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Klattx, M.A., Schröder-Turk, G.E. and Mecke, K. (2017) Mean-intercept

anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative. Medical Physics.

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Mean-intercept anisotropy analysis of porous media.

II. Conceptual shortcomings of the MIL tensor definition and

Minkowski tensors as an alternative

Michael A. Klatt,1, 2, ∗ Gerd E. Schroder-Turk,3 and Klaus Mecke2

1Karlsruhe Institute of Technology (KIT), Institute of Stochastics,

Englerstraße 2, 76131 Karlsruhe, Germany

2Institut fur Theoretische Physik, Universitat Erlangen-Nurnberg,

Staudtstr. 7, 91058 Erlangen, Germany

3Murdoch University, School of Engineering & IT,

90 South Street, Murdoch, WA 6150, Australia

(Dated: April 7, 2017)

This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1002/mp.12280This article is protected by copyright. All rights reserved.

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Abstract

Purpose: Structure-property relations, which relate the shape of the microstructure to physical

properties such as transport or mechanical properties, need sensitive measures of structure. What

are suitable fabric tensors to quantify the shape of anisotropic heterogeneous materials? The mean

intercept length is among the most commonly used characteristics of anisotropy in porous media,

e. g., of trabecular bone in medical physics. Yet, in this series of two papers we demonstrate that

it has conceptual shortcomings that limit the validity of its results.

Methods: We test the validity of general assumptions regarding the properties of the mean-

intercept length tensor using analytical formulas for the mean-intercept lengths in anisotropic

Boolean models (derived in part I of this series), augmented by numerical simulations. We discuss

in detail the functional form of the mean intercept length as a function of the test line orientations.

Results: As the most prominent result, we find that, at least for the example of overlapping

grains modeling porous media, the polar plot of the mean intercept length is in general not an

ellipse and hence not represented by a second-rank tensor. This is in stark contrast to the common

understanding that for a large collection of grains the mean intercept length figure averages to an

ellipse. The standard mean intercept length tensor defined by a least-square fit of an ellipse is

based on a model mismatch, which causes an intrinsic lack of accuracy.

Conclusions: Our analysis reveals several shortcomings of the mean intercept length tensor

analysis that pose conceptual problems and severe limitations on the information content of this

commonly used analysis method. We suggest the Minkowski tensors from integral geometry as

alternative sensitive anisotropy measures. The Minkowski tensors allow for a thorough, compre-

hensive, and systematic approach to quantify various aspects of structural anisotropy. We show the

Minkowski tensors to be more sensitive, in the sense, that they can quantify the remnant anisotropy

of structures not captured by the mean intercept length analysis. If applied to porous tissue and

microstructures, this improved structure characterization can yield new insights into the relations

between geometry and material properties.

Keywords: shape analysis, fabric tensors, mean intercept length, mean chord length, Minkowski tensors,

anisotropic porous media

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From fractures in rocks to the trabecular bone structure, fabric tensors are needed to

characterize complex and disordered microstructure in both natural and man-made materi-

als1,2. Physical insight is often best achieved via a better understanding of the geometrical

properties. Prominent examples are relations between mechanical stability and anisotropy

of trabecular bone3. To describe anisotropic heterogeneous materials, tensorial structural

characteristics are needed to determine both the strength of the anisotropy and the preferred

orientation.

The mean intercept length (MIL) tensor is one of the most common approaches to quan-

tify the anisotropy of composite materials2,4. In the first paper of this series, we have

discussed analytical formulae for the MIL for the commonly used Boolean model for disor-

dered microstructures; (the Boolean model is also known as a fully penetrable grain system,

a homogeneous system of overlapping particles, a Poissonian penetrable grain model, or a

Poisson germ-grain model5). In the present paper, we discuss severe short-comings of the

MIL analysis for (bone-)microstructure characterization.

In a “mean-intercept analysis” of a heterogeneous two-phase medium, parallel test lines

are drawn through the sample. They intersect the interface between the two phases. The

isolated segment within one phase is called an intercept. The MIL L is the mean length

of these intercepts1. In stochastic geometry, the intercepts are called chords and hence the

mean intercept length is also known as the mean chord length5.

If the MIL varies with the orientation of the test lines, the medium has an anisotropic

distribution of the interface. The orientation of a test line is either described by a unit vector

u along the test line or by the angle ω between the test lines and the x-axis of the system.

Usually, the polar diagram of the MIL L(ω) is plotted, i. e., the MIL for each orientation,

where a deviation from a circular shape implies interfacial anisotropy; see Fig. 2 in paper I.

In 1974, Whitehouse examined the structure of trabecular bone6 and found empirically

that the polar diagram of the MIL L(ω) is similar to an ellipse. The standard MIL analysis

fits an ellipse to the MIL figure, which can then be represented by a tensor7

L−2(ω) = utMu (1)

with u = (cos(ω), sin(ω))t. The second-rank tensor M is positive definite, i. e., there exists

exactly one positive definite square root, which is indicated by√

M in the following. The

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MIL tensor is then often also defined by2:

H =(√

M)−1

. (2)

In the first paper of this series, we presented how the MIL can be analytically derived for

these overlapping grains and discuss the orientation and intensity dependence of the MIL.

Here we show that the MIL figure for anisotropic Boolean models is, in general, not an

ellipse, and therefore cannot necessarily be represented by a second-rank tensor; see Fig. 2

in paper I. A fit of an ellipse to the MIL figure would induce an unknown systematic error

and, e. g., strongly depend on the sampling of the test lines. There are severe limitations to

the MIL analysis:

1. We show that the MIL figure is in general distinctly different from an ellipse [see

Eq. (6) and Figs. 3 and 8, and Fig. 2 in paper I].

2. Moreover, standard line or intersection counting techniques to determine the MIL are

time-consuming, sensitive to noise, and depend on variations in the implementation,

see Refs.8,9.

3. Higher than second-rank tensors are sometimes needed to characterize the anisotropy

and predict, e. g., mechanical properties, see Ref.10,11.

4. The MIL analysis implicitly assumes an additional two-fold rotation axis in the het-

erogeneous material, see Ref.6.

5. The MIL analysis is limited to interfacial anisotropy, see Ref.4,9,12.

6. Here, we show how even systems with an obviously anisotropic interface can appear

perfectly isotropic w.r.t. the MIL; see Figs. 6 and 8.

While the MIL analysis was among the first measures of anisotropy and is today a standard

tool to characterize anisotropy in medical bone morphology, its inherent drawbacks call for

a more sensitive and especially systematic approach to morphology quantification.

We show how a correction of the model mismatch in the MIL analysis naturally leads to

a much more general family of integral geometric measures, the Minkowski tensors13. The

Minkowski tensors are sensitive, robust, and comprehensive shape measures based on a solid

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mophological theoretical basis in stochastic and integral geometry. Fast, linear-time algo-

rithms are available. They can be interpreted as moment tensors of the volume or interface

distributions and have already been successfully applied to many physical systems as sensi-

tive measures of anisotropy14–19. We here show that these measures fullfil all requirements

for an anisotropy analysis of microstructured hard- and soft tissues, like bone, but are not

plagued by the substantial drawbacks of the MIL approach.

In Sec. I, we compare the analytic results to simulations for an exemplary class of models

with an adjustable anisotropy. There we find that the MIL figure is not an ellipse and that

it cannot be represented by a second-rank tensor. We then discuss the further limitations

of the anisotropy quantification by the MIL. In Sec. II, we introduce the Minkowski tensors

as robust tensorial shape descriptors from integral geometry. We show how they allow for a

more sensitive anisotropy analysis compared to the MIL approach19.

I. MEAN INTERCEPT LENGTH OF ANISOTROPIC BOOLEAN MODELS

In the first paper of this series, we calculated the MIL for a very general class of Boolean

models. Here we discuss in detail the properties, information content and limitations of

the MIL. We demonstrate this for a parametric class of Boolean models ranging from an

isotropic model to perfectly aligned grains. An adjustable parameter chooses the orientation

bias of overlapping ellipses or rectangles. We explicitly evaluate the MIL as a function of

the orientation of the test lines and compare the results to numerical estimates.

A. Orientation biased Boolean models

To model two-dimensional heterogeneous materials with varying anisotropy, Schroder-

Turk et al. 15 and Horrmann et al. 20 introduced a model with overlapping grains (either

rectangles or ellipses) where both the aspect ratio, i. e., the elongation of the particles, and

the standard deviation of the orientation distribution, i. e., its anisotropy, is variable. Here,

for simplicity, the aspect ratio is fixed to 1/2.

The specific choice for the parametrized family of anisotropic orientation distributions of

the grains is depicted in Fig. 1. The angle θ between the main axis of the grain and the

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0

0.5

1

−π2 −

π4 0 π

4π2

P(θ)

θ

σ = 0.91σ = 0.50 θ

x

σ = 0.91

σ = 0.50

FIG. 1. Anisotropic Boolean models: the angle θ between the main axis of the grain and the

x-axis characterizes the orientation of the single grain. On the right-hand side, three differently

anisotropic probability distribution P(θ) are plotted: an isotropic, i. e., uniform distribution with

standard deviation σ = π/√

12 ≈ 0.91, a distribution with preferred direction θ = 0 and a standard

deviation σ = 0.50, and δ(θ) for perfect alignment with the x-axis. On the left-hand side, two

samples of Boolean models either isotropic (top) or with orientation bias (bottom) are depicted.

x-axis follows a cosine distribution:

P(θ) = Zα cosα(θ) (3)

with Zα = Γ(1 + α/2)/(√πΓ(1/2 + α/2)) with θ ∈ (−π/2, π/2]; see Fig. 1. The anisotropy is

varied by choosing the parameter α: α = 0 produces to a uniform, i. e., isotropic, distribution,

and α = ∞ leads to δ distribution, i. e., all grains are aligned along the x-axis. For a more

intuitive characterization of the anisotropy distribution, we parametrize it in the following

by the standard deviation σ of the orientation probability distribution P(θ), which is a

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invertible function of the parameter α:

σ2=

∫ π2

− π2

dθ θ2P(θ) .

In this parameterization, σ = 0 corresponds to perfect alignment of the major axes with the

x-axis and σ = π/√

12 ≈ 0.91 to an isotropic, i. e., uniform orientation distribution.

B. Monte Carlo sampling

The key results of this series of articles are obtained by analytical derivation without any

numerical input required. Numerical calculations and Monte Carlo methods are solely used

to demonstrate the correctness of these analytic results.

To simulate a sample of a Boolean model, first the number of particles inside the simu-

lation box has to be determined. It is a random number following the Poisson distribution.

Its mean equals the mean number of particles, i. e., the intensity ρ times the size of the

simulation box. The particles are then randomly distributed inside the quadratic simulation

box using periodic boundary conditions.

We simulate Boolean models with a varying degree of anisotropy, i.e., orientation bias.

The anisotropy parameter is either α = 0 (σ ≈ 0.91), α = 3 (σ ≈ 0.50), or α = ∞ (σ = 0).

The particles are either ellipses or rectangles. Their semi-axes lengths are p = 1 and q = 1/2.

The linear size of the simulation box is 50p.

The Boolean model is then intersected with parallel test lines with constant spacing for

different angles ω between the test lines and the x-axis; see Fig. 2 in paper I. The number of

intersections within the unit square are counted; the total length of the test lines is divided

by the number of intersections; averaging over many samples provides the MIL L.

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C. The MIL tensor is in general not a tensor

According to Eq. (19) in paper I, the mean intercept length L of the Boolean model as a

function of the orientation ω of the test lines is analytically given by:

L(ω) =A

S ⊥[ω]︸ ︷︷ ︸

orientation

dependent

· eρA

2ρA︸︷︷︸

intensity

dependent

, (4)

where A is the area of a single grain and S ⊥[ω] is the average length of the projection of

a single grain to a line perpendicular to the test line in direction ω. Note that the MIL is

proportional to the inverse of S ⊥[ω]. This already alludes that in general the MIL cannot

be completely represented by a second-rank tensor.

The analytic calculation of the MIL for Boolean models with rectangles or ellipses as

a function of the orientation ω of the test lines (see Fig. 2 in paper I) using Eq. (4) is

now straightforward. The length of the perpendicular projection is for a single ellipse E or

rectangle R with orientation θ given by

S ⊥[ω; ellipse] = 2

√

p2 sin2(ω − θ) + q2 cos2(ω − θ) (5)

S ⊥[ω; rectangle] = 2 (p sin |ω − θ| + q cos |ω − θ|) , (6)

respectively. The average over the orientation distribution is

S ⊥[ω] =

∫ π/2

−π/2dθ P(θ) · S ⊥[ω; K] . (7)

For σ = 0 (α = ∞) the orientation distribution P(θ) is a δ distribution and S ⊥[ω] is equal

to S ⊥[ω; K] with θ = 0. For σ = π/√

12 (α = 0) the integral over the isotropic orientation

distribution can be solved explicitly; for rectangles, S ⊥[ω] = 4

π(p+q) and for ellipses, S ⊥[ω] =

4

πpE[1 − q2

/p2] with E[m] =∫ π/2

0dϕ

√

1 − m sin2 ϕ the complete elliptic integral of the second

kind.

Inserting Eq. (7) and the area of an ellipse (V[E] = πpq) or a rectangle (V[R] = 4pq) in

Eq. (4) provides the MIL L(ω).

Usually, in a MIL anisotropy analysis the polar plot of the MIL L(ω) is assumed to be an

ellipse (or an ellipsoid in three dimensions)2,6,7. The polar representation r(ω) of an ellipse

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with the origin at the center of the ellipse is

r(ω) =1

√

b−2 sin2 ω + a−2 cos2 ω(8)

with semi-axes a and b. If the MIL polar plot was an ellipse, the functional form of L(ω)

would be of the form in Eq. (8). From Eqs. (5)–(7) we find that an anisotropic polar diagram

of the inverse of the MIL can only be an ellipse for aligned overlapping ellipses. For other

anisotropic Boolean models, the polar figure is distinctly different.

Figures 2 and 3 compare the MIL figures of overlapping ellipses and rectangles:

• Figure 2 is a polar diagram of the MIL of the Boolean model with ellipses. The MIL

is computed for perfectly aligned ellipses σ = 0, for partially aligned ellipses σ = 0.50

and for an isotropic system σ = 0.91 – for each system a sample of a Boolean model

is depicted above the polar plot. The analytic curves (lines) are compared to the

numerical estimates (dots; the error bars are smaller than the point size). Because

the polar plot of the MIL is by definition point symmetric w.r.t. the origin6, the

numerical data is shown only for ω = [−π/4; 3π/4]. The numerical data is in perfect

agreement with the analytic formula. Only in the special case of aligned ellipses, the

polar diagram of the MIL is indeed an ellipse. Inserting Eq. (5) with θ ≡ 0 in Eq. (4)

yields a polar representation of L(ω) according to Eq. (8) with a = eρA/(4qρ) and

b = eρA/(4pρ). If the ellipses are only partially aligned (σ = 0.50), the polar diagram is

not exactly an ellipse, but for a Boolean model with ellipses the deviation of the MIL

figure from an ellipse is only small.

• The deviations become obvious for non-elliptical grain shapes. Figure 3 is the same

polar diagram for rectangles; see also the inset of Fig. 2 in paper I, which shows the

MIL figure of rectangles following an orientation distribution with standard deviation

σ = 0.22. Again the numerical data is in perfect agreement with the analytic formula.

If the rectangles are aligned, the polar diagram is a rhombus, and also for weaker

alignment the MIL figure is distinctly different from an ellipse. (Another example of

a non-elliptic polar plot is presented in Sec. II B in Figure 8; parabola shaped grains

with fixed orientation lead to an even non-convex MIL figure.) For the Boolean model,

a least square fit of an ellipse to the data is not justified. It would induce unknown

systematic errors. Equation (1) does not hold. The polar plot of L−2(ω) deviates from

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-1

0

1

-3 -2 -1 0 1 2 3

L(ω)u

y

L(ω)ux

σ = 0.91

σ = 0

σ = 0 σ = 0.50 σ = 0.91

FIG. 2. Polar plot of the MIL of a Boolean model with ellipses for three differently anisotropic

samples. The anisotropy is characterized by the standard deviation σ of the orientation distri-

bution: σ = 0 corresponds to perfect alignment and σ = 0.91 to an isotropic system. The lines

depict the analytic curves given by Eq. (4) using Eqs. (3)–(7). Although similar, MIL figure for

σ = 0.50 is not an ellipse. Samples of according Boolean models are shown above; the mean solid

area fraction is Φ = 0.5.

the characteristic dumbbell shape. In general, the MIL figure cannot be represented

by a second-rank tensor. In this sense, the MIL tensor is actually not a tensor.

• The non-tensorial nature of the MIL figure contradicts a common assumption that a

large number of objects not all parallel to each other produce on average an elliptic

global MIL figure. The latter turns out to be not only sensitive to the degree of

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-1

0

1

-3 -2 -1 0 1 2 3

L(ω)u

y

L(ω)ux

σ = 0.91

σ = 0

σ = 0 σ = 0.50 σ = 0.91

FIG. 3. Polar plot of the MIL of a Boolean model with rectangles; for details see Fig. 2. The polar

plot of the MIL is a rhombus and a least square fit of an ellipse to the data is not justified.

alignment, but also to the shape of the individual grains, as discussed in the first

paper of this series.

For example, Luo et al. already derived that for a planar N-net system the polar plot

of the MIL is a convex polygon21. We have shown here that for a realistic model of porous

media22–26 the MIL figure is far from being an ellipse.

A similar analysis of further stochastic models for bone geometries (such as level sets of

random fields) or indeed an analysis of high-resolution tomography data is needed to shed

light on the tensorial or non-tensorial nature of the MIL for realistic bone structures. There

are first indications, for example, Ketcham and Ryan 27 reported statistically significant

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deviations from an ellipsoidal shape in experimental measurements of the MIL of trabecular

bone.

The fabric tensor is based on a least-square fit that is not justified and suffers from an

unknown systematic error. The model mismatch causes an intrinsic lack of accuracy. The

analysis of the MIL as a function of the orientation of the test lines is well-defined, but the

following sections, which discuss its characterization of the Boolean model, reveal inherent

drawbacks of the MIL analysis, calling for an alternative approach to sensitively measure

anisotropy.

D. Intensity dependence and anisotropy index

The main purpose of the MIL analysis is to quantify relative variations of the MIL with

orientations u of the test lines, but also the absolute size of the intercept lengths is interesting

for relating structure to the mechanical properties25,28,29 or in the reconstruction of two-

phase random media30,31. As we have shown in the first paper of this series, the intensity

dependence and the orientation dependence of the MIL factorize for Boolean models. The

MIL normalized by the orientation dependent factor is a function of the solid area fraction

Φ only and independent of the grain distribution, i. e., the explicit Boolean model:

L

L(1)=

eρV

2ρV= − 1

2(1 − Φ) ln(1 − Φ). (9)

Figure 4 plots this normalized MIL L

L(1) as a function of the solid area fraction Φ and compares

the analytic curve to the numerical estimate. Also the simulated results show the perfect

agreement for isotropic or aligned Boolean models with either ellipses or rectangles. As

expected, the MIL diverges for both vanishing porosity Φvoid = 1 −Φ (Lvoid → 0 and Lsolid →∞) and vanishing solid area fraction Φ (Lvoid → ∞ and Lsolid → 0).

Because of the separation in an orientation and an intensity dependent factor in Eq. (19)

in paper I, the ratio of the MIL in y- and in x-direction, which is the preferred direction of

the orientation distribution function P(θ), is independent of the intensity and thus of the

solid area fraction Φ. The factor from Eq. (9) cancels out. This ratio βMIL := L(π/2)/L(0) is

therefore a measure of the inherent anisotropy of the grain distribution. This is shown and

compared to the simulation results in Fig. 5. In the case of aligned ellipses, where the MIL

figure is an ellipse, this ratio is equivalent to various definitions of the degree of anisotropy

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0

10

20

0 0.2 0.4 0.6 0.8 1

L/L

(1)

Φ

Ellipses σ = 0.91Ellipses σ = 0

Re tangles σ = 0.91Re tangles σ = 0

FIG. 4. (Color online only) MIL L as a function of the solid area fraction Φ. The orientation and

intensity dependence factorize. If the MIL is normalized by its orientation dependent factor [given

by Eq. (20) in paper I], it becomes independent of the grain distribution. The solid line shows the

analytic curve from Eq. (9). It is compared to the numerical estimates for both Boolean models

with ellipses (dots) and with rectangles (squares) with aspect ratio 1/2 for test lines in x-direction.

In each case isotropic samples σ = 0.91 (orange) and Boolean models with aligned grains σ = 0

(blue) are simulated. The error bars are smaller than point size, and the numerical estimates are

in perfect agreement with the analytic curve. The intensity dependence of MIL is the same for all

homogeneous Boolean models.

(DA); common definitions27,28,32,33 are βMIL = 1/DA or DA = 1− βMIL. This index quantifies

an interfacial anisotropy of the system determined by both the aspect ratio of the grains

and the orientation distribution function P(θ). βMIL = 0 corresponds to perfect anisotropy,

i. e., an effectively one-dimensional heterogeneous material in y-direction, simply a stacking

of layers (or lines) in x-direction with varying height; therefore, L(0) → ∞. βMIL = 1

corresponds to an isotropic model w.r.t. the MIL analysis.

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0

0.2

0.4

0.6

0.8

1

0 0.25 0.5 0.75 1

βMIL(Φ

)

Φ

σ = 0.91

σ = 0.50

σ = 0

0.5

0.75

1

0 0.4 0.8

βMIL(σ)

σ

Ellipses

Re tangles

FIG. 5. The ratio βMIL of the MIL in x- and in y-direction as a function of the solid area fraction Φ

for differently anisotropic systems. The anisotropy is characterized by the standard deviation σ of

the orientation distribution; see Fig. 2. The lines depict the analytic curves, and dots and squares

represent Boolean models with ellipses or rectangles with aspect ratio 1/2. Because of the separation

in an orientation and an intensity dependent factor the ratio βMIL is constant for all porosities Φ.

The inset shows βMIL as a function of the anisotropy σ. It cannot distinguish overlapping rectangles

from ellipses if they are aligned (σ = 0).

However, the MIL analysis is insensitive, in the sense, that a Boolean model can appear

perfectly isotropic in the MIL analysis, although the system is obviously anisotropic, as

discussed below. Already Figure 5 shows that the index βMIL cannot, for example, distinguish

the anisotropy of ellipses or rectangles (with semi-axes q and p) with a fixed orientation

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(σ = 0); in both cases, βMIL =p/q.

In Sec. II B, we even present an Boolean model with an anisotropic interface, but with a

perfectly isotropic MIL anisotropy index βMIL; see Fig. 8.

E. Deficiencies of anisotropy characterization using the MIL analysis

Apart from the model mismatch for the standard MIL fabric tensor, we have found in

Sec. ID that it appears to be a rather insensitive structural descriptor. In this section, we

look in detail at the strengths and disadvantages of the MIL approach in general.

First, it might be insufficient to consider only a second-rank tensor and thus assume, like

in Eq. (1), that the material is orthotropic6,7. For some materials higher rank tensors are

needed to characterize the anisotropy and predict mechanical properties10,11.

One possible approach to characterize directional data by arbitrary rank fabric tensors

is to approximate the polar figure by dumbbell shaped polynomials. A statistical test can

then select the appropriate model and thus detect significant higher rank contributions.

Following the work of Kanatani 34, but slightly generalizing the concept to non-normalized

functions f (u), a distribution of directional data, i. e., a polar function f (u), can be approx-

imated by dumbbell shaped polynomials

Pk(F, u) :=

k∑

i1 ,...ik=1

Fi1,...ik ui1 . . . uik .

For rank k = 2 the polynomial is P2(F, u) = utFu, i. e., the same as in Eq. (1) for L−2. A least

square fit of the expansion to the empirical distribution minimizes∫

Sd−1du [Pk(F, u)− f (u)]2,

assuming an isotropic distribution of test lines.

However, then all tensors F of any rank k are needed in order to perform a statistical

model selection, e. g., a likelihood-ratio test, to choose the appropriate model for the data.

This might be impractical for applications, but we can derive a connection between these

least-square fits and the moment tensors N of f (u):

Ni1 ...ik := E[ui1 . . . uik ] =

∫

Sd−1

du f (u) · ui1 . . . uik .

For example, for a point symmetric function f (u), the second-moment tensor is equivalent

to the covariance tensor.

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The fabric tensor F is proportional to

F ∝ N − 1

5tr(N) 1 in 3 D,

F ∝ N − 1

4tr(N) 1 in 2 D,

where 1 is the identity tensor (or unit tensor).

In contrast to the least-square fits, which need the assumption of a model and are unjus-

tified without the model selection, the use of moment tensors is a well-defined alternative

(see also Sec. I F).

A fundamental restriction of the MIL figure is its limitation to interfacial anisotropy. A

structure may appear perfectly isotropic w.r.t. its boundary, while being, e. g., obviously

anisotropic w.r.t. its volume distribution4,9,12. The MIL analysis does not allow for a sys-

tematic analysis of anisotropy w.r.t. different geometrical properties.

Moreover, the MIL analysis is insensitive even in detecting interfacial anisotropy. Not

only, that the anisotropy index βMIL defined above cannot distinguish aligned rectangles from

aligned ellipses, but obviously anisotropic systems can appear perfectly isotropic w.r.t. the

MIL analysis, e. g., for a Boolean model with aligned Reuleaux triangles or any other curve

of constant width. Further famous examples are the British twenty and fifty pence coins,

which are equilateral heptagons with curved sides such that the width is constant in all

directions. Figure 6(a) shows a Boolean model with aligned Reuleaux triangles. A Reuleaux

triangle is bounded by three circular arcs whose sides are the sides of an equilateral triangle.

Constant width means that the length S ⊥ of its perpendicular projection is independent of

the angle ω between the test lines and the fixed orientation of the Reuleaux triangle. From

Eq. (4) then follows that the MIL is a constant and the MIL figure a circle—see Fig. 6(b).

The Boolean model with aligned grains appears perfectly isotropic in the MIL analysis.

Furthermore from a practical point of view, standard line or intersection counting tech-

niques to determine the MIL are time consuming, sensitive to noise9. Moreover, variations

in the implementation can strongly affect the results8. The lineal approach very strongly

depends on the distribution of orientations ω of test lines. Even for the natural choice of

uniform orientation sampling, i. e., with a constant spacing between the orientations ω, the

principal directions can strongly vary if a relatively low number of test lines is used27.

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(a)

0.8

0

0.8

0.8 0 0.8

L(ω)u

y

L(ω)ux

MIL

S⊥ ≡ 1

ω = 30◦ ω = 0◦

(b)

FIG. 6. Insensitivity of the MIL analysis for a Boolean model with aligned Reuleaux triangles:

an obviously anisotropic sample (a) produces a circle in the MIL figure (b), i. e., appears perfectly

isotropic, because the Reuleaux triangle has a constant width, i. e., the projected length S ⊥ is

independent of the angle ω between the test lines and the fixed orientation of the Reuleaux triangle.

From Eq. (4) follows that the MIL is a constant L(ω) = const.

F. Generalized anisotropy measures

Several extensions of the MIL tensor9,35 and various alternative lineal measures4,12,27,32,33,36

have been proposed. However, they do not offer a systematic approach to characterize dif-

ferent geometric aspects. A porous medium can, for example, be isotropic w.r.t. the volume

but anisotropic w.r.t. the interfacial distribution or vice versa15.

Here we derive how an approach to correct and improve the MIL analysis naturally leads

to replacing it by a more general framework, the family of the Minkowski tensors.

A least-square fit of an ellipse to the MIL figure is in general not justified, neither a fit

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of utFu to L−2(u). In contrast to this, the covariance is, as explained above, an alternative,

well-defined fabric tensor9

Cov(L · u) =

∫

Sd−1

du L2 u ⊗ u (10)

with u ⊗ u the tensor product of the normal vectors, which is represented by a matrix with

ni ·n j as the entry in row i and column j. Note that the mean of L ·u vanishes because of the

point symmetry w.r.t. the origin. The covariance tensor is closely related to the so-called

orientation matrix27,37, but depending on the convention a different normalization might be

applied.

The definition of the covariance tensor avoids the systematic errors of an inappropriate

least-square fit, but suffers from larger statistical fluctuations27. A more robust geometrical

measure is needed, which should also resolve the other drawbacks discussed in Sec. I E.

The MIL is proportional to the average of the inverse of the number of intersections m of

the test lines with the interface between solid and void phase

L = E

[

Lm

]

∝ E[

1

m

]

with L the total length of the test lines and E[.] denoting the expectation w.r.t. different

realizations. The MIL analysis is related to the covariance tensor Cov(E[

1

m

]

·u) w.r.t. varying

directions u.

It is natural to consider to replace the mean of the inverse E [1/m] by the average of the

number of intersections m. Note, that this is not simply the inverse of E [1/m], because the

mean of the inverse is strictly greater than the inverse of the mean (except if m is fixed, i. e.,

not a random number)

E

[

1

m

]

>1

E [m],

which is an example of Jensen’s inequality (which holds not only for the inverse but for

convex functions in general)38. A standard estimator of E [m] is the arithmetic mean, and

for E [1/m] it is the inverse of the harmonic mean.

While this replacement seems to be a small step, it is actually a big step from an integral

geometric point of view. In contrast to the average of the inverse, the expected number of

intersections is an additive quantity. It means that for two disjoint systems with average

number of intersections E [m1] and E [m2], the expected total number of intersections is

simply E [m1 + m2] = E [m1]+ E [m2]. Alesker’s theorem states that any additive, continuous

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and motion-covariant tensor-valued functional on convex bodies can be expressed by a linear

combination of Minkowski tensors multiplied with appropriate unit tensors39,40. In short,

this means that the tensor

Cm :=

∫

Sd−1

du E [m1] u ⊗ u (11)

can be expressed by Minkowski tensors. The tensor Cm is equal to a linear combination of

a Minkowski tensor W0,2

1(which is defined below, see Tabs. I and II) and the unit tensor

multiplied by the surface area (or the perimeter for d = 2). The coefficients are explicitly

given by so-called Crofton formulas for Minkowski tensors41,42. For local versions of these

Crofton formulas, see Ref.43.

The change from a single rather insensitive measure Cov(E[

1

m

]

· u) to the additive alter-

native opens up the broad generalization to the whole family of Minkowski tensors, versatile

and robust shape descriptors from integral geometry13. They can provide a systematic ap-

proach to structure characterization and anisotropy quantification14,15.

Note that there are also several other common structure tensors that are closely related

to the Minkowski tensors and which can easily be generalized to become Minkowski ten-

sors34,44,45. Such generalizations would then allow to take advantage of the mathematical

foundations and theorems for Minkowski tensors, as well as their implementations that are

ready to use.

II. ALTERNATIVE ANISOTROPY MEASURES BASED ON MINKOWSKI TEN-

SORS

The scalar Minkowski functionals and their generalization, the Minkowski tensors, are

defined as volume or surface integrals and thus allow for an intuitive characterization of

random spatial structures. They correspond, for example, to tensors of inertia or characterize

the distribution of the normal vectors on the boundary of the heterogeneous material.

The Minkowski tensors allow for a systematic and sensitive anisotropy analysis of differ-

ent geometrical aspects, like volume, surface, or curvature. They are defined in both two

and three dimensions for arbitrary rank and need no assumptions about the heterogeneous

material or its symmetry19.

Integral geometry provides a rigorous, mathematical foundation of the Minkowski tensor

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formalism as well as insights for applications. In mathematics literature, the Minkowski

functionals are also called intrinsic volumes, and the Minkowski tensors are also called tensor

valuations.

A. Definition and applications

The Minkowski functionals are integrals over either a domain K or its boundary ∂K (see

Fig. 7) appropriately weighted with its curvature. For two and three dimensions they are

listed in Tables I and II. There is an intuitive interpretation of the Minkowski functionals:

in two dimensions, they are the area, the perimeter, and the Euler characteristic and in

three dimensions the volume, the surface area, the integrated mean curvature, and the Euler

characteristic. The latter is a topological constant, i. e., it is a measure of connectivity. For

a compact body K, the Euler characteristic is in two dimensions the number of components

minus the number of holes; in three dimensions, it is equal to the number of components

minus the number of tunnels and plus the number of spherical cavities.

These scalar measures are naturally generalized to the Minkowski tensors by including an

integral over the tensor products of the position vector r and the surface normal vector n; see

0

r

u

Kux

uy

FIG. 7. The Minkowski tensors of a domain K are tensorial shape measures that are defined as

volume and surface integrals. On the left-hand side, a position vector r in K and a normal vector

u at the boundary ∂K are depicted. Their tensor products are used in the integrals in Tables I and

II. On the right-hand side, the distribution of normal vectors for domain K is shown.

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Fig. 7. The mutually linear independent functionals and Minkowski tensors are summarized

in Tables I and II for two or three dimensions, respectively.

The Minkowski tensors using the position vector are closely related to tensors of inertia

where the mass is located in the region of integration and probably weighted by the curvature.

For example, there are intuitive interpretations of the Minkowski tensors of polytopes in three

dimensions: W2,0

0contains the information of the tensor of inertia of the solid object, W

2,0

1of

a hollow object where the mass is located in the shell, W2,0

2and W2,0

3if the mass is distributed

on the edges or vertices, but weighted with the opening angels.

The Minkowski tensor W2,0

1is proportional to the moment (or covariance) tensor of the

distribution of normal vectors – see Fig. 7; W2,0

1is both equal to the moment and the

covariance tensor, because the mean W0,1

1=

∫

∂KdA n vanishes for closed bodies due to the

envelope theorem72. W0,2

2is proportional to the according moment tensor weighted by the

curvature distribution. In contrast to the tensors of inertia, the moment tensors of the

normal distributions are translation invariant.

The Minkowski functionals and tensors are robust, efficient, and versatile structure mea-

sures, which have been successfully applied in physical19,46 and biological systems47,48 on all

length scales from nuclear physics49,50, over condensed and soft matter51,52 or plasma53, to

astronomy and cosmology54–56, and to pattern analysis57.

The Euler characteristic has already been widely applied in medical physics to characterize

the connectivity of trabecular bone4,28,58,59, and Rath et al. 60 have also used the other

Minkowski functionals to characterize the structure of trabecular bone. However, while the

scalar functionals are rotation invariant, the Minkowski tensors provide anisotropy indices

that are explicitly designed to quantify anisotropy14,15,61, in a more robust fashion than the

MIL analysis.

Minkowski tensors are defined for two and three dimensions, actually for arbitrary dimen-

sions d. Rotational integral geometry provides local stereological estimators of Minkowski

tensors and relations for Minkowski tensors of planar sections62–64.

Horrmann et al. 20 analytically derived the expectation value of the translation invariant

Minkowski tensors, like W0,2

1, for Boolean models. The global average of the Minkowski

tensors can also be expressed by local characteristics of a single grain. This allows for an

approximation of the average shape of the typical grain based on the Minkowski tensors of the

Boolean model. The anisotropy index is, like for βMIL, independent of the solid area fraction.

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Functionals Tensors

Area Moment solid

W0 =

∫

KdA W

k,00=

∫

KdA rk

Perimeter Moment hollow Normal dist.

W1 =

∫

∂Kdl W

k,01=

∫

∂Kdl rk W

0,k1=

∫

∂Kdl nk

Euler charact. Moment vertices

W2 =

∫

∂Kdl κ W

k,02=

∫

∂Kdl κ rk

TABLE I. Definition of the Minkowski functionals and tensors of rank k in two dimensions evaluated

for a domain K, where κ is the local curvature on ∂K, r are the position vectors in K, and n are the

normal vectors on the boundary ∂K; see also Fig. 7. rk=⊗k

i=1r = (ri1 . . . rik )i1 ...ik (or nk

=⊗k

i=1n)

is the tensor product; for k = 2, for example, r2= r⊗ r is represented by a matrix with ri · r j as the

entry in row i and column j.

It quantifies inherent anisotropy of the Boolean model. For parametric anisotropic Boolean

models like those defined in Sec. IA, Schroder-Turk et al. 15 and Horrmann et al. 20 have

estimated the Minkowski tensors for pixelated or triangulated representations, respectively.

Moreover, Horrmann et al. 20 have derived and tested an unbiased estimator of the model

parameters based on the measurement of a Minkowski tensor.

For Boolean models, not only the mean values of the Minkowski functionals and ten-

sors but also their second moments and joint probability distributions are known analyti-

cally54,65–67.

The Minkowski functionals have already been used to adjust a Boolean model to an

experimental structure, resulting in an excellent match of the mechanical and transport

properties26,68. They have also been used to predict properties of nanoscale flow through

porous materials69.

A comprehensive introduction to the Minkowski tensors as anisotropy indices and exem-

plary applications can be found in Refs.14,15.

We show here how these shape characteristics resolve the drawbacks of the MIL analysis

pointed out above.

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Functionals Tensors

Volume Moment solid

W0 =

∫

KdV W

k,00=

∫

KdV rk

Surface area Moment hollow Normal dist.

W1 =

∫

∂KdA W

k,01=

∫

∂KdA rk W

0,k1=

∫

∂KdA nk

Mean curvature Moment wireframe Curvature dist.

W2 =

∫

∂KdA

κ1+κ22

Wk,02=

∫

∂KdA

κ1+κ22

rk W0,k2=

∫

∂KdA

κ1+κ22

nk

Euler charact. Moment vertices

W3 =

∫

∂KdA κ1κ2 W

k,0

3=

∫

∂KdA κ1κ2 rk

TABLE II. Definition of the Minkowski functionals and tensors of rank k in three dimensions

evaluated for a domain K, where κ1 and κ2 are the principle curvatures on ∂K, r are the position

vectors in K, and n the normal vectors on the boundary ∂K; see also Fig. 7. rk=⊗k

i=1r =

(ri1 . . . rik )i1 ...ik (or nk=⊗k

i=1n) is the tensor product; for k = 2, for example, r2

= r⊗r is represented

by a matrix with ri · r j as the entry in row i and column j.

B. Sensitive measures of anisotropy

In contrast to the line or intersection counting techniques for the MIL analysis which are

sensitive to noise, the Minkowski tensors are defined as volume and surface integrals, which

are robust against noise. The Minkowski tensors are additive and conditional continuous,

i. e., continuous on convex bodies. Therefore, small convex fluctuations due to noise do not

greatly affect the outcome.

Because of the additivity, the tensor valuations can be computed by summing up local

contributions listed in a look-up table19,56,57. Such an algorithm scales linearly with the sys-

tem size. Its computation time is of the order O(Np) with Np the number of pixels, while for

standard implementations of the MIL analysis the computation time is of the order O(NL ·Np)

with NL the number of test lines9. Free software for the Minkowski tensors based on fast

linear-time algorithms are available for both two and three dimensions and for both pixelated

and triangulated data. The free software papaya and karambola for two or three dimen-

sions, respectively, is available at http://www.theorie1.physik.fau.de/research/software.html

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Exemplary analyses of different systems are also presented there.

Tables I and II show how the family of Minkowski tensors allows for a systematic analysis

of anisotropy w.r.t. volume, surface, or curvature. While the sample might be isotropic w.r.t.

one of the measures, the Minkowski tensors can still detect the anisotropy w.r.t. another

quantity15. Examplary systems of discs that form voids in the plane4,9 are always isotropic

w.r.t. the MIL tensor, because the MIL analysis can only detect interfacial anisotropy.

However, such a system might be anisotropic w.r.t. its mass-distribution. The volume

Minkowski tensor W2,0

0, which is related to the tensor of inertia, can quantify this anisotropy.

We have introduced the anisotropy index βMIL similar to previously defined indices for

the Minkowski tensors15. For example, β0,2

1is the ratio of the minimal to maximal eigenvalue

of W0,2

1. Figure 8 compares the two anisotropy indices for different Boolean models.

The Minkowski anisotropy index β0,2

1only coincides with βMIL for aligned rectangles, for

which both are equal to the aspect ratio. For ellipses with aspect ratio 1/2 and a varying

bias in the orientation distribution they provide slightly different degrees of anisotropy.

Therefore, in contrast to βMIL, the Minkowski index β0,2

1does discriminate a Boolean model

with ellipses with a fixed orientation from aligned overlapping rectangles or rhombuses with

the same aspect ratio.

The higher sensitivity of the Minkowski analysis can also be demonstrated by a Boolean

model with parabola shaped grains, which is depicted at the top of the right-hand side in

Fig. 8. The boundary of such a grain is formed by two parabolas, y = 1 − x2 and y = x2 − 1

for x ∈ [−1; 1] (for a grain centered at the origin). A parabola is not symmetric in x and

y. Therefore, the interface of the grains and thus of the Boolean model is anisotropic in x-

and in y-direction. This is also visualized in Fig. 8 by green normal vectors at the boundary.

However, the MIL is the same for test lines along x and y: L(x) = L(y), because the length

of the projections along these lines are the same. This can be seen in the polar MIL plot at

the bottom of the right-hand side of Fig. 8; therefore, βMIL = 1, the MIL anisotropy index

indicates perfect interfacial anisotropy. In contrast to this, the index of the second-rank

Minkowski tensor W0,2

1captures the interfacial anisotropy: β0,2

1< 1 (△).

In contrast to the MIL analysis, no assumption about the heterogeneous material or

its symmetry are needed for the Minkowski tensors, because the latter can be defined for

arbitrary rank k. For example, a tensor of rank four is needed to distinguish cubic from

spherical symmetry, i. e., characterize the anisotropy of a Boolean model with aligned cubes.

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0.4

0.6

0.8

1

0.5 0.75 1

β0,2

1

βMIL

isotropi

ellipses

re tangles

aligned rhombuses

-1.5

0

1.5

-1.5 0 1.5

L(ω)u

y

L(ω)ux

MIL

FIG. 8. (Color online only) Comparison of the Minkowski tensor anisotropy index β0,21

to the MIL

degree of anisotropy βMIL for different Boolean models: aligned rectangles with varying aspect ratio

(dashed blue line), ellipses with aspect ratio 1/2 but varying bias in the orientation distribution (solid

line colored yellow), the dots correspond to particles with fixed orientations, rectangles (�), ellipses

(◦), rhombuses (⋄), and parabola shaped grains (△). The latter appear perfectly isotropic w.r.t.

βMIL, but the Minkowski index β0,21

detects the interfacial anisotropy. A sample of a Boolean model

with the parabola shaped grains is depicted at the top of the right-hand side; at the bottom, the

polar MIL diagram is depicted with L(0) = L(π/2). It is obviously not an ellipse.

With tensors of uneven rank also directed anisotropy can be quantified distinguishing n from

−n.

Moreover, the higher rank tensors allow for a very sensitive and comprehensive charac-

terization of anisotropy. In Sec. I E, we showed that the MIL figure is a circle for Boolean

models with grains of constant width even if they are perfectly aligned. A Boolean model

with aligned Reuleaux triangles appears perfectly isotropic w.r.t. the MIL.

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0.4

0.6

0.8

1

0 0.25 0.5

A

n

i

s

o

t

r

o

p

y

i

n

d

i

e

s

σ

βMIL, β0,21

q3 from W 0,31

isotropi aligned

FIG. 9. Anisotropy indices for overlapping Reuleaux triangles with differently anisotropic orienta-

tion distributions. The anisotropy is parametrized by the standard deviation σ of the distribution;

σ = 0 corresponds to perfect alignment and σ = π/3√

3 ≈ 0.60 to random orientations. Because

the MIL figure is a perfect circle (see Fig. 6), not only the index βMIL ≡ 1 is perfectly isotropic,

but any arbitrary rank representation. While the index β0,2

1of the second-rank Minkowski tensor

W0,21

is also constantly equal to unity, the tensor W0,31

of rank three is sensitive to the interfacial

anisotropy. The corresponding anisotropy index q3 is plotted as a function of the standard devi-

ation σ. q3 < 1 detects the anisotropy of the systems with partially aligned Reuleaux triangles.

Samples of anisotropic or isotropic Boolean models with Reuleaux triangles are depicted at the

bottom.

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This insensitivity of the MIL analysis is also demonstrated in Fig. 9: the MIL anisotropy

index βMIL is shown to be equal to unity for both isotropic and anisotropic systems. It

cannot distinguish the aligned grains from the randomly oriented ones.

Because of the threefold symmetry of the grains, the system appears isotropic for any

second-rank tensor. Therefore, also the index β0,2

1of the Minkowski tensor W

0,2

1is constant

to unity. For the MIL, all of its tensor representations of arbitrary rank appear perfectly

isotropic because the MIL figure is a perfect circle. In contrast to this, the Minkowski tensor

W0,3

1of rank three is sensitive to the anisotropy and discriminates the isotropic Boolean

model from that with aligned grains.

The irreducible representation34,70,71 of the third rank Minkowski tensor W0,3

1provides

a scalar index q3 for anisotropy of rank three19. The tensor W0,3

1can be rewritten as the

covariance tensor of the distribution of normal vectors multiplied by the total perimeter,

which is useful for an intuitive explanation of the irreducible representation of the Minkowski

tensors W0,s

1in two dimensions. It is given by the Fourier coefficients Ek of this distribution

function multiplied by the perimeter. An anisotropy index of arbitrary rank k can then be

defined by

qk := 1 − |Ek|E0

. (12)

The second rank anisotropy index q2 is basically equivalent to the ratio β0,2

1with q2 =

2β0,2

1/(1 + β

0,2

1). The third rank anisotropy index q3 indicates anisotropy detected by the

tensor of rank three. If the system is isotropic, W0,3

1vanishes and the anisotropy index

q3 = 1 is equal to unity. However if q3 < 1, the system is anisotropic. For more details on

the irreducible representation of the Minkowski tensors in two dimensions, see Sec. 2.4.2 in

Ref.19.

Figure 9 compares the MIL anisotropy index βMIL to those of the Minkowski tensors, β0,2

1

for W0,2

1and q3 for W0,3

1for overlapping Reuleaux triangles with differently anisotropic orien-

tation distributions. In contrast to the two-fold symmetric ellipses and rectangles, where the

orientation distribution of the angle θ is defined on (−π/2, π/2] (see Sec. IA), the Reuleaux

triangles are three-fold symmetric and the probability distribution of the orientation θ can

be restricted to (−π/3, π/3]. A uniform distribution on (−π/3, π/3] produces a perfectly

isotropic Boolean model. We here choose the anisotropic orientation distributions to be

uniform distributions on intervals (−α, α] with parameter |α| ≤ π/3 to adjust the anisotropy:

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TABLE III. Comparison of a standard MIL analysis to Minkowski tensors. For both approaches,

mean values and in part even second moments and complete probability distributions can be

derived analytically for a variety of mathematical models, like the Boolean model or Gaussian

random fields19. However, there are both numerical and fundamental shortcomings of the MIL

analysis that are rectified by Minkowski tensor methods.

MIL analysis Minkowski Tensors Ref.

Assumption of elliptical Defined for any two-phase mediumFig. 3, Sec. II

MIL figure causes model mismatch Rigorous foundation in integral geometry

Algorithmic complexity O(#pixels · #lines) O(#pixels) Sec. II B

Symmetry assumptions Two-fold rotation axisHigher-rank tensors avoid

e.g., Fig. 9implicit assumptions

Shape information

Only interfacial anisotropyDistribution of volume, surface orientation,

Tables I and IIor curvature

Perfectly isotropic forSensitive detection of anisotropy Fig. 9

bodies of mean width

Complete additive shape description Ref.39,40

α = π/3 corresponds to isotropy and α = 0 to perfect alignment.

Because the MIL figure is a circle even if all grains are aligned in one direction, the MIL

analysis cannot distinguish models with anisotropic interfaces from those with isotropic

interfaces. In contrast to this, the Minkowski tensors do detect and characterize this inter-

facial anisotropy. While the second-rank Minkowski tensor must remain isotropic due to the

three-fold symmetry of the grains, the Minkowski tensor anisotropy index q3 recognizes the

different orientation distributions of the grains, see Fig. 9. If all grains are aligned in the

same direction (σ = 0), the index is distinctly anisotropic q3 = 1 − 2/π ≈ 0.36 < 1.

III. CONCLUSIONS

Orientation-dependent mean intercept length analyses are the most commonly used tool

to quantify anisotropy in porous bone materials2,4,6. Here, we have clarified which morpho-

logical information is contained in these measures.

In contrast to a common assumption, we found that the MIL figure is in general not

an ellipse for anisotropic heterogeneous media; see Eq. (6) and Figs. 3, 8, and Fig. 2 in

paper I. The assumption of Eq. (1) is not full-filled, and the MIL figure cannot necessarily

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be represented by a second-rank tensor. A least-square fit would be based on a model

mismatch and cause an intrinsic lack of accuracy. The deviations are substantial enough to

adversely affect numerical anisotropy analyses, strongly depending on the fitting procedure.

We explicitly showed for a realistic model of porous media how the MIL figure can be

very different from an ellipse. Note, that also in experimental measurements of the MIL of

trabecular bone statistically significant deviations from an ellipsoid have been reported27.

We also discussed further inherent disadvantages of the MIL analysis, like their sensitivity

to noise, but their insensitivity to anisotropy, in the sense that there are systems both with

obvious volume or even interfacial anisotropy, which appear perfectly isotropic w.r.t. the MIL

figure; see Figs. 8 and 6. Moreover, from a practical point of view, standard implementations

of the MIL analysis are very time consuming and sensitive to noise9.

These drawbacks can be corrected for by the family of Minkowski tensors, see Table III.

We showed that, on the one hand, only a seemingly small change is needed from the covari-

ance tensor of the MIL figure to a linear combination of Minkowski tensors. However, this

step from the inverse of the number of intersections to the number of intersections has, on

the other hand, great implications from both a practical and a fundamental point of view.

For the Minkowski tensors are additive, continuous for convex distortions, and can be defined

as volume and surface integrals; they are therefore robust against noise, and there are fast

linear-time algorithms. Free software is available; see Sec. II B. They are more sensitive and

detect the anisotropy in the systems that appeared to be isotropic w.r.t. MIL analysis; see

Figs. 8 and 9. Probably most importantly, the Minkowski tensors allow for a comprehensive

and systematic analysis of different types of anisotropy w.r.t. the volume, the surface, or

the curvature.

A characterization of porous tissue and microstructures using the Minkowski tensors

could provide new insights into the relations between geometry and material properties.

The improved structure characterization makes it possible to extract information that might

hitherto have been overlooked. This could be especially interesting, for example, for cancel-

lous bone, where experimental measurements could be used to relate the anisotropy of its

elasticity to the geometrical anisotropy sensitively quantified by the Minkowski tensors.

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ACKNOWLEDGMENTS

Financial support by the Deutsche Forschungsgemeinschaft (DFG) through the Re-

search Unit “Geometry and Physics of Spatial Random Systems” (GPSRS) under grants

ME1361/11, SCHR1148/3, HU1874/3-2, and LA965/6-2 is gratefully acknowledged.

DISCLOSURE OF CONFLICTS OF INTEREST

The authors have no relevant conflicts of interest to disclose.

∗ Electronic mail: michael.klatt@kit.edu

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