Image Anal Stereol 2009;28:129-141
Original Research Paper
3D MORPHOLOGICAL MODELLING OF A RANDOM FIBROUS
NETWORK
CHARLES PEYREGA1,DOMINIQUE JEULIN1,CHRISTINE DELIS ´EE2AND J´ERˆOME
MALVESTIO2
1Centre de Morphologie Math ´ematique, Math ´ematiques et Syst `emes, Mines ParisTech, 35 rue Saint Honor ´e,
77300 Fontainebleau, France,2Unit´e des Sciences du Bois et des Biopolym `eres, 351, cours de la Lib ´eration -
33405 Talence CEDEX, France
e-mail: charles.peyrega@mines-paristech.fr, dominique.jeulin@ensmp.fr, delisee@us2b.pierroton.inra.fr,
malvestio@us2b.pierroton.inra.fr
(Accepted September 14, 2009)
ABSTRACT
In the framework of the Silent Wall ANR project, the CMM and the US2B are associated in order to
characterize and to model fibrous media studying 3D images acquired with an X-Ray tomograph used by
the US2B. The device can make 3D images of maximal 23043voxels with resolutions in the range of
2µm to 15 µm. Using mathematical morphology, measurements on the 3D X-Ray CT images are used
to characterize materials. For example measuring the covariance on these images of an acoustic insulating
material made of wooden fibres highlights the isotropy of the fibres orientations in the longitudinal planes
which are perpendicular to the compression Oz axis. Moreover, it is possible to extract other morphological
properties from these image processing methods such as the size distribution either of the fibres or of the
pores by estimating the morphological opening granulometry of the considered medium. Using the theory
of random sets introduced by Georges Matheron in the early 1970’s, the aim of this work is to model such
a fibrous material by parametric random media in 3D according to the prior knowledge of its morphological
properties (covariance, porosity, size distributions, etc.). A Boolean model of random cylinders in 3D stacked
in planes parallel to each other and perpendicular to the Oz compression axis is first considered. The
granulometry results provide gamma distributions for the radii of the fibres. In addition, a uniform distribution
of the orientations is chosen, according to the experimental isotropy measurements in the longitudinal planes.
Finally the third statistical factor is the length distribution of the fibres which can be fitted by an exponential
distribution. Thus it is possible to estimate the validity of this model first by trying to fit the experimental
transverse and longitudinal covariances of the pores with the theoretical ones taking into account the statistical
distributions of the dimensions of the random cylinders. The second method to validate the model consists
in comparing morphological measurements (density profiles, covariance, opening granulometry, tortuosity,
specific surface area) processed on real and on simulated media.
Keywords: 3D images, Boolean model, fibrous media, mathematical morphology, random media.
INTRODUCTION
Fibrous materials are commonly used for thermal
and acoustical insulating in buildings. The Silent Wall
ANR1project’s objective is to build an acoustical
insulating wall made of fibrous media, with innovating
acoustical properties, developed in a context of
environmental efficiency and competitiveness by using
natural raw materials such as wood and other cellulose
fibres. In this purpose, morphological measurements
were performed (Peyrega et al. , 2009a) to characterize
the Thermisorel , which is a wooden fibre board
100% naturally papermaking processed from recycled
wood, used as a reference material in the Silent
Wall project. This paper focuses on modelling the
microstructure of such a fibrous material.
When 3D images were not available yet, 2Dimages from confocal microscope acquisitions were
used to fit models of random fibrous media. The
method proposed by Jeulin (2000), Cast ´eraet al.
(2000), Michaud et al. (2000) and Delis ´eeet al.
(2001) consists in generating a Boolean model of
Poisson lines, dilated by spheres with a random
radius (Matheron, 1967; Serra, 1982). This method is
effective to simulate 3D stacks of very long fibres after
identification from 2D projected images, when their
length is large compared to the images. Modelling a
fibrous medium with 3D Poisson lines was also made
later by Schladitz et al. (2006) to study its acoustical
properties. However, the dimensions of the fibres of
Thermisorel in the present paper are finite compared
to those of the sample (about 600 ×600×360 voxels
with resolutions ranging from 2 µm to 15 µm). That is
why a different model is proposed here.
1http://us2b.pierroton.inra.fr/Projets/Silent Wall/description.htm
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As the fibres of Thermisorel have finite
dimensions, they could be modelled by a Boolean
model of random cylinders whose radii, lengths
and orientations are estimated by the morphological
measurements performed by Peyrega et al. (2009a).
Moreover, these cylinders are stacked in the xOy
planes perpendicular to the axis of compression Oz,
according to the industrial papermaking process of
this material. The covariance measurements of the
fibres (Matheron, 1967; Serra, 1982) highlight the
global anisotropy of the material, and the isotropy
of their orientations in the xOy planes. Thus, it
seems to be realistic to simulate uniformly distributed
orientations between 0 and πin these planes. The
opening granulometry of the fibres (Matheron, 1967;
Serra, 1982) allows us to extract the distribution of
their radii. Automatic morphological fibre analysis
using 2D images with the MorFi system2brings the
distribution of their lengths out.
After having presented the 3D X-Ray CT
images of Thermisorel used for the morphological
characterization, the Boolean model of random
cylinders will be introduced. Afterwards two methods
to extract the distribution of the radii will be presented
(the first one from the granulometry, and the second
one from a minimization of the mean square error
using the transverse covariance of the external pores).
At last an estimation of the distribution of the lengths
will be introduced.
3D IMAGES OF THERMISOREL
The 3D images are obtained with the US2B X-
Ray tomograph, with resolutions equal to 2 µm, 5µm,
9.36µm and 15 µm per voxel, depending on the
wanted observation scale. The grey level images are
segmented by manual thresholding, after smoothing
the noise by a low-pass filter (Fig. 1). Several
structures are observable in the Thermisorel (Fig.
2a). The fibrous phase is composed of isolated fibres
and of clusters, the sticks. The external porosity and
the internal one, i.e., the lumens into the fibres where
the sap flows in the tree, compose the porous phase.
With a 9 .36µm resolution, the lumens are not
observable, but for 5 µm they should be filled in order
to isolate the external porosity. An algorithm to fill
the lumens is proposed by Lux (2005). It consists
in the succession of morphological operations on
the segmented image. The final result is represented
in Fig. 2b on an image acquired in Grenoble on
the beamline ID 19 of the European SynchrotronRadiation Facility (ESRF) with a resolution of
5µm/voxel.
(a)
 (b)
Fig. 1. Binarized X-Ray CT images of Thermisorel ;
(a) Source: ESRF; resolution: 5 µm/voxel;
dimensions: 5.9 mm ×5.9 mm ×1.68 mm, (b) Source:
US2B; resolution: 9.36 µm/voxel; dimensions:
5.6 mm ×5.6 mm ×3.4 mm.
(a)
 (b)
Fig. 2. Lumen filling of Thermisorel (resolution:
5µm/voxel; source: ESRF ID19). (a) Segmented 2D
slice of Thermisorel in the xOy plane, (b) Filled
lumens.
As the production process of Thermisorel is
a papermaking one, the fibrous mat is compressed
along the Ozaxis. The fibres are thus isotropically
oriented in the xOy planes perpendicular to Oz. This
is observable on the measurements of the covariance
C(h)of the fibres (Peyrega et al. , 2009a), showing
identical covariances in the xOy planes, whatever the
orientation of the vector /vectorh, which is a consequence of
the transverse isotropy. However, the fibrous medium
is anisotropic in the other directions of 3D space, since
the correlation length in the Ozdirection is shorter than
in the xOyplane.
2http://cerig.efpg.inpg.fr/dossier/EFPG-innovations/page10.htm
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Image Anal Stereol 2009;28:129-141
BOOLEAN MODEL OF RANDOM
CYLINDERS
THE BOOLEAN MODEL
In order to model a Thermisorel -like fibrous
medium, with fibres having finite lengths, the Boolean
model with random cylinders as primary grains seems
to be suitable. The Boolean model was introduced
by Matheron (1967) and used by Jeulin (2000) in a
modelling of random textures for materials. The first
step consists in generating a Poisson point process.
Random primary grains A/primeare implanted on
Poisson points xkwith the intensity Θi.e., the average
number of generated points per unit volume. The
overlap of these grains is possible. Let Abe the random
set generated by the grains (Eq. 1),
A=/uniondisplay
xkA/prime
xk. (1)
REMINDER ON THEORETICAL
PROPERTIES OF THE BOOLEAN
MODEL
Ais the set of fibres with p=P(x∈A). The
probability Q(K)for a compact set Kto be included
in the set Ac(with q=P(x∈Ac) =1−p) is given by
Matheron (1967) and Serra (1982) as a function of the
erosion /circleminusand dilation ⊕operations in Eq. 2 and Eq.
3. Let µnbe the average Lebesgue measure (average
volume in 3D), and ˇK={−x,x∈K}be the transposed
set of K.
Q(K) =P(K⊂Ac) =P(x∈Ac/circleminusˇK), (2)
Q(K) =e−Θµn(A/prime⊕ˇK)=qµn(A/prime⊕ˇK)
µn(A/prime). (3)
The covariance ( Q(h)), linear erosions ( Q(l)), and
erosions by balls ( Q(B(r))) are particular cases for K
of the expression given in Eq. 2 and Eq. 3.
Covariance Q(h)
Ais the set of fibres. Let Q(h)(Eq. 4) be the
covariance of the porous medium, i.e., of the set Ac.
The Eq. 5 gives the theoretical expressions of Q(h)
for the Boolean model. Let K(h)be the geometrical
covariogram of the grain A/prime,K(h) =µn(A/prime∩A/prime
−h).
The normalized covariogram r(h)is defined by r(h) =
K(h)/K(0), with K(0) =µn(A/prime),
Q(h) =P(x∈Ac,x+h∈Ac), (4)
QBooleanModel (h) =q2eΘ(K(h))=q2−r(h).(5)From Eq. 5 it is then possible to deduce the
porosity q(Eq. 6), which leads to the intensity Θof the
Poisson point process from which the Boolean model
has been generated (Eq. 7).
q=e−Θµn(A/prime), (6)
Θ=−ln(q)
µn(A/prime). (7)
Linear erosions Q(l)
Considering the porous phase, the theoretical
expression Q(l)is given by Eq. 8 for a random convex
grain A/primein which we consider r/prime(0) = ( dr/dh)h=0.
QBooleanModel (l) =e−Θµn(A/prime⊕l)=q1−l r/prime(0).(8)
Erosions by balls Q(B(r))
The probability for a ball with radius r,B(r)to be
included in the pores is given by Eq. 9:
QBooleanModel (r) =e−Θµn(A/prime⊕B(r)). (9)
In the case of convex grains, like random cylinders,
theµn(A/prime⊕B(r))can be expanded according to
the Steiner’s formula involving the Minkowski’s
functionals of A/primeand of B(r). It is therefore a
polynomial of degree 3 in r. According to the Steiner’s
formula, it is possible to write Eq. 10 which implies
Eq. 11. Let us consider V(K)the volume of a compact
setK,A(K)its surface area, and M(K)its integral
mean curvature, with R=E[R]andL=E[L].
µn(A/prime⊕B(r)) = V(A/prime) +M(B(r))S(A/prime)
4π
+M(A/prime)S(B(r))
4π+V(B(r)),(10)
µn(A/prime⊕B(r)) =4
3πr3+π/parenleftbig
L+πR/parenrightbig
r2
+/bracketleftBig
2πR(R+L)/bracketrightBig
r+πE/bracketleftbig
R2/bracketrightbig
L.(11)
These expressions, as well as the exponential
behavior of QBooleanModel (l)are used as to test the
assumption of a Boolean model with convex grains.
TEST OF THE VALIDITY OF A BOOLEAN
MODEL FOR THE FIBROUS NETWORK
In order to validate the assumption of a
Boolean model of random cylinders to simulate a
Thermisorel -like material, several morphological
characteristics of the fibrous network have been
compared to the theoretical results. As a first step,
these measurements are the linear erosions Q(l)
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(Fig. 3 with a logarithmic scale) and the erosions by
rhombicuboctaedra Q(B(r))(Fig. 4 with a logarithmic
scale, and Fig. 7).
 1e-006 1e-005 0.0001 0.001 0.01 0.1 1
 0  100  200  300  400  500  600  700  800  900Fraction of the volume of PORES (LOG Scale)
Sizes l of LINEAR EROSIONS (MICRONS)Q(l) of REAL Thermisorel after LINEAR EROSIONS
Q(l)_X
Q(l)_Y
Q(l)_Z
Fig. 3. Linear erosions of Thermisorel in the Ox, Oy
and Oz directions, shown in logarithmic scale (source:
sample Fig. 1b).
In the Ox and Oy longitudinal directions log(Q(l))
cannot be fitted exactly by a straight line (Fig. 3).
This result is due to the fact that fibres are not strictly
convex. They should be cylinders with some slight
bending, which modifies the behavior of log (Q(l))in
the directions of the xOy planes. In the Oz direction,
log(Q(l))can be fitted by a straight line. However, we
will neglect these points for those fibres, and use the
‘perfect’ cylinders assumption.
 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1
 0  50  100  150  200  250Fraction of the volume of PORES (LOG Scale)
Sizes r of EROSIONS by RhombiC 24-C (MICRONS)Q(r) of REAL Thermisorel after EROSIONS by RC
Q(r)_Thermisorel
Fig. 4. Erosions by rhombicuboctaedra B(r) of
Thermisorel , shown in logarithmic scale (source:
sample Fig. 1b).
In Fig. 4, the experimental curve log (Q(B(r)))
(B(r)being a ball of radius r) should be fitted by a
theoretical polynomial of degree 3 in r(Eq. 11).BOOLEAN MODEL OF RANDOM
CYLINDERS WITH A TRANSVERSE
ISOTROPY
We consider now as primary grains a population of
cylinders with a random radius Rand a random length
L. A given grain is completely known from these two
characteristics and from its orientation. With respect
to the already mentioned transverse isotropy of the
network, we will consider fibres orthogonal to the Oz
axis, and with a uniform distribution of orientations
in the xOy planes. In this context, the most general
random fibre model would require the knowledge
of the bivariate distribution of Rand L,f(r,l). For
simplification, and in absence of simultaneous data on
the same fibres, we will assume here that these two
random variables are independent.
Knowing the distributions for Rand for L, it is
possible to compute the theoretical covariance in the
transverse direction (along Oz), or in any horizontal
direction in the xOy planes (longitudinal covariance).
The input of the observed size distributions enables us
to check the validity of the model, by comparison
of the measured and calculated covariances.
Alternatively, we can also fit the parameters of the
distribution from the experimental covariances, by a
least squares minimization.
Consider first the transverse covariance of the
pores, QZ(h). From Eq. 5, it is expressed as a
function of the transverse reduced covariogram of the
cylindrical fibre, rZCylinder(h).
(a)
 (b)
Fig. 5. Transverse geometrical covariogram of the
cylinder. (a) Cylindrical primary grain A’, (b)
Geometrical covariogram of the disc (red area).
Let f1(r)be the distribution of the random radii
of the fibres. The transverse geometrical covariogram
of a fibre is given as a function of the average length
Lfibresand of the average geometrical covariogram of a
population of discs with a random radius Rfollowing
the distribution f1(r)byKZCylinder(h) =LfibresKZDisc(h).
Therefore we have:
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Image Anal Stereol 2009;28:129-141
rZCylinder(h) =KZDisc(h)
KZDisc(0). (12)
The normalized transverse geometrical
covariogram of the fibres does not depend on their
lengths, as a result of the independence between
Rand L. In Eq. 12, KZDisc(h)is deduced from the
geometrical covariogram of a disc KZDisc(h,r)(Fig. 5)
and from f1(r). We have, if 0 ≤h≤2r(ifh>2r,
KZDisc(h,r) =0):
KZDisc(h,r) =2r2/parenleftBigg
arc cos/parenleftbiggh
2r/parenrightbigg
−h
2r/radicalBigg
1−/parenleftbiggh
2r/parenrightbigg2/parenrightBigg
,(13)
and
KZDisc(h) =/integraldisplay+∞
h/2KZDisc(h,r)f1(r)dr. (14)
Consider now the longitudinal covariance QxOy(h),
given for a vector /vectorhin the horizontal xOy planes
(due to the transverse isotropy this vector can be
considered as parallel to the Oxaxis). Noting f2(l)
the distribution function of the random length L, the
average longitudinal geometrical covariogram of the
fibre KxOy Cyl(h)is given by Eq. 16 which is illustrated
in Fig. 6.
(a)
 (b)
Fig. 6. Longitudinal geometrical covariogram of the
cylinder (red volume). (a) Geometrical covariogram,
xOy projection, (b) Geometrical covariogram, yOz
projection.
The red volume in the Fig. 6 corresponds to the
geometrical covariogram of the cylinder and is defined
by the Eq. 15, for hX≤l(forhX>l,KxOy Cylinder(h,θ) =
0),
KxOy Cylinder(hX,hY,l) = ( l−hX)KZDisc(hY).(15)
Considering hX=hcos(θ),hY=hsin(θ)and the
uniform distribution of the orientations θof the fibresit is then possible to write the Eq. 16, f2(l)being the
distribution of the random length Lof the fibres,
KxOy Cyl(h) =/integraldisplay+∞
(hsinθ)/2KZDisc(h,r)f1(r)dr
·1
π/integraldisplayπ
0/integraldisplay+∞
hcosθ(l−hcosθ)f2(l)dldθ.(16)
The theoretical expression of the longitudinal
covariance of the pores is obtained by using the Eq.
17 into the Eq. 5,
rxOy Cylinder(h) =KxOy Cyl(h)
KxOy Cyl(0). (17)
IDENTIFICATION OF THE
PARAMETERS OF THE MODEL
AND EXPERIMENTAL RESULTS
After having defined the theoretical properties of
the Boolean model, the objective is to fit them with the
experimental measurements.
DISTRIBUTION FUNCTIONS
Knowing the porosity q, the first step consists in
estimating the intensity Θof the Poisson point process
from which the Boolean model originates. In the case
of cylinders as primary grains A/prime, with radii Rand
lengths L, both independent random variables, we can
write the Eq. 18 in the Eq. 7:
µn(A/prime) =V(A/prime) =πE/bracketleftbig
R2/bracketrightbig
E[L]. (18)
In the second step the distribution functions of R
and Lare estimated. The opening granulometry by
rhombicuboctaedra (Fig. 7 and Fig. 9) gives access to
a ‘volume’ weighted distributions of the radii of the
fibres (Matheron, 1967; Serra, 1982).
Fig. 7. Rhombicuboctaedron.
The morphological measurements presented by
Peyrega et al. (2009a) show that the distribution in
‘volume’ of the radii of the fibres gα1,b1(r)can be fitted
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PEYREGA CET AL :3D morphological modelling of a random fibrous network
by a gamma distribution (Eq. 19) with parameters α1=
E[R]2/Var(R)andb1=E[R]/Var(R),
gα,b(r) =bα
Γ(α)rα−1e−br, (19)
Γ(α) =/integraldisplay+∞
0tα−1e−tdt. (20)
Moreover it is possible to link ‘volume’ g(r), and
‘number’ f1(r)weighted granulometries as gamma
distributions with the following formula (Eq. 21,
Peyrega et al. , 2009a). Then we can write the Eq. 22
in the Eq. 18,
f1(r) =g[α1−2,b1](r), (21)
E/bracketleftbig
R2/bracketrightbig
=α(1+α)
b2=[α1−2](1+ [α1−2])
b2
1.(22)
For the Thermisorel material, we obtain the
following parameters: α1=6.53 and b1=0.164µm−1.
It appears that the distribution of Lfollows
approximately an exponential distribution f2(l)as
obtained with the MorFi system. Therefore it comes
for the ‘number’ distribution f2(l):
f2(l) =1
E[L]e−l/E[L]. (23)
For the Thermisorel material, we obtain E[L] =
1654µm. In both cases, the quality of the fit can be
measured by the mean square error (MSE).
FITTING THE TRANSVERSE
COVARIANCE
The Fig. 8 represents the transverse covariance
QZ(h)of the pores. The thick plain red curve gives
the experimental covariance. We can notice that
this covariance reaches its theoretical sill without
oscillations, which is required for a Boolean model. It
shows that in the present case a hard-core point process
for the location of the centres of fibres, instead of a
Poisson point process, would be irrelevant. The blue
middle dashed curve is the theoretical prediction of the
covariance obtained with the fitted gamma distribution
of the radii of the fibres with parameters α=4.53 and
b=0.164µm−1, which gives a MSE =1.16×10−5for
QZ(h).
Alternatively, we can fit the parameters of the
distribution by minimizing the mean square error
between the experimental and the theoretical QZ(h)
curves in the linear zone, i.e., for 0 ≤h≤56.16µm.
This method consists in initializing both αand b
parameters and then to make them vary aroundthe values estimated from the granulometry. The
theoretical covariance is then calculated using the
corresponding gamma distribution fin ‘number’. The
couple of parameters (αMSE,bMSE)minimizing this
error is then chosen.
Here, for f1we obtain αMSE =4.98 and bMSE =
0.159µm−1, for a minimized MSE =0.84×10−5
for QZ(h), which gives parameters close to the
direct estimation obtained from the opening size
distribution. The Fig. 9 shows the corresponding
gamma distributions fitting the experimental ‘volume’
granulometry of the fibres. Both distributions are
very close to each other and have quite similar scale
parameters b. This shows that the two estimations of
the parameters (from the granulometry and from the
transverse covariance) are consistent, which validates
the model.
 0.3 0.35 0.4 0.45 0.5 0.55
 0  50  100  150  200COVARIANCE of the PORES
h (MICRONS)TRANSVERSE COVARIANCE of the PORES of THERMISOREL
Qz(h) of the PORES
Qz(h)_B_Fit; alpha_R=4.53 ; b_R=0.164
Qz(h)_MSE; alpha_R=4.98 ; b_R=0.159
Sill Qz(0)^2 : 0.3183
Fig. 8. Transverse covariance Q Z(h)of Thermisorel .
Thick plain red: experimental data; middle dashed
blue: theoretical values with a data-estimated gamma
distribution for ‘r’; thin dashed green: theoretical
values with a MSE-estimated gamma distribution.
 0 0.01 0.02 0.03 0.04 0.05
 0  20  40  60  80  100  120Probability density
r (MICRONS)DISTRIBUTION of R_FIBRES of thermisorel1/Gamma Dist.
DIST. of R_FIBRES (openings by RC)
Gamma_Dist; alpha_1=6.53; b_1=0.164
Gamma_MSE; alpha_1=6.98; b_1=0.159
Fig. 9. Volume weighted distribution of the radii of
the fibres of Thermisorel and fitting by gamma
distributions. Thick plain red: experimental data;
middle dashed blue: data-estimated gamma
distribution for ‘r’; thin dashed green: gamma
distribution with MSE-estimated parameters.
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Image Anal Stereol 2009;28:129-141
FITTING THE LONGITUDINAL
COVARIANCE
Considering the longitudinal covariance in the Ox
direction, the Fig. 10 shows that the Boolean model of
random cylinders correctly fits the experimental data
as well with a MSE =0.27×10−5forQX(h)with
parameters α=4.53 and b=0.164µm−1forf1and
E[L] =1654µm for f2.
 0.3 0.35 0.4 0.45 0.5 0.55
 0  500  1000  1500  2000  2500COVARIANCE of the PORES
h (MICRONS)LONGITUDINAL COVARIANCE of the PORES of THERMISOREL
Qx(h) LONG. COV. of the PORES
Qy(h) LONG. COV. of the PORES
Qx(h)Expo ; Mean_L=1654.2 Microns
Sill Qx(0)^2 : 0.3183
Fig. 10. Longitudinal covariances Q X(h)and Q Y(h)
of Thermisorel . Thick plain red and medium
dashed blue: experimental data; thin dashed green:
theoretical values with data-estimated gamma and
exponential distributions respectively for R and L.
EXAMPLE OF 3D SIMULATION
For illustration a 3D simulation of the
Thermisorel fibrous network is shown in Fig. 11
and Fig. 12 to be compared respectively to Fig. 1
and Fig. 13. This simulation was processed in 4
minutes for about 18000 fibres with a desktop
computer Intel Core 2 Extreme X6800 2.93 GHz.
The input parameters of this simulation are those
estimated from Fig. 1b (porosity q=0.5642; gamma
distribution of the radii of the cylinders α=4.53 and
b=0.164µm−1; mean length E[L] =1654µm).
As a consequence of the Boolean model,
rectangles can be observed on the 2D slices in the xOy
projection (Fig. 12a), and sections of ellipses in the
zOy planes (Fig. 12b).
Fig. 11. Boolean model of cylinders (Dimensions:
5.6 mm ×5.6 mm ×3.4 mm. Resolution:
9.36µm/voxel.)
(a)
 (b)
Fig. 12. Boolean model of cylinders (2D slices of Fig.
11). (a) Transverse xOy projection, (b) longitudinal
zOy projection.
(a)
 (b)
Fig. 13. Real Thermisorel (2D slices of Fig. 1b).
(a) Transverse xOy projection, (b) longitudinal zOy
projection.
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COMPARISON OF
MORPHOLOGICAL
MEASUREMENTS
In order to validate the Boolean model of random
cylinders, morphological measurements have been
processed on both real (Fig. 1b) and simulated
(Fig. 11) Thermisorel .
DENSITY PROFILES
The porosity of the simulated fibrous material
is equal to 56.30% and is consistent with the
experimental porosity equal to 56.42%. Moreover, the
area fraction profiles of fibres in the Ox, Oy and Oz
directions are homogeneous for both media (Figs. 14
and 15).
 0 10 20 30 40 50 60 70
 0  1000  2000  3000  4000  5000% of the voxels of the image
h (MICRONS)Area fraction of the FIBRES of REAL Thermisorel in 3 dir.
Ox
Oy
Oz
Volume fraction of the FIBRES = 43.58 %
Fig. 14. Profiles of area fraction of real fibres in the
Ox, Oy and Oz directions.
 0 10 20 30 40 50 60 70
 0  1000  2000  3000  4000  5000% of the voxels of the image
h (MICRONS)Area fraction of the FIBRES of SIMULATED Thermisorel in 3 dir.
Ox
Oy
Oz
Volume fraction of the FIBRES = 43.70 %
Fig. 15. Profiles of area fraction of simulated cylinders
in the Ox, Oy and Oz directions.COVARIANCE
Transverse covariance
As shown in Fig. 16, the transverse covariances
of real fibres and simulated cylinders (in the Oz
direction) are almost superimposed. Thus both media
have transverse characteristic lengths of the same
order.
 0.15 0.2 0.25 0.3 0.35 0.4
 0  200  400  600  800  1000  1200COVARIANCE of the FIBRES/CYLINDERS
h (MICRONS)TRANS. COV. of the FIBRES of REAL/SIM. Thermisorel
Profile Z_SIM
Profile Z_REAL
Sill of the COVARIANCE = C(0)^2 = 0.19
Fig. 16. Transverse covariance of fibres and simulated
cylinders.
Longitudinal covariance
Concerning the longitudinal covariances of real
fibres and simulated cylinders (in the xOy planes), the
4 curves in Fig. 17 are almost superimposed as well.
Therefore both media have longitudinal characteristic
lengths of the same order, with fibres and cylinders
isotropically oriented in the xOy planes, since the
covariances in the Ox and Oy directions are practically
identical for both materials.
 0.15 0.2 0.25 0.3 0.35 0.4
 0  200  400  600  800  1000  1200COVARIANCE of the FIBRES/CYLINDERS
h (MICRONS)LONG. COV. of the FIBRES of REAL/SIM. Thermisorel
Profile X_SIM
Profile Y_SIM
Profile X_REAL
Profile Y_REAL
Sill of the COVARIANCE = C(0)^2 = 0.19
Fig. 17. Longitudinal covariance of fibres and
simulated cylinders.
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Image Anal Stereol 2009;28:129-141
OPENING GRANULOMETRY
The Fig. 18 and Fig. 19 represent the volume
weighted opening granulometry respectively of the
fibres and of the pores of real Thermisorel . The
openings are processed in 3D with rhombicuboctaedra
as structuring elements. For both figures the
corresponding fitted gamma distributions are plotted
as well, with α=E[R]2/Var(R)andb=E[R]/Var(R).
 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
 0  20  40  60  80  100  120Probability density
r (MICRONS)DIST. of R_FIBRES of REAL_Thermisorel
DISTR. of R_FIBRES (openings by RC)
Gamma_Dist; alpha_1=6.53; b_1=0.164
Fig. 18. Volume weighted opening granulometry of real
fibres.
 0 0.005 0.01 0.015 0.02 0.025
 0  50  100  150  200Probability density
r (MICRONS)DIST. of R_PORES of REAL_Thermisorel
DISTR. of REAL R_PORES (openings by RC)
Gamma_Dist; alpha_1=4.15; b_1=0.076
Fig. 19. Volume weighted opening granulometry of real
pores.
The volume opening granulometries of the
simulated cylinders and of the simulated pores
(respectively Fig. 20 and Fig. 21) are slightly different
from those of real Thermisorel . Concerning the
real fibres and the simulated cylinders, both fitted
gamma distributions have quite similar parameters
(difference of about 0.1 for αand difference of about
0.01µm−1forb). Concerning the porous media, both
fitted gamma distributions are different from each
other (Figs. 19 and 21), but there is no theoretical
requirement to use here gamma distributions. However
both red curves have similar evolutions and the samelocal maxima in r=37.44µm and r=65.52µm, and
the same local minima in r=28.08µm,r=46.8µm
andr=56.16µm. These results validate the method
used in this study which consists in processing the
3D opening granulometry directly on the real global
fibrous medium to deduce the ‘number’ distribution
of the radii of the fibres, without extracting them
individually.
 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
 0  20  40  60  80  100  120Probability density
r (MICRONS)DIST. of R_CYLINDERS of SIM_Thermisorel
DISTR. of R_CYLINDERS (openings by RC)
Gamma_Dist; alpha_1=6.66; b_1=0.155
Fig. 20. Volume weighted opening granulometry of
simulated cylinders.
 0 0.005 0.01 0.015 0.02 0.025
 0  50  100  150  200Probability density
r (MICRONS)DIST. of R_PORES of SIM_Thermisorel
DISTR. of SIM. R_PORES (openings by RC)
Gamma_Dist; alpha_1=5.49; b_1=0.119
Fig. 21. Volume weighted opening granulometry of
simulated pores.
MORPHOLOGICAL TORTUOSITY
The next characteristic to be compared is the
morphological tortuosity, which corresponds to the
ratio between the geodesic and the Euclidian distances
between two parallel faces of the image according
to a direction of propagation. The geodesic distance
is processed either in the fibrous or in the porous
phase. The minimum tortuosity is then equal to 1 by
definition. The method to estimate it from 3D image
processing is described by Peyrega et al. (2009a)
and uses the algorithm proposed by Decker et al.
(1998), which is coupled with a Fast Marching method
137
PEYREGA CET AL :3D morphological modelling of a random fibrous network
(Sethian, 1996; Petres et al. , 2005) to estimate the
geodesic lengths of the paths in the media.
In the fibrous media
In the fibrous media, both Fig. 22 and Fig. 23
show a common trend. On the one hand the histograms
of tortuosities in the transverse Ox and Oy directions
are almost superimposed because of the isotropy of
the orientations of the fibres and cylinders in the xOy
planes. On the other hand, this implies lower mean
tortuosities in every direction in these planes than in
the Oz direction where higher tortuosities are reached.
Fig. 22. Morphological tortuosity of real fibres in the
Ox, Oy and Oz directions.
Fig. 23. Morphological tortuosity of simulated
cylinders in the Ox, Oy and Oz directions.
The transverse tortuosities of cylinders have mean
values and standard deviations relatively similar to
those of real fibres. However in the Oz direction,
the real fibres are less tortuous, because they are
not strictly straight and not strictly oriented in the
xOy planes contrarily to the simulated ones. They are
slightly curved.In the porous media
The tortuosities in both materials are globally
lower and close to 1 in the pores (Figs. 24 and 25),
which proves that the pores are connected by straight
paths, which will have an influence on the general
acoustic behavior of the material.
Fig. 24. Morphological tortuosity of real pores in the
Ox, Oy and Oz directions.
Fig. 25. Morphological tortuosity of simulated pores in
the Ox, Oy and Oz directions.
Like for the fibrous media, the tortuosities of the
porous media according to Ox and Oy are lower than
in the Oz direction and are similar for both real and
simulated media. Moreover both histograms according
to Ox and Oy in the simulated pores (Fig. 25) are
quite superimposed because of transverse isotropy.
Contrarily to the fibrous media, the tortuosities
according to Oz in the real pores have mean values
higher than those in the simulated material.
SPECIFIC SURFACE AREA
The specific surface area is the ratio between the
contact surface between fibrous and porous media and
the volume of the sample. This parameter has a direct
138
Image Anal Stereol 2009;28:129-141
influence on the thermal and acoustical heat transfer.
It has been estimated by the Crofton formula. It is
equal to 0 .016µm−1for real Thermisorel and equal
to 0.017µm−1for simulated cylinders. Both values are
of the same order.
IMPROVEMENT OF THE MODEL
The first morphological measurements processed
on this Boolean model of random cylinders seem to
validate this method to build artificial Thermisorel -
like fibrous networks. However, as shown in the
previous part, the tortuosity in the Oz direction, do
not correctly fit the real material. This was in part
explained by the fact that real fibres are not strictly
oriented in the xOy planes. In spherical coordinates
(Fig. 26), this means that their orientations φare not
strictly equal toπ
2. On the contrary, they are randomly
distributed aroundπ
2because of the compression of the
sample.
Fig. 26. Spherical coordinates.
INTRODUCING A DISTRIBUTION OF
ORIENTATIONS IN 3D
In order to simulate these random orientations φ,
we use the distribution proposed by Schladitz et al.
(2006) which is explicited in Eq. 24, with θ∈[0,2π[,
andφ∈[0,π[. We can notice that this distribution
of orientations is independent from θ. The anisotropy
parameter βis equal to 1 for isotropically distributed
cylinders in 3D. In order to simulate a compression
of them along the Oz axis, βis increased. Thus the
previous model with cylinders oriented only in the xOy
planes has a very high β,
p(θ,φ) =1
4πβsin(φ)
[1+ (β2−1)cos2(φ)]3/2. (24)
Thus Fig. 27a and Fig. 27b respectively represent
the probability density and the distribution functions of
the distribution in Eq. 24 for β=1, 3, and 10.
 0 1 2 3 4 5
 0  20  40  60  80  100  120  140  160  180Probability density
Phi (DEGREES)DIST. of PHI angle according to parameter Beta
Beta = 1 (ISOTROPIC)
Beta = 3
Beta = 10(a)
 0 0.2 0.4 0.6 0.8 1 1.2
 0  20  40  60  80  100  120  140  160  180Probability
Phi (DEGREES)REPARTITION FUNCTION of PHI according to Beta
Beta = 1 (ISOTROPIC)
Beta = 3
Beta = 10
(b)
Fig. 27. Distribution of orientations φin 3D for
β= 1, 3 and 10. (a) Probability density function, (b)
distribution function.
MORPHOLOGICAL MEASUREMENTS
ON THE IMPROVED MODEL
An improved sample having the same parameters
than Fig. 11 and having a βparameter equal to 10 is
studied in this part. The same measurements than in the
previous section have been processed on this sample.
The density profiles, the covariance and the opening
granulometry give similar results than for the sample
on Fig. 11. The specific surface area is the same as
well.
The only difference between both simulations lies
in the tortuosities of the cylinders. As shown in Fig.
28 the histogram of the tortuosities of the simulated
cylinders in the Oz direction is closer from the one of
the real material (Fig. 22) than the previous model is
(Fig. 23). However, we can notice that this new model
is not improved for the tortuosities of the simulated
pores according to Oz, (Fig. 29), which do not
correctly fit the histogram of the real pores in the same
139
PEYREGA CET AL :3D morphological modelling of a random fibrous network
direction (Fig. 24). Finally this modified version of the
model still correctly fits the real Thermisorel for the
tortuosities in the Ox and Oy directions whatever the
medium considered (cylinders or simulated pores).
 0 1 2 3 4 5 6 7
 1  1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9% of the voxels of the paths linking the 2 facesTortuosityTortuosity of SIMULATED Thermisorel into the CYLINDERS
Tort-X-CYL ; average: 1.0232 ; std-dev: 0.012
Tort-Y-CYL ; average: 1.0214 ; std-dev: 0.012
Tort-Z-CYL ; average: 1.2617 ; std-dev: 0.068
Fig. 28. Morphological tortuosity of simulated
cylinders in the Ox, Oy and Oz directions for the
improved model with β=10.
 0 2 4 6 8 10 12 14
 1  1.05  1.1  1.15  1.2  1.25  1.3% of the voxels of the paths linking the 2 facesTortuosityTortuosity of SIMULATED Thermisorel into the PORES
Tort-X-PORES-C ; average: 1.0259 ; std-dev: 0.007
Tort-Y-PORES-C ; average: 1.0259 ; std-dev: 0.007
Tort-Z-PORES-C ; average: 1.0462 ; std-dev: 0.017
Fig. 29. Morphological tortuosity of simulated pores in
the Ox, Oy and Oz directions for the improved model
withβ=10.
CONCLUSION
It was shown that a Boolean model of random
cylinders could describe properly a 3D fibrous
network. In order to validate this assumption, it
was necessary to fit the experimental measurements
with the theoretical expressions of morphological
parameters of the pores such as the transverse and
longitudinal covariances.
The second step consisted in comparing
morphological measurements of real and simulated
media. The area fraction of fibres and cylinders are
homogeneous for both materials. Moreover theirtransverse and longitudinal covariances have similar
evolutions and characteristic lengths. The opening
granulometries of the fibres and of the cylinders
can be fitted by gamma distributions with similar
parameters. The morphological tortuosities in the Ox
and Oy directions of fibrous and porous media for both
materials have quite the same histograms. However the
tortuosities in the Oz direction are different for both
materials whatever the phase considered, which can be
explained by the geometrical differences between the
cylinders and the real fibres, which are curved and not
strictly oriented in the xOy planes. The last parameter
which validates the model is the specific surface area
which presents similar values for real and simulated
media.
In order to improve the model, a distribution of
orientations of the cylinders in 3D has been used.
The morphological characteristics of this new material
such as the density profiles, the covariance, the
opening granulometry, the specific surface area, and
the morphological tortuosities according to Ox and Oy,
still correctly fit the real Thermisorel . Moreover the
tortuosities of the cylinders in the Oz direction are
closer from the histogram of the real material than
the one of the previous model is. However there are
not any improvement concerning the tortuosities of the
simulated pores according to Oz.
In a next step, the simulations will be used as input
in the prediction of the acoustic properties of fibrous
networks, with the aim to find microstructures with an
optimized physical behavior.
ACKNOWLEDGEMENTS
This article is an extended version of a
communication (Peyrega et al. , 2009b) to the 10th
European Congress of Stereology and Image Analysis
(ECS10, 2009 June 22-26; Milan, Italy).
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