FREE” PARAMETERS
Following the aforementioned remarks on radius
we define a criterion for choosing the suitable radius
188

Image Anal Stereol 2016;35:181-193
0 1 2 3 4 5 6
2.5 3.0 3.5 4.0 4.5 5.0
angles
Supportfunctions
0 1 2 3 4 5 6
2.5 3.0 3.5 4.0 4.5 5.0
angles
Supportfunctions
0 1 2 3 4 5 6
-0.4 -0.2 0.0 0.2 0.4
angles
Testfunction
0 1 2 3 4 5 6
2.5 3.0 3.5 4.0 4.5 5.0
angles
Supportfunctions
0 1 2 3 4 5 6
2.5 3.0 3.5 4.0 4.5 5.0
angles
Supportfunctions
0 1 2 3 4 5 6
-0.6 -0.4 -0.2 0.0 0.2 0.4
angles
Testfunction
Fig. 5. Support functions of 100 independent and randomly chosen cells from Boolean model (left) and Repulsive
model (middle) with black lines representing mean values of these functions, and graphical representation of
envelope test for these two models without alignment (upper line) and with alignment for C=0.9 (lower line).
(expressed again in pixels) as
R= max
i=1;2;:::
fi : PD(i) dg; (4)
whered is some predefined pixel difference level.
Note that the values PD(i) depend on random
covering, so it is a random variable and therefore the
value R is random, too. In the section on simulation
studies, we have a lot of input realisations of the same
random set, thus we use the mean value of PD(i) of
all these realisations to find the suitable radius R. In
real applications where usually only one realisation
is available, we can take R calculated from only one
covering of that realisation since our simulation study
suggests that the values PD(i) for a given realisation
do not change significantly.
Since we are comparing two data sets D
A
and D
B
it may happen that the corresponding optimal radii R
A
and R
B
are different. In that case we define R as
R= minfR
A
;R
B
g: (5)
RESULTS
SIMULATED DATA
We applied the methodology described in Material
and Methods section to a few preselected pairings
between three different simulated realisations of
random sets. One random set was a Boolean model
(see Definition 5) and the other ones were random
disc Quermass-interaction processes (see Definition 6)
with parameters chosen so that the processes
formed clusters or mutually repulsive components,
respectively.
More precisely, the Boolean model used for the
simulation study had centres of discs in the window
25 25, intensity of the disc centres equal to 0.4
and uniform distribution of radii on the interval
(0:5;1) (see the left picture in Fig. 6). The second
simulated data set was given by realisations of
Quermass-interaction process with parameters q
1
=
0:65, q
2
= 1:1 and q
3
= 0 with respect to the
mentioned Boolean model. Since this process produces
realisations with larger area and smaller perimeter
compared to the reference process, it tends to create
clusters (see the middle picture in Fig. 6). Therefore,
we will refer to it as the cluster process in the
rest of the text. On the other hand, the third data
set simulated as Quermass-interaction process with
parameters q
1
= 1, q
2
= 1 and q
3
= 0 prefers
smaller area and larger perimeter. Realisations are
usually small non-overlapping components (see the
right picture in Fig. 6) and therefore the process will
be referenced as the repulsive process in the rest of the
text.
189

GOTOVAC V ET AL: Dissimilarity of random sets
We simulated 400 realisations for each of
the mentioned processes. All realisations were
transformed to matrices of 400 400 black and white
pixels (an example of such a matrix for the Boolean
model is shown in Fig. 3b) and these matrices played
the role of the input data.
To explore the sensitivity of the methodology
within and between classes of processes the following
approach was taken. First, the input matrices were
divided into two groups of 200 realisations for
Boolean model, cluster process and repulsive process,
respectively, and dissimilarity was studied on these
groups separately for different processes. Additionally,
we considered two groups of 200 realisations of
different processes, namely Boolean vs cluster,
Boolean vs repulsive and cluster vs repulsive process,
and studied the dissimilarity again.
CHOOSING THE OPTIMAL RADIUS
In order to choose optimal radii, i.e., to determine a
suitable pixel difference leveld for approximations of
data sets by unions of discs, we studied the behaviour
of the approximations for different values of radii and
their corresponding pixel difference levels. Data sets
introduced in the Results section were covered by discs
with identical radii using the method described in the
Material and Methods section and taking the values
of radii r = 3;:::;15 while the corresponding pixel
differences PD(r) were calculated. Consequently the
V oronoi tessellations on unions of covering discs were
constructed as described in the Material and Methods
section. Then we studied the quality of coverings for
chosen pixel differences levels d = 10%;20%;30%
by visual comparison of the shape of original and
approximating sets. Moreover, we explored the shape
of cells in the corresponding V oronoi tessellations as
well as their quantities.
For a realisation of the Boolean model, three
different approximations using radii r = 4;7;9
corresponding to pixel differences levels d =
10%;20%;30%; respectively, are presented in Fig. 7.
By visual comparison of the similarity of the original
set and V oronoi tessellations of its approximations, we
have inferred that for a pixel difference level of 10%
cells of the tessellation are too small thus do not give
us enough information on the inner structure of the
original set while for a pixel difference level of 30%
the approximation of the original data set is not good
enough in description of its shape. However, for a pixel
difference level of 20% covering discs approximate
original set shape more appropriately and thereby the
corresponding tessellation provides good information
about inner structure of the original set, too. Thus, we
have chosen the pixel difference leveld = 20%.
It is easy to notice that the optimal radius gives
us information about the structure of the underlying
random set and it could be tempting to conclude
that the difference between optimal radii in models
could be used as the criterion for the final decision
on dissimilarity. However, one should be careful in
using it as the criterion for the final decision since
different realisations of the same process may have
different optimal radii and, moreover, since covering is
random, even one realisation may result in significant
differences of PD for the same radii, see Fig. 8, leading
to different optimal radii.
Fig. 8. The original set in resolution 43 43, its
reduced form and two ways of covering by radius 10,
where in the first case, we have 268 white pixels, i.e.,
PD(10) = 268=43
2
= 0:14, and in the second case,
we have 606 white pixels so we obtain PD(10) =
606=43
2
= 0:33.
NUMERICAL RESULTS OF ENVELOPE
TEST
As mentioned in the previous subsection, we
considered 400 realisations of each of the Boolean
model, cluster process and repulsive process and
studied the dissimilarity between realisations of the
same processes as well as between realisations of
different processes. Therefore, we had six different
combinations of pairs in the study.
First, we had to choose the optimal radius for each
of studied process. Using pixel a difference level of
20% as described in Material and Methods section
we obtained from (4) the optimal radii R = 7 for
the Boolean model, R = 13 for the cluster process
and R= 5 for the repulsive process. Recall that these
values were derived so that in (4) we considered the
mean pixel difference PD(R) calculated from all 400
190

Image Anal Stereol 2016;35:181-193
Fig. 6. Examples of realisations of Boolean model with intensity of the disc centres 0.4 in the window 25 25
and 0 otherwise, and distribution of radii U(0:5;1) (left picture), Quermass-interaction process with parameters
q
1
= 0:65,q
2
= 1:1 andq
3
= 0 (middle picture) and Quermass-interaction process with parametersq
1
= 1,
q
2
= 1 andq
3
= 0 (right picture).
Fig. 7. Approximations by covering using discs of r= 4 (left figure, corresponding to d = 10%), r= 7 (middle
figure, corresponding tod = 20%) and r= 9 (right figure, corresponding tod = 30%) for the planar set and its
digital approximation from Fig. 3.
realisations of the given process while we obtained the
following characteristics for PD (in %):
radius PD
mean
PD
sd
number of
realisations
having
PD < 20%
when
covering
with the
corresponding
radius
Boolean 7 19.81 1.03 252 (63%)
Cluster 13 18.55 2.07 323 (81%)
Repulsive 5 19.78 0.64 284 (71%)
Due to different radii between processes Eq. 5
was applied leading to the following radii for different
combinations:
Boolean Cluster Repulsive
Boolean 7 7 5
Cluster 13 5
Repulsive 5
Then we applied the envelope test introduced
in the Material and Methods section to two groups
of 200 realisations for each combination obtaining
appropriate p-values. More precisely, for each of the
six combinations and appropriate groups, realisations
were covered by discs with radii derived with
Eq. 5. These coverings were translated to V oronoi
tessellations. Consequently, we randomly sampled
100 independent (i.e., non-neighbouring) cells of the
tessellation and calculated their support functions.
Finally, the envelope test as described in the Material
and Methods section was applied to pairs of these
support functions, both with and without previous
alignment performed for C= 0:9:
Histograms of corresponding p-values are shown
in Fig. 9. We can observe that comparing groups of
realisations of the same processes, with or without
191

GOTOVAC V ET AL: Dissimilarity of random sets
alignment, p-values are approximately uniformly
distributed which means that the test considers the
cells to be identically distributed, in other words
that the inner structure is the same so the sets are
similar enough. On the other hand when assessing
dissimilarity of different processes, p-values are
mostly very close to zero, so the hypothesis of identical
distribution of the cells is significantly often rejected,
so we can conclude that the similarity measure is very
low, i.e., the original sets are not similar. The effect of
alignment can be best seen when observing histogram
of p-values for testing Boolean vs repulsive models,
where p-values of the test have significantly lower
values after performing alignment (without alignment
48:5% of the p-values were less than 0.05 while 64:5%
of p-values were less than 0.05 when using alignment).
Since the p-values of test are already very close to
zero when comparing Boolean vs cluster and cluster
vs repulsive models (all less then 0.05), the effect of
alignment is not so noticeable.
DISCUSSION
Assessing dissimilarity of random sets could
have big implications in applied science but is,
unfortunately, hard to achieve. Especially if theoretical
results are of interest due to high complexity of some
random sets.
Nevertheless, some advancement can be achieved
if heuristic results are considered. In this paper
we have presented one of such results. We have
presented a method based on covering of random sets
with unions of convex compact sets and consequent
application of envelope test on support functions of
these approximating sets. To justify our choice we
have conducted a simulation study on different types
of some common models of random sets and obtained
valuable results.
The presented method, however, suffers from some
problems. The main weakness, in our humble opinion,
is the covering of the input data set by a union of
discs, especially the choice of suitable radius and pixel
difference level, i.e., the measure of inaccuracy of the
approximation. Here we had to make several decisions
based on visual comparisons and results obtained by
additional simulation studies. Despite the fact that
the obtained numerical results are satisfactory, this
part has the potential to be improved. One of the
more straightforward approaches to try to improve the
current choice of free parameters is in the case were
multiple realizations for processes are available. Here,
a researcher could use the machine learning approach
with training and test sets to derive (optimized) best
parameters.
Additionally, the workflow presented in this paper
can be represented through modules: cover a set,
from the covering of the original set derive convex
compact sets representing an approximation, derive
some characteristics of the approximation and use
these characteristics for the dissimilarity measure.
Therefore, the workflow is suitable to changes only
within modules. If a better covering algorithm can be
achieved or a different test devised they can easily be
used instead of the original modules.
One of the common problems in practice, that
was not investigated as a special case in this paper, is
the existence of edge-effects, i.e., the case when the
observation window contains only parts of the original
set and the other parts lying outside the window are
not observed (an example is presented in Fig. 3a). In
our simulation study we tried to avoid this problem by
ignoring the cells lying on the edge of the observation
window, sampling only those that were fully contained
in the window. For random sets with significant parts
observed on the edges appropriate adjustments should
be devised. We have left this for future work.
Also, in future work additional ways for
testing equality of distributions of convex compact
approximations might be applied and further
simulation studies of the current model would be
beneficial to better understand properties of the
introduced method. This will, hopefully, lead us
to suggest suitable modifications or convenient
alternatives to the current method.
ACKNOWLEDGEMENTS
This work was supported by The Czech Science
Foundation (GA
ˇ
CR), project No.13-05466P.
REFERENCES
Chiu SN, Stoyan D, Kendall WS, Mecke J (2013).
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Diggle PJ (1981). Binary mosaics and the spatial pattern of
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Ebeida MS, Davidson AA, Patney A, Knupp PM, Mitchell
SA, Owens JD (2011). Efficient maximal Poisson-disk
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Hall P (1985). Counting methods for inference in binary
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Helisov´ a K (2014). Modeling, statistical analyses and
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192

Image Anal Stereol 2016;35:181-193
BooleanvsCluster
p-values
Frequency
0.00 0.01 0.02 0.03 0.04 0.05
0 50 100 150 200
withalignment
withoutalignment
BooleanvsRepulsive
p-values
Frequency
0.0 0.2 0.4 0.6 0.8 1.0
0 20 40 60 80 100 120
withalignment
withoutalignment
ClustervsRepulsive
p-values
Frequency
0.00 0.01 0.02 0.03 0.04 0.05
0 50 100 150
withalignment
withoutalignment
BooleanvsBoolean
p-values
Frequency
0.0 0.2 0.4 0.6 0.8 1.0
0 5 10 15 20 25
withalignment
withoutalignment
ClustervsCluster
p-values
Frequency
0.0 0.2 0.4 0.6 0.8 1.0
0 5 10 15 20 25
withalignment
withoutalignment
RepulsivevsRepulsive
p-values
Frequency
0.0 0.2 0.4 0.6 0.8 1.0
0 5 10 15 20 25
withalignment
withoutalignment
Fig. 9. Histograms of p-values for testing all pairs of simulated processes, i.e., Boolean-cluster (upper left),
Boolean-repulsive (upper middle), cluster-repulsive (upper right), Boolean-Boolean (lower left), cluster-cluster
(lower middle) and repulsive-repulsive (lower right); 200 realisations of each process were used.
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