ImageAnalStereol2009;28:179-185
OriginalResearchPaper
MESH FREE ESTIMATIONOF THESTRUCTUREMODEL INDEX
JOACHIMOHSER1,CLAUDIAREDENBACH2ANDKATJASCHLADITZ3
1HochschuleDarmstadt,FachbereichMathematikundNaturwi ssenschaften,Sch¨ offerstraße3,D-64295
Darmstadt, Germany;2TechnischeUniversit¨ atKaiserslautern,FachbereichMat hematik,
Erwin-Schr¨ odinger-Straße,D-67653Kaiserslautern,Ger many;3Fraunhofer-Institutf¨ ur Techno-und
Wirtschaftsmathematik, AbteilungBildverarbeitung,Fra unhofer-Platz1,D-67663Kaiserslautern,Germany.
e-mail: jo@h-da.de,redenbach@mathematik.uni-kl.de,ka tja.schladitz@itwm.fraunhofer.de
(AcceptedSeptember30,2009)
ABSTRACT
Thestructure modelindex(SMI)is a meansof subsumingthe to pologyof a homogeneousrandomclosed set
underjustonenumber,similartotheisoperimetricshapefa ctorsusedforcompactsets. Originally,theSMIis
defined as a functionof volume fraction, specific surface are a and first derivative of the specific surface area,
wherethederivativeisdefinedandcomputedusingasurfacem eshing. The generalised Steinerformulayields
howevera derivativeofthespecificsurfaceareathatis– upt o aconstant–thedensityoftheintegralofmean
curvature. Consequently, an SMI can be defined without refer ring to a discretisation and it can be estimated
from3Dimage data without need to mesh the surface but using the numb erof occurrencesof 2 ×2×2pixel
configurations, only. Obviously, it is impossible to comple tely describe a random closed set by one number.
In this paper, Boolean models of balls and infinite straight c ylinders serve as cautionary examples pointing
out the limitations of the SMI. Nevertheless, shape factors like the SMI can be valuable tools for comparing
similarstructures. Thisisillustratedonreal microstruc turesofice,foams,andpaper.
Keywords: image analysis, integral of mean curvature, intr insic volume densities, random closed set, shape
factor.
INTRODUCTION
Nowadays, a variety of imaging techniques – first
of all computed tomography, but also so-called FIB
tomography, electron tomography or atomic force
microscopy – are able to produce high quality 3D
images of microstructures, increasing the demand
for subsequent quantitative analysis. Assuming
macroscopic homogeneity, the microstructure can
be modelled by a stationary (or macroscopically
homogeneous)randomclosedset.
In many applications, geometric characteristics of
the random closed set have to be estimated from the
given image. A very attractive set of such global
geometric characteristics are the densities of the
intrinsic volumes (or quermassintegralsor Minkowski
functionals). In 3D, they are, up to constants, volume
fractionVV, surface area density SV, density of the
integral of mean curvature MV, and Euler number
densityχV.
These characteristics can be estimated efficiently
from observations in digital binary (black-and-white)
images based on discretised Crofton intersection
formulae (Ohser et al., 2009; Ohser and Schladitz,
2009).These classical integral geometric formulae
allow to compute the intrinsic volumes by calculation
of Euler numbers in lower dimensional intersections
and subsequent integration over all positions of theintersecting affine subspaces. The Euler numbers in
turn can be determined efficiently using the Euler-
Poincar´ eformula for all 2 ×2×2pixel configurations
and exploiting additivity. The core of the algorithm
first outlined by Lang et al.(2001) consists in a
convolutionofthebinaryimagewitha2 ×2×2mask,
resulting in an 8 bit grey value image, coding the
2×2×2 pixel configurations in the original binary
image. Subsequently, the grey value histogram of this
image is multiplied by a vector of suitable weights to
derive the desired intrinsic volume. It is particularly
noteworthy that the size of the grey value histogram
does not depend on image size or content. Moreover
all measurements are deduced directly from the pixel
configurations there is no need to approximate the
surfacebyasurfacemesh.
A further characteristic for macroscopically
homogeneous random sets Ξin the Euclidean space/CA3is thestructuremodelindex (SMI) definedas
fSMI=6VVS′
V
S2
V,
whereS′
Vdenotes the ’first derivative’ of the
surface density SV. The SMI was first suggested by
Hildebrand and R¨ uegsegger (1997a;b) for evaluating
bonestructure.
Inordertodefinethederivative S′
V,Hildebrandand
R¨ uegseggeruseasurfacemeshing.Roughly,the mesh
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OHSERJET AL:Meshfree SMI
ismovedoutwardsslightlythusdefiningadilationof Ξ
byasmallball.Thederivativeisthenapproximatedby
the differencequotient. In this paper, S′
Vis definedvia
the Steiner formula. As a consequence, the SMI can
be expressed in terms of the densities of the intrinsic
volumes which in turn allows to estimate the SMI
from the grey valuehistogram of the convolvedimage
described above. The definition for the SMI without
referring to a discretisation as well as the mesh free
estimatorarederivedin the sectionbelow.
Being a shape factor, the SMI can clearly not
capture completely the random closed set under
consideration. This is emphasised by the possible
ranges for the SMI of Boolean models of balls and
infinite straight cylinders. On the other hand, the
SMI is helpful for comparing structures that are
sufficiently similar like the pore systemsin Greenland
firnfromdifferentdepthsorforassessingthedegreeof
closednessoftechnicalfoams.
DERIVATIONOFTHEMESHFREE
ESTIMATOR
The class of all compact convex sets (convex
bodies) in /CA3is denoted by K. Furthermore, we use
thesymbol Rfortheconvexring consistingofallfinite
unions of convex bodies. Finally, we introduce the
extended convexring Sconsisting of all sets X⊂ /CA3
such that X∩Kis an element of Rfor eachK∈K.
Denote by Bra ball with radius rand centred in the
origin. Let B(x,r)be a ball with centre xand radius r,
thatisB(x,r) =Br+x.
TheSteiner formula expresses the volume of the
parallelset K⊕Brofaconvexbody Katdistance r>0
asapolynomialofthe intrinsic volumesof KandBr,
V(K⊕Br) =3
∑
k=0r3−kκ3−kVk(K),r≥0,K∈K,
(1)
(Schneider,1993,p.197).Here κkdenotesthevolume
of thek-dimensional unit ball. The intrinsic volumes
Vk,k=0,...,3,are defined by (1). They are up to
constantsvolume V=V3,surfacearea S=2V2,integral
ofmeancurvature M=πV1,andEulernumber χ=V0,
andthe integralofmeancurvatureis closelyrelatedto
the meanwidth M=2πb.
TheSteinerformulacanbeextendedtotheconvex
ringR. We use Schneider’s index function j (X,x,y)
of a setX∈Ratxwith respect to y, as defined in
Schneider and Weil (2008, Section 14.4). Using theEulernumber χ, theindex jis definedas
j(X,x,y)=

lim
δ↓0lim
ε↓0χ(X∩B(x,δ)∩B(y,/bardblx−y/bardbl−ε)),
ifx∈X,
0,otherwise
for allX∈Randx,y∈ /CA3. It follows from the
additivity of the Euler number that the index jis
additive in its first argument, too. Now, introducing
localparallelsets with multiplicity we definethe local
measureρr(X,·)by
ρr(X,A) =/integraldisplay/CA3cr(X,A,y)dy,r>0
withcr(X,A,y) =∑
x∈A\{y}j/parenleftbig
X∩B(y,r),x,y/parenrightbig
,
for Borel sets A⊆ /CA3. Here the sum is taken over
onlyfinitelymanysummandsdifferentfrom zero.The
functional ρrinherits the additivity from the index j.
The Steiner formula for the local functional ρrand
its extension on the convex ring is given in Schneider
(1993, Section 4.4) and Schneider and Weil (2008,
Section 14.4). Here we will use the special case A=/CA3, only.Thatis, weconsiderthe functional
ρr(X) =ρr(X, /CA3). (2)
Let now Ξbe a macroscopically homogeneous
randomclosedseton /CA3with realisationsof Ξalmost
surely belonging to the extended convex ring S. We
assume that Ξis observed through a compact and
convex window Wwith nonempty interior. Moreover,
assume that Ξfulfils the integrability condition/BX2#(Ξ∩K)<∞foranyconvexbody K∈K,where#X
denotestheminimalnumber msuchthattheset Xhasa
representation X=K1∪...∪KmwithK1,...,Km∈K.
The volume density VV,3ofΞis the expectationof
thevolumefraction of ΞinW,
VV,3(Ξ) =
/BXV3(Ξ∩W)
V(W),
V(W)>0.Thisdefinitionofthevolumedensitycanbe
extendedtothedensitiesoftheotherintrinsicvolumes.
The realisations of Ξintersected with aW,a>0
are poly-convex sets. Hence, the intrinsic volumes
Vk(Ξ∩aW),k=0,...,3,existandthe intrinsicvolume
densitiesV V,kofΞcanbedefinedbythelimits
VV,k(Ξ) =lim
a→∞
/BXVk(Ξ∩aW)
V(aW),k=0,1,2,
(cf.Schneider and Weil, 2008). In 3Dthe intrinsic
volume densities are (up to multiplicative constants)
thevolume density V V=VV,3, thesurface density or
specific surface area S V=2VV,2, thedensity of the
integralofmeancurvatureM V=πVV,1,andthedensity
oftheEulernumber χV=VV,0.
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ImageAnalStereol2009;28:179-185
The functional ρr(X)as definedin Equation (2) is
additive,translationinvariantandlocallyboundedand,
hence,its density
ρV,r(Ξ) =lim
a→∞
/BXρr(Ξ∩aW)
V(aW), (3)
exists and satisfies a Steiner-type formula, too, as ρr
does:
ρV,r(Ξ) =3
∑
k=0r3−kκ3−kVV,k(Ξ),r≥0,(4)
(seeSchneiderandWeil,2008,p.428).Nowitfollows
that
/bracketleftbiggd
drρV,r(Ξ)/bracketrightbigg
r=0=κ1VV,2(Ξ) =SV
and
/bracketleftbiggd2
dr2ρV,r(Ξ)/bracketrightbigg
r=0=2κ2VV,1(Ξ) =2MV.
In this sense we formally write S′
V=2MVand the
structure modelindex fSMIis givenby
fSMI=12VVMV
S2
V, (5)
which has a similar structure as a shape factor for
convex bodies:Consider the three isoperimetric shape
factors for K∈K
f1(K) =6√πV(K)/radicalbig
S3(K),f2(K) =48π2V(K)
M3(K),
f3(K) =4πS(K)
M2(K).
These shape factors are normalised such that
f1(Br) =f2(Br) =f3(Br) =1. Deviations from 1
describe various aspects of deviations from ball
shape. The shape factor f4(K) =f2(K)/(3f2
3(K)) =
3V(K)M(K)/S2(K)derived from f2andf3is
analogousto thestructuremodelindexwhichthuscan
be seenasashapefactorforrandomsets.
Note however, that S′
Vin this sense is in general
not the same as the derivative of SVderived from
an infintesimal dilation as defined by Hildebrand and
R¨ uegsegger (1997a). Roughly speaking, this is due to
the fact that the index function used in the definition
of the functional ρcounts signed surface points and
thus does not describe a dilation in the case of
overlapping grains. Nevertheless, the fSMIas defined
here coincides with the orignal SMI by Hildebrand
and R¨ uegsegger (1997a) for non-overlapping grains.For an example where the two concepts differ see the
following section.
For a system Ξof non-overlapping balls of
constant radius rwith ball density (mean number of
ballspervolumeunit) λwe get
fSMI(Ξ) =12VV(Ξ)MV(Ξ)
S2
V(Ξ)=12λ4
3πr3λ2π2r
λ216π2r4=4.
A macroscopically homogeneous system of plate-
like structures can only be obtained as a dilated
randomhyperplanesystem.Thuswehave MV=0and
consequently fSMI=0,too.
The third special case considered by Hildebrand
and R¨ uegsegger(1997a;b) are sphericalcylinders. Let
Ξbe a random system of non-overlapping infinite
spherical cylinders of constant radius r, that is a
random system of dilated straight lines. This can be
achieved e.g. by dilating a system of lines parallel to
the z-axis whose feet form a hard core point process
inthex-y-plane. LetΞhavelengthdensity(meantotal
lengthpervolumeunit) λ. ThenVV(Ξ) =λA=λπr2,
SV(Ξ) =λL=λ2πr,andMV(Ξ) =λπ,whereAandL
denote the section area and the section circumference
ofthecylinders.Thus
fSMI(Ξ) =12λπr2λπ
(λ2πr)2=3.
SMIFORBOOLEANMODELS
LetΞbe a homogeneous and isotropic Boolean
model in /CA3with typical grain X0and density λ.
That is,Ξ=/uniontext∞
i=1(xi+Xi), whereΦ={xi}∞
i=1is a
homogeneous Poisson point field, the Xiare i. i. d.
likeX0, isotropic, and independent of Φ. LetV,S,b
denotetheexpectationsofthevolume,thesurfacearea,
and the mean width of the grain X0, respectively, i.e.,
V= /BXV3(X0),S=2 /BXV2(X0)andb=1
2
/BXV1(X0).Then
thevolumefraction VV,thesurfacedensity SV,andthe
density of the integral of the mean curvature MVofΞ
can be expressed in terms of λ,V,S, andbbyMiles’
formulae:
VV(Ξ) =1−e−λV,
SV(Ξ) =e−λVλS,
MV(Ξ) =e−λV/parenleftbigg
2πλb−π2λ2
32S2/parenrightbigg
,
(Miles,1976),whichisaspecialcaseofSchneiderand
Weil(2008,p.389).By definitionoftheSMIwe get
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OHSERJET AL:Meshfree SMI
fSMI(Ξ) =12(1−e−λV)e−λV/parenleftbigg
2πλb−π2λ2
32S2/parenrightbigg
·e2λV1
λ2S2
=24π(eλV−1)/parenleftBigg
b
λS2−π
64/parenrightBigg
.
The dilation of a Boolean model is the same as
the Boolean model of the dilated grains ( cf.Chadœf
et al., 2008). Thus a derivative of SVcan also be
deducedfrom Miles’ formula. As already noted in the
previous section, this is not the same as S′
Vdefined
above. Consider the special case of the typical grain
X0=Brbeingaballofconstantradius r>0.Wehave
SV(Ξ) =e−λ4
3πr34πλr2
andthus
SV(Ξ)′=e−λ4
3πr3/parenleftbig
8πλr−(4πλr2)2/parenrightbig
while
2MV(Ξ) =e−λ4
3πr3/parenleftbig
8πλr−(π2λr2)2/parenrightbig
.
Nevertheless, all following considerations regarding
the significance of fSMIhold analogously for the SMI
in the sense of Hildebrand and R¨ uegsegger (1997a),
too.
In the special case of the typical grain X0=Br
being a ball of constant radius r>0 this further
simplifiesto
fSMI(Ξ) =3(eλ4
3πr3−1)/parenleftbigg1
λπr3−π2
8/parenrightbigg
.
Thus lim λ→∞fSMI(Ξ) =−∞. On the other hand,
fSMI(Ξ)>0 forλ<8/π3r3. Hence, already for one
model–theBooleanmodelwithconstantballradius–
fSMIcanattain awide rangeofvalues.
As a second example we consider now a Boolean
cylinder model formed by a Poisson line field with
each line dilated by a convex body K∈K. The
union of the resulting cylinders is a (generalised)
Boolean model. More precisely, let Φ={Li}∞
i=1be a
macroscopically homogeneous and isotropic Poisson
point field in the space of straight lines in /CA3. Let
K1,K2,...be i. i. d. convex bodies. Then the random
closed set Ξ=/uniontext∞
i=1(Li⊕Ki)is a Boolean model of
infinite straightcylinders.
In the special case of the Ki=Bribeing balls,
formulae for the densities of the intrinsic volumes
in terms of the model parameters were derived by
Davy(1978),seealsoOhserandSchladitz(2009).Lettheribe i. i. d. as r0and denote by A=π /BXr2
0the
mean cylinder section area and L=2π /BXr0the mean
circumference. The density of the Poisson line field is
λ.TheMiles’formulae forVV,SVandMVnowbecome
VV=1−e−λA
SV=λLe−λA
MV=/parenleftbigg
πλ−π2
32(λL)2/parenrightbigg
e−λA.
SpiessandSpodarev(2009)provedthesurfacedensity
formula for the anisotropic case and general Ki.
Hoffmann(2007a;b)derivedfurther generalisationsto
inhomogeneousPoissonline fields.
Inthecaseofthetypicalgrain Ki=Brbeingaball
of constant radius r>0 the Miles’ formulae further
simplify to
VV=1−e−λπr2
SV=e−λπr22πλr
MV=e−λπr2πλ/parenleftbigg
1−π3
8λr2/parenrightbigg
,
whichgivesforthe SMI
fSMI(Ξ) =12(1−e−λπr2)πλ/parenleftBig
1−π3
8λr2/parenrightBig
e−λπr2
4π2λ2r2/parenleftbig
e−λπr2/parenrightbig2
=3/parenleftBig
eλπr2−1/parenrightBig/parenleftbigg1
λπr2−π2
8/parenrightbigg
.
Thus, lim λ→∞fSMI(Ξ) =−∞whilefSMI(Ξ)>0 for
λ<8/π3r2.
Given the ranges for the SMI for Boolean models
of balls and infinite straight cylinders derived above,
there are clearly sets of parameters such that the
systems of overlapping balls and the cylinder system
havethesameSMI.AsaspecialcaseconsiderBoolean
modelsΞbofballswith pointdensity λbandfixedball
radiusrbandΞcofcylinderswithlengthdensity λcand
crosssectionradius rc.ThenfSMI(Ξb) =fSMI(Ξc) =0
if
λb=8
π3r3
bandλc=8
π3r2c.
Theseequationshold for instance for λb=1000,rb=
0.064,λc=286.68, andrc=0.03. Realisations of
these two models, which are obviously not plate-like,
areshownin Fig. 1.
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ImageAnalStereol2009;28:179-185
(a)
 (b)
Fig.1.VisualisationsofrealisationsofBooleanmodels
of spheresandcylinderswith theoreticalvalue f SMI=
0.(a)Booleanmodelofspheres,estimated /hatwidefSMI(Ξb) =
−0.022, (b) Boolean model of cylinders, estimated
/hatwidefSMI(Ξc) =−0.200.
APPLICATION
Thesectionaboveshowsthattheinformativevalue
offSMIis rather restricted. This holds however for all
shape factors. Nevertheless, fSMIis surely a valuable
tool for comparing similar microstructures. In the
following wewill discussseveralexamples.
GREENLAND FIRN(SINTERED SNOW)
During the densification of polar firn, significant
changes of the microstructure can be observed ( cf.
Freitaget al., 2004, and references therein). These
include a decrease of porosity with increasing depth
butalsochangesofthetopologicalstructureofthepore
space from a connected system of pore channels to a
systemofisolatedsphericalairbubbles.TheSMImay
be usedasameansto characterisethesechanges.
As an example we analysed several samples of
firn from the firn core B26 which was drilled during
the North Greenland traverse of the Alfred Wegener
Institute Bremerhaven in 1995. The borehole was
located at 77◦15’N, 49◦13’W. Five firn samples taken
from different depths within the ice core were imaged
using a portable µCT scanner (1074SR SkyScan)
inside a cold room at −25◦C. The analysis is based
on grey value images consisting of 4003pixels with
a pixel size of 40 µm. From these, binary images of
the pore system of the firn were obtained by global
thresholdingby Freitag etal.(2004). Visualisationsof
the firn samplesareshownin Fig.2.
(a) 56 m
 (b) 60 m
(c) 69 m
 (d) 72 m
(e) 74m
Fig. 2.Visualisations of reconstructed tomographic
images of firn samples from five different
depths. Samples and imaging: J. Freitag, Alfred-
Wegener Institute for Polar and Marine Research,
Bremerhaven.Theporesystemis visualised.
The estimated values given in Table 1 show an
increase of fSMIwith increasing depth. The starting
value around 3 indicates a cylindrical structure. The
visualisations of the deeper samples already show a
numberofisolatedsphericalpores.Therefore,afurther
increaseof fSMItowards4canbeexpectedwhengoing
deeperwithin thefirn core.
Table1.SMIforfirnsampleswith differentporosities.
depth[m] porosity[%] /hatwidefSMI
56 15.71 2.954
60 13.16 3.135
69 10.11 3.373
72 9.06 3.410
74 7.83 3.503
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OHSERJET AL:Meshfree SMI
TECHNICALFOAMS
Foams are used in an increasing number of
application areas including filters, heat exchangers
or sound absorbers. They are divided into open-cell
foams consisting of a connected system of struts
and closed-cell foams whose cells are bounded by
membrane-likewalls.However,alsomixedformswith
varying proportions of closed walls can be observed.
Thedegreeofclosednessofafoamplaysanimportant
role for its macroscopic properties. Therefore, easy
waysforitscharacterisationarehighlydesirable.Here,
weproposetheSMIasameasurefortheclosednessof
a foam.
(a)
 (b)
(c)
 (d)
Fig. 3.Visualisations of reconstructed tomographic
images of four foam samples. (a) Open aluminium
foam, sample: m-pore GmbH, imaging: Fraunhofer
IZFP, (b) Open nickel-chromium foam, sample:
RecematInt.(RCM-NC-2733.10),imaging:RJLMicro
& Analytic, (c) Partly open ceramic foam, sample:
FOSECO GmbH, imaging: Fraunhofer IZFP, (d)
Closed polymer foam, sample and imaging: R.
Schlimper,FraunhoferIWM.
WeestimatedtheSMIfromtomographicimagesofthe
followingfoamsamplesshowingdifferentproportions
ofclosedwalls:
–an open aluminium foam, 820 ×820×278pixels,
pixelsize64 .57µm,
–a nickel-chromium foam, used as sound absorber,
800×1600×1600pixels,pixelsize3 .14µm,–a ceramic foam, used for filtering metal melts,
670×670×270pixels,pixelsize70 .88µm,
–a closed polymethacrylimide (PMI) foam, used
as lightweight core material for sandwich
applications, 480 ×480×360 pixels, pixel size
10.21µm.
Visualisations of the tomographic images of the
foamsamplesareshowninFig.3.TheestimatedSMIs
are given in Table 2. None of the estimated SMIs is
nearthevalueforanidealcylinderstructure.However,
the two open foams have a significantly higher SMI
than the (partially) closed samples and the SMI of
the closed PMI foam is close to the expected value
0. Moreover, the difference between the strut shape
between the aluminium foam (round) and the nickel-
chromium foam (trilobal) is clearly reflected by the
SMI, too.
Table2.SMIforthefoam samples.
sample /hatwidefSMI
aluminium 2.188
nickel-chromium 1.695
ceramic 0.429
polymer 0.207
PAPER
The microstructure of paper determines important
properties like tensile strength or filtration properties.
Therefore, it has been studied for a long time,
however mainly based on 2d images. Here we use a
3Dimage of a recycling paper sample obtained by
synchrotron-based phase contrast microtomography
at beamline ID22 of the European light source
ESRF (employing an effective pixel size of 0.7 µm
corresponding to approx. 2 µm spatial resolution,
13 keV monochromatic photon energy). For further
details on the experimental setup and the phase
retrievalalgorithm applied see Weitkamp et al.(1999)
and Paganin et al.(2002), respectively. Most of the
cellulose fibres are collapsed and the paper contains
variousadditives.This causesthe microstructure to be
very irregular. Nevertheless,estimation of the SMI on
seven 2603pixel subvolumes yields the values 0.19,
0.37, 0.50, -0.08, 0.2, 0.14, 0.29, still indicating a
ratherplatelike structure.
Fig.4showsvisualisationsofthetwosampleswith
theminimal andmaximalestimatedSMI.
184
ImageAnalStereol2009;28:179-185
(a)/hatwidefSMI=0.50
 (b)/hatwidefSMI=−0.08
Fig. 4.Visualisations of two 2603pixel subvolumes of
the reconstructed tomographic image of a recycling
paper sample. Sample: Papiertechnische Stiftung
(PTS), imaging:A.Rack,ESRF,ID-22.
DISCUSSION
InspiredbyHildebrandandR¨ uegsegger’sstructure
model index we define in this paper a shape factor
formacroscopicallyhomogeneousrandomclosedsets.
This definition can be formulated without use of
discretised realisations. Second, we derive a simple
estimator for our SMI based on 3Dimage data.
Contrary to the originally proposed estimator, ours
does not require a surface meshing. Finally we prove
that the SMI is of value for the comparison of
similar microstructures while suffering from the same
shortcomingsasshapefactors forcompactsets.
Given the analogy between the SMI and the
isoperimetric shape factors pointed out above, other
shapefactorscouldbeexplored,too.However,adirect
replacement of V, S, and M in the formulae for the
isoperimetric shape factors by the respective densities
yields scale dependent quantities. In Hildebrand and
R¨ uegsegger (1997a), two other shape factors, the
structure volume exponent and the structure surface
exponent, are suggested. These additionally depend
on the thickness of the structure which is assumed to
be constant throughout the sample. This assumption,
however,is typicallynotmetbyrealmicrostructures.
ACKNOWLEDGEMENTS
Joachim Ohser and Claudia Redenbach were
supported by the FH3-programme of the German
Federal Ministry of Education and Research under
project grant 1711B06.Katja Schladitz was supported
by the German Federal Ministry of Education and
Research project 01 SF 0708-III-4a (Fraunhofer-
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