an integral-geometric approach for the euler–poincaré characteristic of spatial images

An integral-geometric approach for the Euler–Poincarecharacteristic of spatial images

W. NAGEL*, J. OHSER† & K. PISCHANG‡*Institute of Stochastics, Friedrich Schiller University of Jena, D-07740 Jena, Germany†Institute of Industrial Mathematics, Erwin–Schrodinger-Straße, D-67663 Kaiserslautern,Germany‡Institute of Materials Science, Dresden University of Technology, Mommsenstraße 13,D-01069 Dresden, Germany

Key words. Euler number, image analysis, integral geometry, stereology.

Summary

The determination of the Euler–Poincare characteristic of aset can be based on observations of a digitized image of thatset. In the present paper the correctness of the method isproved due to a strict integral-geometric approach. Ourresult also provides a link to the methods which are used inimage analysis and are based on graph theory.

Introduction

The estimation of the Euler–Poincare characteristic (Eulernumber) of a spatial set is of current interest in stereologyas well as in image analysis. The importance of the Eulernumber is emphasized in Klain & Rota (1997). In stereologythe estimation is based mostly on observations on parallelplanar sections, see Ohser & Nagel (1996) and referencestherein. Observations on lattices of points are treatedmainly in the context of image analysis, and there theestimation of the Euler number is based on the Euler formulaof graph theory, see Serra (1969), Serra (1982; p. 204),Jouannot et al. (1996) and Lohmann (1998; p. 45f).

The definition of the Euler–Poincare characteristic for aset in the Euclidean space was given by Hadwiger (1957) inthe context of integral geometry. This definition looks veryattractive with respect to a stereological application. Theproblems which prevent the immediate use of Hadwiger’sformula are discussed by Ohser & Nagel (1996).

The purpose of the present paper is to develop pure integral-geometric formulas for the Euler number where the set isobserved on a point lattice. This is due to a general formulaproved by Ohser & Nagel (1996). Special attention is paid toformulate sufficient conditions which guarantee that theformulas are exact and the derived estimators are unbiased.

The result also elucidates the link between the integral-geometric and the graph-theoretic approaches.

Observations of a set on point lattices can also be used toestimate other characteristics of this set such as volume,surface or integral of mean curvature. This is notconsidered here but it was thoroughly studied by Serra(1982), Ohser et al. (1998) and Ohser & Mucklich (2000;Chapter 4).

The digitized image of a set

Throughout this paper we assume that a discretized three-dimensional image of a spatial structure is available.

In materials science such spatial discretized images aregenerated by computer assisted tomography and similartechniques using X-ray scattering, magnetic resonance orisotope emission. (Note that magnetic resonance can beapplied in the case of non-metallic materials only.) Examplesare given in Figs 1 and 2, which show visualizations ofspatial images of sintered copper powders where thespherically shaped particles have random diameters between750 and 1000 mm and between 125 and 350 mm,respectively. Both images are of size 455 × 455 × 241 andthe lattice distances are ∆1 ¼ ∆2 ¼ ∆3 ¼ 10 mm. The data forthese spatial images were obtained by means of X-raytomography.

There are various possible discretizations in the three-dimensional space. Here we consider the most commoncuboidal lattices where the unit cell forms a cuboid, seeFig. 3. The edge lengths ∆1, ∆2 and ∆3 of the unit cell arereferred to as the lattice distances into the x-, y- and thez-direction, respectively. Their reciprocal values are therespective lateral resolutions.

The spatial lattice is given by the set L of its lattice points

L ¼ fxi j k ¼ ði∆1; j∆2; k∆3Þ; i; j; k integerg.

Journal of Microscopy, Vol. 198, Pt 1, April 2000, pp. 54–62.Received 8 April 1999; accepted 19 October 1999

q 2000 The Royal Microscopical Society54

Correspondence to: W. Nagel. Fax: þ 49 3641 946252.

Let X ⊆ R3 a subset of the three-dimensional Euclideanspace. Its digitized spatial image can be understood as amatrix B ¼ (bijk) with the components

bi j k ¼ 1Xðxi j k Þ ð1Þ

where 1X is the indicator function of the set X. As the valuesof this function are either 1 or 0, the image is a binary one.We remark that the result of the discretization of X forms aset of points, and taking into account certain connectivitiesof the voxels, the discretized set X can be interpreted as aspatial graph.

In the analysis of digitized images, the detection of certainpixel configurations is crucial. For our problem it isappropriate to investigate 2 × 2 × 2-configurations as indi-cated in Fig. 4. In order to facilitate the algorithms all thepossible configurations are coded by a one-to-one mappingonto the set {0, . . ., 255}

gi j k :¼ bi j k þ 2biþ1;j;k þ 4bi;jþ1;k þ 8biþ1;jþ1;k þ 16bi;j;kþ1

þ 32biþ1;j;kþ1 þ 64bi;jþ1;kþ1 þ 128biþ1;jþ1;kþ1: ð2Þ

Thus, from B a new matrix G ¼ (gijk) is calculated. Theabsolute frequency of a certain 2 × 2 × 2-configuration

which is coded by the value , [ {0, . . ., 255} can be writtenas

h, ¼X

i

Xj

Xk

d,ðgi j kÞ; , ¼ 0;…;255 ð3Þ

with

d,ðmÞ ¼1 if , ¼ m

0 if , Þ m:

�For our purposes, the digitized image of the set X can be

q 2000 The Royal Microscopical Society, Journal of Microscopy, 198, 54–62

Fig. 1. Microstructure of a spherical powder sintered at 1025 8Cfor 1 h under vacuum. During this process, so-called sinter necksbetween the particles formed; these increase the spatial connectiv-ity of the copper particles. The number of necks and their curva-ture radii depend on the activity of the powder. The copper phasecan be modelled as a homogeneous random set observed in acuboidal window. This figure shows a visualization of a spatialimage obtained by means of computer assisted tomography. Thedata of the copper phase are available as a binary matrix B. Thepurpose is the estimation of the specific Euler number whichdescribes spatial connectivity of the copper spheres.

Fig. 2. Microstructure of a spherical powder sintered at 1025 8Cfor 1 h under vacuum. In this case the mean particle size was smal-ler than in the microstructure shown in Fig. 1. Thus, the spatialconnectivity of the particles is much higher and the sinteringstate differs from that in Fig. 1.

Fig. 3. (a) A cuboidal lattice L of 5 × 5 × 5 voxels; (b) a cell of thelattice. The ordinary cuboidal lattice of uniform lattice distance isthe most usual one applied in discretization of spatial microstructures.

EULER – POINCARE CHARACT ER ISTIC OF SPATIAL IMAGES 55

given in the condensed form of the vector h ¼ (h,). Note thatthe length of the vector h does not depend on the set X andalso not on the window where this set is observed.

The estimators

A general basic formula

The determination of the Euler number as well as a series ofother functionals can be based on formulae of integralgeometry. The application of the classical Crofton formulafor the estimation of volume, surface and the integral ofmean curvature is described in detail by Ohser & Mucklich(2000; Chapter 4).

The integral-geometric approach to the Euler numbergoes back to Hadwiger (1957). It cannot immediately beapplied to digitized images, contrary to the first impression.This was discussed by Ohser & Nagel (1996). In that paperalso a new integral-geometric formula was proved whichnow will be adapted to discretization. We recall it here.

For d ¼ 3, 2, 1 let Rd denote the d-dimensional Euclideanspace and (E, E∆) a pair of parallel planes (if d ¼ 3), of parallellines (if d ¼ 2) or of points (if d ¼ 1), respectively, with a fixeddistance ∆ > 0 and a given (normal) direction. In our context,these distances will be the lattice distances ∆3, ∆2 and ∆1.This means that in a d-dimensional reference space there areconsidered pairs of parallel (d – 1)-dimensional hyperplanes.

A shift of the pair (E, E∆) in its normal direction by a realvalue t [R is denoted by (Et, Etþ∆). When X is intersectedwith Et and Etþ∆ there occur section profiles. In order toapply set operations, these profiles are shifted back toE ¼ E0, i.e. we consider

Yt :¼ ðX ∩ EtÞ¹t; and Ytþ∆ ¼ ðX ∩ Etþ∆Þ¹ðtþ∆Þ: ð5Þ

Let xd denote the Euler number in the d-dimensional space,d ¼ 1, 2, 3. Finally, S denotes a line segment of length ∆

which is orthogonal to (E, E∆) in Rd. The morphologicalopening and the morphological closing of a set X withrespect to S are denoted by X W S and X X S, respectively. Forthe definition and illustration of these operations see Serra(1982) and Ohser & Nagel (1996).

Theorem 1: (Ohser & Nagel, 1996)

Let X ⊂ Rd be a finite union of convex bodies and (E, E∆) afixed pair of parallel hyperplanes as described above. If X ismorphologically open and morphologically closed with respect tothe segment S, i.e. if

X◦S¼ X ¼ X•S; ð5Þ

then

xdðXÞ ¼1∆

�∞

¹∞xd¹1ðYt ∪ YtþDÞ ¹ xd¹1ðYtÞh i

dt ð6Þ

and, equivalently

xdðXÞ ¼1∆

�∞

¹∞xd¹1ðYtÞ ¹ xd¹1ðYt ∩ Ytþ∆Þh i

dt: ð7Þ

The prerequisite X W S ¼ X ¼ X X S is sufficient but notnecessary for the validity of the given formulas. Aninterpretation and discussion were given in Ohser & Nagel(1996). It seems to be a hard problem to find conditionswhich are sufficient and necessary and which fit better topractical situations.

It should be emphasized that the formulae – as well asHadwiger’s original formulae – are true for any fixeddirection of the pair of hyperplanes (E, E∆). That means onthe one hand that observations on planes with one directionare sufficient and on the other hand that the correspondingapplication of the formulae for different directions yieldidentical values of the integrals.

If there is a series of equidistant parallel hyperplanes – aso-called serial section – then the integrals in Eqs (6) and(7) can be rewritten as sums of integrals over boundedintervals. Under the given assumptions, these integrationscan be left out completely, and it remains a sum for a fixedposition of the serial section. At the first glance this may besurprising, because it is in contrast to formulas for the otherfunctionals such as volume, surface and integral of meancurvature. The disappearance of the integrals in our contextcan be due to the fact that the Euler number is a topologicalcharacteristic and not a metric one.

Corollary 1: Let X ⊂ Rd be a finite union of convex bodies and(Ek∆), k integer, a fixed (infinite) sequence of parallel equidistant

Fig. 4. (a) The notation used for the vertices of the unit cell; (b) spatial representation of the filter mask used to generate the gijk as described in(2); (c) an examplary 2 × 2 × 2-configuration associated with the code g ¼ 99. The points indicated by X are assigned to the voxels covered byX, whereas W are the voxels that hit the complementary set Xc.

56 W. NAGEL ET AL .


hyperplanes. If X is morphologically open and morphologicallyclosed with respect to the segment S, i.e. if Eq. (3) is fulfilled,then

xdðXÞ ¼X

k

xd¹1ðYk∆ ∪ Yðkþ1Þ∆Þ ¹ xd¹1ðYk∆Þh i

ð8Þ

and, equivalently,

xdðXÞ ¼X

k

xd¹1ðYk∆Þ ¹ xd¹1ðYk∆ ∩ Yðkþ1Þ∆Þh i

ð9Þ

The proof of the corollary is given in the Appendix.

Application to digital spatial images

In order to estimate the Euler number of a three-dimensional set X which is given by its digital imageB ¼ (bijk), this discretization is considered as an intersectionwith a point grid. This suggests the iterated application ofTheorem 1 for d ¼ 3, 2, 1 one after the other. That results ina practicable formula and moreover, it links our integral-geometric approach with the formulas of graph theorywhere cells, faces, edges and nodes are counted.

There are two standard situations considered in stereol-ogy, the design-based and the model-based approaches. Inthe design-based approach it is assumed that the set ofinterest X ⊂ R3 is fixed and that the point lattice L – which isconsidered as the design for observations – is in randomposition with respect to X. The absence of integrals in Eqs(8) and in (9) yields that the value of the sums occurringthere is the same for any position of the point lattice, i.e. this

position has not to be randomized. Therefore, in order todetermine the Euler number it is superfluous to choose thedesign-based approach instead of considering a fixedbounded set X and a fixed point lattice L.

A fixed bounded set X

It is assumed that a fixed bounded set X ⊂ R3 and a pointlattice L are given. With respect to this situation thefollowing numbers are defined:n(X, L) – number of nodes (i.e. of lattice points or voxels) in X,e(X, L) – number of cell edges which are completely contained in X,f(X, L) – number of cell faces which are completely contained in X,c(X, L) – number of cells which are completely contained in X.

Denote S the set of all edges, face diagonals, and spatialdiagonals of the unit cell (0, ∆1) × (0, ∆2) × (0, ∆3). Then wehave:

Theorem 2: If X ⊂ R3 is a finite union of convex bodies and L afixed point lattice such that

X◦S ¼ X ¼ X•S for all S [ S

then

x3ðXÞ ¼ nðX; LÞ ¹ eðX; LÞ þ f ðX; LÞ ¹ cðX; LÞ

¼X255

,¼0

h,c, ¼ hh;ci ð10Þ

with the vectors h given in Eq. (2) and c ¼ (c,) according toTable 1, and where h·,·i signifies the inner product.

The proof is given in the Appendix.


Fig. 5. A set X with two holes. Hence, its Euler number is x2(X) ¼ ¹1. This set is morphologically open as well as morphologically closed withrespect to segments S1 ¼ [(0, 0), (0, ∆1)] and S2 ¼ [(0, 0), (∆2, 0)], respectively. It is also morphologically open as well as morphologicallyclosed with respect to the unit cell S1 × S2. Nevertheless, a two-fold application of the intersection formula (7) (which provides a formulaanalogous to (17) for x2 as a sum of values of x0) would yield the value x2(X) ¼ 0. This error is due to the fact that X is not morphologicallyopen with respect to the diagonal S3 ¼ [(0, ∆1), (∆2, 0)].


It is not quite obvious that the two versions of the‘estimator’ are equal. In the first formula nodes, edges, facesand cells are taken into account while in the second one the2 × 2 × 2-neighbourhood configurations are evaluated. Thefirst formula is the well-known Euler formula for polyhe-drons. This formula is widely used in digital image analysisin order to determine the Euler number. It is noteworthythat here we derived this formula in a purely integral-geometric approach (with Theorem 1 as the key result)for sets X in Euclidean space R3. The assumptionX 8 S ¼ X ¼ X X S for all S [ S guarantees the coherence ofthe Euler numbers of both the set X and its complement Xc.Thus, with the given assumption the set as well as itscomplement can be treated with one and the sameneighbourhood system (6- or 12-neighbourhood in R3) inorder to determine the respective Euler numbers.

Our assumption excludes configurations which Serracalls diagonal ones. The assumption implies that somecomponents of the vector h have to be 0, e.g. h99, cf.Fig. 4(c).

Discussion of the requirement on X

The assumption that X W S ¼ X ¼ X X S for all S [ S may lookstrange to the reader and probably the impression arises thatour method is not applicable in most of the practical situations.

The condition expresses an assumption which is generallymade when a set is digitized: if two adjacent points of thelattice belong to the set X, then it is assumed that the wholesegment joining these points belongs to the set too. Therespective prerequisite is made for the complement Xc. Such anassumption is made more or less explicitly and more or lessprecisely by all authors dealing with problems of discretization.

For the purpose of determining the Euler number, all pairsof points of the elementary cell of the cuboidal lattice have

to be considered as adjacent ones. Therefore, our assump-tion refers to all edges, face diagonals, and spatial diagonalsof the elementary cell. The sets X which fulfil theassumption are all—not necessarily convex—polytopeswith the following properties: (i) all the edges are parallelto one of the 13 directions given by the edges and diagonalsof the elementary cell, (ii) the length of any edge is at leastas large as the length of the corresponding parallel edge ordiagonal of the elementary cell, (iii) there are no ‘sharp’vertices nor edges in the sense that all tangent lines in oneof the 13 directions touch the polytope in a segment whichis at least as long as the corresponding edge or face diagonalof the elementary lattice cell.

In particular, the assumption excludes configurationswhich Serra calls diagonal ones. On the one hand, our wayof formulating the prerequisites allows strict mathematicaltreatment and proofs of the theorems. On the other hand,one can get an idea of the errors one makes in estimatingthe Euler number when an arbitrary set and the resolutionof the discretization lattice are given. This error can bezero even if the condition on X is not fulfilled, i.e. in thelanguage of mathematics, our condition is sufficient butnot necessary for the validity of Theorem 2. As a roughgeneral rule one can say that there is no error in thedetermination of the Euler number x3(X) if the morpho-logical openings and closings with all the segments S [ Sdo not change the Euler number of X. An example is aball which is large enough such that in any position itcontains at least one complete elementary cell of thediscretization lattice.

If the assumption of Theorem 2 is not fulfilled then therearises the well-known problem of digital image analysis thatthe estimates

x3ðXÞ ¼ hh;ci and x3ðXcÞ ¼ hh;cci

Table 1. The coefficients of the vector c.

0 1 0 0 0 0 0 ¹ 1 0 1 0 0 0 0 0 0 0 . . . 150 0 0 ¹ 1 0 ¹ 1 0 ¹ 2 0 0 0 ¹ 1 0 ¹ 1 0 ¹ 1 16 . . . 310 1 0 0 0 0 0 ¹ 1 0 1 0 0 0 0 0 0 32 . . . 470 0 0 0 0 ¹ 1 0 ¹ 1 0 0 0 0 0 ¹ 1 0 0 48 . . . 630 1 0 0 0 0 0 ¹ 1 0 1 0 0 0 0 0 0 64 . . . 790 0 0 ¹ 1 0 0 0 ¹ 1 0 0 0 ¹ 1 0 0 0 0 80 . . . 950 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 96 . . . 1110 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 112 . . . 1270 1 0 0 0 0 0 ¹ 1 0 1 0 0 0 0 0 0 128 . . . 1430 0 0 ¹ 1 0 ¹ 1 0 ¹ 2 0 0 0 ¹ 1 0 ¹ 1 0 ¹ 1 144 . . . 1590 1 0 0 0 0 0 ¹ 1 0 1 0 0 0 0 0 0 160 . . . 1750 0 0 0 0 ¹ 1 0 ¹ 1 0 0 0 0 0 ¹ 1 0 0 176 . . . 1910 1 0 0 0 0 0 ¹ 1 0 1 0 0 0 0 0 0 192 . . . 2070 0 0 ¹ 1 0 0 0 ¹ 1 0 0 0 ¹ 1 0 0 0 0 208 . . . 2230 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 224 . . . 2390 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 240 . . . 255

58 W. NAGEL ET AL .


of X and of its complement Xc, respectively, can differ (i.e.are not consistent). Here, cc is the vector of coefficientscc

, ¼ c255¹,, , ¼ 0, . . . , 255. In order to improve theseestimates with respect to their consistency we suggest theuse of

xd3ðXÞ ¼ xd3ðXcÞ ¼ 1

2 ðhh;ci þ hh;cciÞ ¼ hh; 12 ðc þ ccÞi:

One has to be aware of the fact that in the case that theassumption of the Theorem is not fulfilled, i.e. if theresolution of the discretization is not sufficient with respectto the slender structure of X or its complement, by the use ofany method the Euler number can only be estimated andnot be determined exactly.

As nothing is random in this framework it is notappropriate to speak of an (unbiased) estimator as usual.If X fulfils the assumption of the theorem then the Eulernumber can be determined exactly from the observation onthe point lattice.

It should be emphasized here that our formula differsfrom the one given by Serra (1969; p.72) and repeated inSerra (1982; p.204). To illustrate this, consider the setX ¼ [0, ∆1] × [0, ∆2] × [0, m∆3], m ¼ 1, 2, . . . It is a convexset, and hence x3(X) ¼ 1. If we understand Serra’s formularightly we obtain x3(X) ¼ m. This is a contradiction of theresult provided by the Euler formula.

Model-based approach

Now consider the case that the set X is observed inside abounded window W. In a model-based approach the sampleW ∩ X is modelled as the part of a realization of a randomset Y which appears in W. In order to define characteristicsindependent of the window W, it is assumed that therandom set Y is homogeneous, i.e. the distribution of Y isinvariant with respect to translations. Homogeneity impliesthat the realizations X (almost surely) are unbounded sets.Thus, instead of the Euler number x(X) the specific Eulernumber xV: ¼xV(Y) is introduced as the mean Euler numberper unit volume. This specific Euler number can beestimated from observations in a bounded window whereedge effects have to be taken into account.

Let Y denote a three-dimensional random homogeneousset, and W ¼ [0, n1∆1] × [0, n2∆2] × [0, n3∆3] is a cuboidwhere n1, n2 and n3 are positive integers. If for (almost) allrealizations X of Y, the set X ∩ W can be represented as afinite union of convex bodies, i.e. X is a so-called locallyfinite union of convex bodies, then the theorem ‘Satz 2.4.1’from Mecke et al. (1990) can be applied and hence the meanEuler number xV(Y) of Y per unit volume can be expressedas

xVðYÞ ¼E x3ðY ∩ WÞ ¹ x3ðY ∩ ∂ðþÞWÞ� �

VðWÞð11Þ

where E means the expectation, V denotes the volumeand

∂ðþÞW ¼ fðx; y; zÞ [ W : x ¼ n1∆1 or

y ¼ n2∆2 or z ¼ n3∆3g

is a part of the boundary of W, which is used for the purposeof edge correction.

The estimator for xV(Y) is similar to that for the fixed non-random set model up to the edge correction and anormalization to unit volume. There are several techniquesof edge correction. We follow the approach indicated bydefinition (11), which is analogous to that proposed byBhanu Prasad et al. (1989).

Assume that the set can be observed on the set of latticepoints inside W,

LW ¼ f½i∆1; j∆2; k∆3Þ : i ¼ 0;…;n1;

j ¼ 0;…; n2; k ¼ 0; :::; n3g:

Consequently, the vector of absolute frequencies of obser-vable 2 × 2 × 2-neighbourhood configurations analogous toEq. (3) is now the following one:

hW ¼ ðhW, Þ with hW

, ¼Xn1¹1

i¼0

Xn2¹1

j¼0

Xn3¹1

k¼0

d,ðgi j kÞ ð12Þ

For the sake of edge correction define

LðþÞW :¼ L ∩ ∂ðþÞW ¼ fði∆1; j∆2; k∆3Þ :

i ¼ n1 or j ¼ n2 or k ¼ n3g:

Note that there is no three-dimensional cell in LW(þ) and

therefore c(X,LW(þ)) ¼ 0 for arbitrary X.

The volume of the cuboidal sampling window W isV(W) ¼ n1n2n3∆1∆2∆3.

Theorem 3: Let Y be a random homogeneous set which is alocally finite union of convex bodies and

Y W S ¼Y¼ Y X S for all S [ S

almost surely. It is assumed that the set can be observed on thepoint lattice LW and the vector of absolute frequencies of2 × 2 × 2-configurations is hW ¼ (hl

W) as given in Eq. (12).Then the estimator

cxV ¼1

VðWÞ

nnðX; LWÞ ¹ n X; LðþÞ

W

ÿ �� ¹ eðX; LWÞ ¹ e X; LðþÞ

W

ÿ �� þ f ðX; LWÞ ¹ f X; LðþÞ

W

ÿ �� ¹ cðX; LWÞ

oð13Þ

¼hhW ;ci

VðWÞ; for all realizations X of Y; ð14Þ

is unbiased for xV(Y).The proof is sketched in the Appendix. The components of

the vector c are the same as given in Table 1.



Owing to edge effects, estimates of xV(Y) depend on‘direction of measurement’. Estimates based on the coeffi-cients of Table 1 are assigned to the z-direction. However,from statistical point of view, estimates of xV(Y) should beaveraged over ‘all directions’.

Given a cube centred at the origin, we consider thoserotations q of the proper rotation group which keep thiscube unchanged. There are 24 rotations qn and thesymmetry group {q0, . . ., q23} of the cube is also referredto as the octaeder group. Let cxV(qn) denote the estimator ofxV which is assigned to the rotation qn. Each estimatorcxn(qn) can be written in the form (14), i.e. bxn(qn) ¼hhW,cni/V(W) for a correspondingly chosen cn. Because of thetechnique used in the computation of hW, all theseestimators cxV(qn) are unbiased. Nevertheless, the cxV(qn)can take different values even for the same realization of therandom set Y. Thus, the specific Euler number xV should beestimated as the rotation average

fxV ¼1

24

X23

n ¼ 0

cxVðqnÞ ¼1

24

X23

n ¼ 0

hhW ;cni

VðWÞ¼

124

hhW ;croti

VðWÞ

which is also unbiased for xV. The vector crot: ¼ S23n¼ 0cn is

given in Table 2.For the material shown in Fig. 1 the estimate of the

specific Euler number is cxV ¼ ¹ 2.33 mm¹3 and for thematerial in Fig. 2 it is cxV ¼¹ 103.7 mm¹3. As the copperforms spheres, neither the copper phase nor the pore spaceof these materials fulfil the conditions of Theorem 3.However, as been indicated in Section 2, the spherediameters are at least 12 times larger than the latticedistances in the discretization. Thus, one can see from theimages in Figs 1 and 2 that not only the diameters of thecopper spheres, but also the widths of the sinter necks andof the pores are large with respect to the lateral resolution

used in the imaging of the material. Hence, according to thegeneral rule formulated in the discussion on Theorem 2,one can expect that the obtained estimates are very close tothe ‘true’ values of xV.

In integral geometry, the integral of the Gaussiancurvature of a convex body is also defined for polytopeswith the help of the Steiner formula. In this context, up to aconstant the integral of the Gaussian curvature appears as aquermassintegral, Minkowski functional, or inner volume,respectively. This definition can be extended to the convexring, i.e. to finite unions of convex bodies. The so-called‘local version of the Steiner formula’ allows the definition ofcurvature measures, see Schneider & Weil (1992) andSchneider (1993). Thus, for homogeneous random sets thecorresponding random Gaussian curvature measure can bedefined, see Stoyan et al. (1995). As for other homogeneousrandom measures the density can be considered as themean value of the measure per unit volume. The density KV

of the Gaussian curvature measure is related to xV byKV ¼ 4pxV. For sets with smooth surfaces, KV is the integralof the Gaussian curvature per unit volume.

With respect to the typical surface element of Y, the meanGaussian curvature K can be introduced as the ratio KV/SV,where SV is the surface density (specific surface area) of therandom set. It can be estimated using

bK ¼ 4pfxVcSV

where cSV is an estimator of SV.For the sinter metals shown in Figs 1 and 2 we have

obtained bK ¼¹ 26 mm¹2 and bK ¼¹ 305 mm¹2, respec-tively, where the corresponding surface densities havebeen estimated by means of the method described in Ohser& Mucklich (2000).

Table 2. The coefficients of the vector crot.

0 3 3 0 3 0 6 ¹ 3 3 6 0 ¹ 3 0 ¹ 3 ¹ 3 0 0 . . . 153 0 6 ¹ 3 6 ¹ 3 9 ¹ 6 6 3 3 ¹ 6 3 ¹ 6 0 ¹ 3 16 . . . 313 6 0 ¹ 3 6 3 3 ¹ 6 6 9 ¹ 3 ¹ 6 3 0 ¹ 6 ¹ 3 32 . . . 470 ¹ 3 ¹ 3 0 3 ¹ 6 0 ¹ 3 3 0 ¹ 6 ¹ 3 0 ¹ 8 ¹ 8 0 48 . . . 633 6 6 3 0 ¹ 3 3 ¹ 6 6 9 3 0 ¹ 3 ¹ 6 ¹ 6 ¹ 3 64 . . . 790 ¹ 3 3 ¹ 6 ¹ 3 0 0 ¹ 3 3 0 0 ¹ 8 ¹ 6 ¹ 3 ¹ 8 0 80 . . . 956 9 3 0 3 0 0 ¹ 8 9 12 0 ¹ 3 0 ¹ 3 ¹ 8 ¹ 6 96 . . . 111

¹ 3 ¹ 6 ¹ 6 ¹ 3 ¹ 6 ¹ 3 ¹ 8 0 0 ¹ 3 ¹ 8 ¹ 6 ¹ 8 ¹ 6 ¹ 12 3 112 . . . 1273 6 6 3 6 3 9 0 0 3 ¹ 3 ¹ 6 ¹ 3 ¹ 6 ¹ 6 ¹ 3 128 . . . 1436 3 9 0 9 0 12 ¹ 3 3 0 0 ¹ 8 0 ¹ 8 ¹ 3 ¹ 6 144 . . . 1590 3 ¹ 3 ¹ 6 3 0 0 ¹ 8 ¹ 3 0 0 ¹ 3 ¹ 6 ¹ 8 ¹ 3 0 160 . . . 175

¹ 3 ¹ 6 ¹ 6 ¹ 3 0 ¹ 8 ¹ 3 ¹ 6 ¹ 6 ¹ 8 ¹ 3 0 ¹ 8 ¹ 12 ¹ 6 3 176 . . . 1910 3 3 0 ¹ 3 ¹ 6 0 ¹ 8 ¹ 3 0 ¹ 6 ¹ 8 0 ¹ 3 ¹ 3 0 192 . . . 207

¹ 3 ¹ 6 0 ¹ 8 ¹ 6 ¹ 3 ¹ 3 ¹ 6 ¹ 6 ¹ 8 ¹ 8 ¹ 12 ¹ 3 0 ¹ 6 3 208 . . . 223¹ 3 0 ¹ 6 ¹ 8 ¹ 6 ¹ 8 ¹ 8 ¹ 12 ¹ 6 ¹ 3 ¹ 3 ¹ 6 ¹ 3 ¹ 6 0 3 224 . . . 239

0 ¹ 3 ¹ 3 0 ¹ 3 0 ¹ 6 3 ¹ 3 ¹ 6 0 3 0 3 3 0 240 . . . 255

60 W. NAGEL ET AL .


References

Bhanu Prasad, P., Lantuejoul, C., Jernot, J.P. & Chermant, J.L.(1989) Unbiased estimation of the Euler–Poincare character-istic. Acta Stereol. 8, 101–106.

Hadwiger, H. (1957) Vorlesungen uber Inhalt, Oberflache undIsoperimetrie. Springer, Berlin.

Jouannot, P., Jernot, J.P. & Lantuejoul, C. (1996) Unbiasedestimation of the specific Euler–Poincare characteristic. ActaStereologica, 15, 45–51.

Klain, D.A. & Rota, G.-C. (1997) Introduction to GeometricProbability. Cambridge University Press, Cambridge.

Lohmann, G. (1998) Volumetric Image Analysis. Wiley, Chichesterand B. G. Teubner, Stuttgart.

Mecke, J., Schneider, R., Stoyan, D. & Weil, W. (1990) StochastischeGeometrie. Birkhauser, Basel.

Ohser, J. & Mucklich, F. (2000) Statistical Analysis of MaterialsStructures. Wiley, Chichester.

Ohser, J. & Nagel, W. (1996) The estimation of the Euler–Poincarecharacteristic from observations on parallel sections. J. Microsc.184, 117–126.

Ohser, J., Steinbach, B. & Lang, C. (1998) Efficient texture analysisof binary pictures. J. Microsc. 192, 20–28.

Schneider, R. (1993) Convex bodies: the Brunn-Minkowski theory.Encyclopaedia of Mathematics and its Application, Vol. 44. Cam-bridge University Press, Cambridge.

Schneider, R. & Weil, W. (1992) Integralgeometrie. B. G. Teubner,Stuttgart.

Serra, J. (1969) Introduction a la morphologie mathematique.Cahiers du Centre de Morphologie Mathematique no. 3, Paris.

Serra, J. (1982) Image Analysis and Mathematical Morphology.Academic Press, London.

Stoyan, D., Kendall, W.S. & Mecke, J. (1995) Stochastic Geometry andits Applications. 2nd edn. Wiley, Chichester.

Appendix

Proof of Corollary 1

With the given notations and assumptions, for a convexbody XX

k

xd¹1ðYk∆ ∪ Yðkþ1Þ∆Þ ¹ xd¹1ðYk∆Þh i

¼ 1 ¼ xdðXÞ

since all items are 0 except one of them. There is exactly oneinteger k such that X ∩ E(kþ1)∆ Þ f and X ∩ E k∆ ¼f. Thisprovides the item 1 and thus the sum 1.

Hence the proof of Theorem 1 in Ohser & Nagel (1996)can be copied almost literally when (1/t)

Rds is replaced by

Sk and the other notions are adapted appropriately. Thisyields Eq. (8). Equation (9) follows immediately from Eq. (7)when the index k is reflected to –k and the additivity of xd istaken into account.

Proof of Theorem 2

Essentially, the proof of Theorem 2 can be based on athreefold application of Theorem 1 and of Corollary 1 for

descending dimensions. When doing so the sufficientconditions for the validity of the formulas have to beadapted step by step. Let X ⊂ R3 be a finite union of convexbodies. Choose the sequence (Ek∆3), k integer, of parallel(two-dimensional) planes, such that E: ¼ E0 is the xy-plane.In order to shorten the formulas the notion Yk: ¼ Yk∆3 willbe used. Thus, Eq. (9) for d ¼ 3 reads as

x3ðXÞ ¼X

k

x2ðYkÞ ¹ x2ðYk ∩ Ykþ1Þ� �

: ð15Þ

According to Theorem 1 and Corollary 1, it is sufficientfor (15) that X W S ¼ X ¼ X X S for the segment S of length ∆3,which is orthogonal to the xy-plane.

Now Eq. (7) with d ¼ 2 is plugged in at the positions in(15) where x2 occurs. In order to do so a sequence ofparallel lines on each of the planes Ek∆3 has to be chosen.This yields a grid of parallel lines. In our approach, it isappropriate to consider (ej∆2, k∆3), j, k integer, where e: ¼ e0,0

is the x-axis. We extend and abbreviate the notation for thesection profiles introduced in Eq. (4) in the following way:

Yj; k :¼ ðX ∩ ej∆2;k∆3Þ¹ðj∆2;k∆3Þ ð16Þ

where ej∆2, k∆3 is the x-axis shifted by the vector (0,j∆2, k∆3), and the index ¹(j∆2, k∆3) means that the sectionprofile is shifted back to the x-axis.

Thus, Eqs (15) and (9) yield

x3ðXÞ ¼X

j

Xk

x1ðYj; kÞ ¹ x1ðYj; k ∩ Yj þ 1;kÞ

¹hx1ðYj; k ∩ Yj; k þ 1Þ

¹ x1ðYj;k ∩ Yj; k þ 1 ∩ Yj þ 1;k þ 1Þi: ð17Þ

To justify these steps one has to check whether the sufficientcondition (5) is fulfilled for each of the sets which occur inthe several items of the sum. For the application of Theorem1 and Corollary 1 to the first item of Eq. (15), it is sufficientthat X W S ¼ X ¼ X X S for the segment S of length ∆2, whichis parallel to the y-axis. Regarding the second item of Eq.(15), one has to investigate what condition (5) means whenit is applied to the set

Yk ∩ Ykþ1 ¼ ðX ∩ Ek∆3Þ¹k∆3 ∩ ðX ∩ Eðkþ1Þ∆3Þ¹ðkþ1Þ∆3

and the segment S of length ∆2, which is parallel to they-axis. A thorough case study shows that this condition isequivalent to the requirement that X W S ¼ X ¼ X X S for thesegment S of length ∆2, which is parallel to the y-axis andthe segments S, which are the facial diagonals of the cuboidface with the nodes (0, 0, 0), (0, ∆2, 0), (0, 0, ∆3), (0, ∆2,∆3). Figure 5 illustrates this condition and also the fact thatit is not equivalent to the condition that X is morpho-logically open and closed with respect to the cuboid face, i.e.



the full rectangle. This last mentioned condition would notexclude possible diagonal configurations, cf. Serra (1982;p. 173).

Now it is straightforward but tedious to apply Theorem 1and Corollary 1 with d ¼ 1 and a sequence of equidistantpoints on each line of the line grid to the four items ofEq. (17). This yields a threefold sum with respect to i, j, k,which contains eight items, the values of which can beeither 1 or 0.

Consider the lattice of points (i∆1, j∆2, k∆3), i, j, k integer.In order to extend the notation of Eq. (16) denote

Yi;j;k :¼ ðX ∩ fði∆1; j∆2; k∆3ÞgÞ¹ði∆1;j∆2 ;k∆3Þ

¼fð0;0;0Þg if ði∆1; j∆2; k∆3Þ [ X

B if ði∆1; j∆2; k∆3Þ Ó X

(for (i∆1, j∆2, k∆3) [R3. Thus, also the intersections ofY-values for different points can be either empty or the one-element set {(0,0,0)}, which has the Euler number 1.

x3ðXÞ ¼ SiSjSk

x0ðYi;j;kÞ

¹ x0ðYi;j;k ∩ Yiþ1;j;kÞ

¹ x0ðYi;j;k ∩ Yi;j þ 1;kÞ

þ x0ðYi;j;k ∩ Yi;j þ 1;k ∩ Yi þ 1;j;k ∩ Yi þ 1;j þ 1;kÞ

¹ x0ðYi;j;k ∩ Yi;j;k þ 1Þ

þ x0ðYi;j;k ∩ Yi;j;k þ 1 ∩ Yi þ 1;j;k ∩ Yi þ 1;j;k þ 1Þ

þ x0ðYi;j;k ∩ Yi;j;k þ 1 ∩ Yi;j þ 1;k ∩ Yi;j þ 1;k þ 1Þ

¹ x0ðYi;j;k ∩ Yi;j;k þ 1 ∩ Yi;j þ 1;k ∩ Yi;j þ 1;k þ 1

∩ Yi þ 1;j;k ∩ Yi þ 1;j;k þ 1

∩ Yi þ 1;j þ 1;k ∩ Yi þ 1;j þ 1;k þ 1Þ ð18Þ

The consequence of condition (5) applied to the severalintersections occurring in Eq. (17) is that X W S ¼ X ¼ X X S

for all edges, facial and spatial diagonals of the unit cell ofthe point grid, i.e. for all S [ S.

As x8 is either 0 or 1, the single items in Eq. (18) indicatewhether or not all points of certain point configurationsbelong to X. In particular, x8(Yi,j,k ∩ Yiþ1,j,k) ¼ 1 if both thepoints (i∆1, j∆2, k∆3) and ((i þ 1)∆1, j∆2, k∆3) and, due tothe assumption, also the edge between these points belongto the set X. Otherwise it is 0. Thus, the items in Eq. (8) canbe interpreted in terms of nodes, edges, faces and cells of thepoint lattice, respectively. This yields the Euler formula.

With bi jk as given in Eq. (1), the items in Eq. (8) can alsobe expressed as

x0ðYi;j;kÞ ¼ bi j k; x0ðYi;j;k ∩ Yiþ1;j;kÞ ¼ bi j k·biþ1;j;kj;…

For all fixed integers i, j, k the item in Eq. (18) provides anevaluation of the configurations of eight points, which canbe considered as the eight voxels of a 2 × 2 × 2-neighbour-hood, as given in Fig. 4b. There are 256 possibleconfigurations, and the corresponding values of the itemare given in Table 1. Note that the assumption X W S ¼

X ¼ X X S for all S [ S excludes several of these configura-tions, e.g. that given in Fig. 4c. Therefore, if the assumptionis not fulfilled, the formal application of Eq. (18) can yield avalue different from x3(X).

Sketch of the proof of Theorem 3

It is a consequence of Theorem 2 that for any realization Xof the random set Y the value of cxV is exactly

1VðWÞ

ðx3ðX ∩ WÞ ¹ x3ðX ∩ ∂ðþÞWÞÞ:

This is due to the construction of hW and of LW(þ). Thus, the

contribution to the Euler number of X ∩ ∂(þ)W is neglectedfor the sake of edge correction. Taking definition (10) ofxV(Y) into account it becomes immediately clear that cxV isan unbiased estimator.

62 W. NAGEL ET AL .


an integral-geometric approach for the euler–poincaré characteristic of spatial images

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