3D complex shape characterization by statistical analysis

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3D complex shape characterization by statistical analysis:Application to aluminium alloys

Estelle Parra Denis a, , Cécile Barat b , Dominique Jeulin a , Christophe Ducottet b

a Ecole Nationale Supérieure des Mines de Paris, 35, rue Saint-Honoré, 77300 Fontainebleau, Franceb Laboratoire Traitement du Signal et Instrumentation,UMR CNRS-UJM 5516, Bâtiment F, 18 rue du Pr.Benoît Lauras, 42000 Saint-Etienne,

France

A R T I C L E D A T A A B S T R A C T

Article h istory:Received 9 August 2006Accepted 18 January 2007

The goal of this paper is to describe a methodology for characterizing 3D complex shapesusing morphological features. First, we provide 3D morphological measurements for understanding complex shapes. Second, we explain the analysis method based onprincipal component analysis. We illustrate our approach on populations of intermetallicparticles of aluminium alloys investigated using X-ray microtomography. In that case, theanalysis provides a description of shapes with a limited number of parameters, with amorphologicalinterpretation foreach of them. We finallydemonstrate the practical interestof our work by comparing two populations extracted from the same aluminium sample at

two deformation stages of a hot rolling process. © 2007 Elsevier Inc. All rights reserved.

Keywords:Aluminium alloys characterizationIntermetallic particles classificationImage processing

Morphological analysisPrincipal component analysis

1. Introduction

The microstructure of a material determines its physicalproperties. Having an understanding of the microstructureformation is a key tool for material scientists to predict themechanical properties of the material and to develop productswith desired properties.

X-ray microtomography can now provide a 3D representa-tion of the microstructure of materials with high resolution ina non destructive way. Image processing is then essential toextract the relevant microstructural components and toperform 3D measurements to characterize quantitatively thematerial's microstructure of interest. These microstructuralcomponents oftenexhibit complex shapes, which makes their analysis difficult.

Many 3D shape analysis algorithms exist in the literature[2,4]. However, most of the time, they only apply to simple 3D

shapes or star-shaped objects. Hence, the analysis of 3Dcomplex shapes like those encountered in material studiesrequired the development of new approaches.

In this paper, we propose a methodology to carry out 3Dcomplex shape analysis using morphological features. Thismethodology is illustrated with the analysis of intermetallicparticles of aluminium alloys. It provides a description of shapes with a limited number of parameters, with a morpho-logical interpretation for each of them.2D or 3D plots can thenbe used to study the shape variability of populations.

In the case of aluminium alloys, such analysis is useful toreveal morphological differences between particles and totrack the deformation of the particles when hot-rolling isapplied to the studied alloy. During this process used totransform aluminium slabs into sheets, the material under-goes important stress and strain and intermetallic particlesthen break up.

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Corresponding author. Tel.: +33 164694805; fax: +33 164694707.E-mail addresses: [email protected] (E.P. Denis), [email protected] (C. Barat), [email protected]

(D. Jeulin), [email protected] (C. Ducottet).

1044-5803/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.matchar.2007.01.012

mailto:[email protected]




http://dx.doi.org/10.1016/j.matchar.2007.01.012

http://dx.doi.org/10.1016/j.matchar.2007.01.012







The processing was made on 3D images of an aluminiumsample of 1 mm 2 ×1 cm at different rolling process stages of the material. X-ray microtomography was performed at theEuropean Synchrotron Radiation Facility (Grenoble, France).

These images have a resolution of 0.7 μ m3 and containthousands of intermetallic particles with volume ranging from 9 μ m3 up to 24.000 μ m3.

The following paper is organized as follows. First, weprovide some Morphological features for 3D complex shapecharacterization.Second, we perform a statistical multivariateanalysis to select a set of parameters adapted to themorphology of particles. Third, results at two stages of thehot rolling process are proposed to demonstrate the practicalinterest of the methodology. Finally, a Conclusion is given.

2. Morphological Features

In this section, we provide a set of morphological parametersfor characterizing 3D complex shapes. The parameters can bedivided into four categories: basic measures, shape indexes,geodesic measures, and mass distribution parameters.

2.1. Basic Measures

They include volume and surface area:

- the volume (V) is calculated as the number of voxels thatform the object.

Fig. 1 – Theoretical graph of λ 2 versus λ 1 .

Fig. 2 – Pairwise correlation analysis.

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- the surface area (S) is estimated by the stereologicalmethod of Crofton [1].

2.2. Shape Indexes

Shape indexes compare a studied shape to a reference one [7]:

- the index of sphericity is defined as: Is =36π V 2 /S2. Asphere will have an index of sphericity equal to 1.- the index of compacity is Ic =6Is / π : It compares the shapeto a cube instead of a sphere, a cube will have an index of compacity equal to 1.

2.3. Geodesic Measures

Parameters of this category arebased on thegeodesicdistance[5], which is an important geometric measure for understand-ing complex shapes of objects. The geodesic distance betweentwo points x1 and x2 belonging to a given shape X is equal to

the length of the shortest path connecting x1 to x2, remaining included in X. It allows to determine:

- the geodesic radius ( Rmin ) of a shape X which correspondsto the smallest ball included in X. It provides informationabout the size of the core of the shape. It is normalized tocorrespond to a shapeindexwhich compares the volume of the minimum ball included in the shape to the one of theobject.- thegeodesicelongation index (IG g ), is a novel index whichwe propose as an extension to 3D of the well knowngeodesic stretching index in 2D [5]. It characterizes theobjectelongation and is defined as: IG g =π Lg

3 / 6V , where Lg is

the geodesic length. Lg is the maximum length of a pathwhich can be drawn within the shape.

2.4. Mass Distribution Parameters

Moments of inertia of an object depend on its shape andcharacterize the distribution of mass within the shape. Theycorrespond to the eigen values of the inertia matrix of theshape [6] (computed from the center of mass of X, under theassumption that the mass is uniformly distributed and thatthe elementary volume is the voxel). They are normalized inorder to be independent of the volume. If I1, I2, I3 denote themoments of inertia, the normalized moments λ 1, λ 2, λ 3 are

defined as: λ i =Ii / I1 +I2 +I3, i=1,2,3. Their sum is equal to 1 andthey are ordered: λ 1 ≥ λ 2 ≥ λ 3. From those equations end thedefinition of inertia moments, the two following inequalitiescan be deduced:

8i ; k iV 0: 5 and k 2z 0: 5d 1 k 1ð Þ

Plotting all these equations leads to a triangle having originalproperties to describe shapes ( Fig. 1). At the triangle vertices,we can distinguish 3 types of mass distribution within 3Dobjects: spherical, flat and needle. Between these extremities,shapes varycontinuously. Alongthe triangle edges, shapes areprolate ellipsoid-typed, oblate ellipsoid-typed or flat ellipse-typed.

3. Shape Statistical Analysis

In the case of the analysis of a complex particles population, astatistical study must be performed. The goal is to provide adescription of shapes with a minimum number of parameters.For that purpose, we propose in this section to use the

principal component analysis [3] (PCA). This analysis and theway we select parameters is illustrated with a population of intermetallic particles.

3.1. Data Matrix

Measurements of the previous morphological parameterswere made on 3500 intermetallic particles for a 10% deformedaluminium alloy. A pairwise correlation analysis betweenparameters presented in Fig. 2 suggests to remove someparameters before applying PCA. Indeed, as we can see on Fig.2, the volume V and surface S parameters are linearlycorrelated, as well as Is and the geodesic radius Rmin .

Consequently, we choose to disregard S and Rmin , as they donot bring any further information.

Finally, the considered parameters are: volume, sphericityindex, geodesic elongation index and the normalized eigenvalues of the inertia matrix ( λ 1, λ 2).

Fig. 3 – PCA correlation circles (A) factorial axes 1 –2 (B)factorial axes 2 –3.

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The data matrix is therefore composed of 3500 particlesand 5 parameters describing each particle. Different types of PCAexist, varying in the waythe data arepresented (centered/not centered – reduced/not reduced). In our case, it is appro-priate to center and reduce the data because data pointsproject uniformly on all axes and they are measured indifferent units.

3.2. Principal Components Analysis

The computation of the PCA of our data matrix returns thefollowing percentages of the variability for each eigen values:e1=45,5%, e2=28,5%, e3=13,8%, e4= 8,0% and e5=4.2%. For thepresent study, we only keep the first three axes since theyrepresent 87.8% of the variability.

To understand the role of the initial variables in theformation of the principle axes, it is usual to project themonto the new axes leading to the correlation circle maps(Fig. 3).

On the correlation circles, we observe that the geodesicelongation index IGg is strongly negatively correlated withaxis 1 and slightly correlated with axis 3. The index of

sphericity Is is strongly positively correlated with axis 1 andclose to IGg on axis 3. The volume is negatively related withaxis 1 and positively related with axis 2. λ 1 and λ 2 are stronglynegatively related with axis 2 and they are mixed. They splitaccording to axis 3 where λ 2 is positive and λ 1 is negative.

The opposite correlation of IGg, V and Is with axis 1 reflectsthat axis 1 characterizes elongation changes and that themore elongated an object is, the larger it is. Axis 2 suggeststhat the larger a particle, the smaller λ 1 and λ 2, which means,according to paragraph 2, that a large particles tend to have aspherical mass distribution, while small ones tend to have aflat or cylindrical mass distribution. Axis 3 allows to distin-guish objects with a flat mass distribution from ones having aneedle mass distribution. The interpretation of correlationcircles is presented with arrows on Fig. 4(C) and (D).

3.3. Analysis of Morphological Differences of AluminiumParticles

The previous interpretationsof the principalaxes areusefultoanalyze the cloud of our data points in order to identify somegroupsof particle shapes. Themaps of the particleson thetwo

Fig. 4 – Data cloud in PC space and shape trends (A) Plane 1 –2 (B) Plane 2–3 (C) Plane 1 –2 with shape trends (D) Plane 2 –3 withshape trends.




first factorial planes are given on Fig. 4. It is clear that theshape of particles varies continuously. No group of particlesstands out.

3.3.1. Analysis of Trends on Plane 1 –2From Fig.4A, we caninfer that small particles(quadrant 2 – 3– 4)aremore numerousthan large ones. In quadrant1, we observea pointed distribution of objects in the opposed direction of λ 1

and λ 2. It expresses that the higher Is , the more compact theobject and naturally, the more spherical its mass distribution(particleB on Fig. 5). Quadrant2 corresponds to particleswithalarge volume. Along the vertical axis, their mass distribution

gets more andmore spherical (particleE).Along thehorizontalaxes, objects gets more elongated (particle A). In quadrant 3,we find elongated particles (particle C). Quadrant 4 containssmall particles having any possible mass distribution.

3.3.2. Analysis of Trends on Plane 1 –2The data cloud on the second factorial plane ( Fig. 4B) ischaracterized by a triangular structure. As a matter of fact,axes 2 and 3 are mainly correlated with the λ 1 and λ 2 variables.The observed triangle corresponds to the theoretical triangleexplained in paragraph 2, up to a scale factor. Shape trendsexplained in Sections 3.3.1 and 3.3.2 are reported on Fig. 4(D).

Fig. 5 – 5 different types of AA5182 particles.

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This plane allows to characterize particles according to thetype of theirmass distribution. The 3 types are illustrated withparticles C, D and E.

4. Application to the Comparison of TwoDeformation Stages

In mechanical studies, it is fundamental to understand thebreak-up of intermetallic particles in aluminium alloys. Wepropose here to use our morphological shape analysis methodto compare the two particle populations of an aluminiumsample at two deformation stages of an hot rolling process.

The first population is the one studied in Section 3. Itcorresponds to the beginning of the process (10% deforma-tion).The secondone corresponds to a more advanced stageof deformation (80%).

The comparison between two particle populations is madepossible by projecting one of them in the PCA representationspace of the other. As the deformation process progresses, thenumber of particles increases. Fig. 6 plots the 80% data pointsin the 10% factorial planes.

It is obvious that there are fewer particles in quadrant 2 of graph 6-a than in graph 4-a. This quadrant corresponds tolarge particles. As expected, large particles tend to disappear,while needle-liked and flat ones tend to appear, which wehave checked on a 2D histogram of plane 2 – 3. Large particlesare indeed the most brittle. As the deformation process goesalong, they break. Their pieces become new smaller particleswith simpler shapes.

5. Conclusion

In this paper, we have presented a set of morphologicalparameters adapted to the characterization of 3D complexshapes. We have shown that applying PCA on the measuredparameters was efficient to characterize morphological differ-ences inside a large population of intermetallic particles andto compare populations of a same sample at two stages of deformation. We now plan to use the results for clustering particles into different classes. Our final goal is to model themicrostructure evolution during hot-rolling process.

R E F E R E N C E S

[1] Crofton. On the theory of local probability. Phiols Trans R SocLond 1868;158:181– 99.

[2] Delarue A, Jeulin D. 3D morphological analysis of compositematerials with aggregates of spherical inclusions. Image AnalStereol 2003;22:153– 61.

[3] Greenacre MJ. Theory and applications of correspondenceanalysis. London: Academic Press; 1984.

[4] Holboth A, Pedersen J, Vedel Jensen E. A deformable templatemodel, with special reference to elliptical models. J MathImaging Vis 2002;17:131– 7.

[5] Lantuejoul C, Maisonneuve F. Geodesic methods in quantita-tive image analysis. Pattern Recogn 1984;17:177.

[6] Parra-Denis E, Ducottet C, Jeulin D. 3D image analysis of nonmetallic inclusions. Proc 9th European congress on Stereology,Zakopane, 10 – 13 may; 2005.

[7] Soille P. Morphological Image Analysis Principles and Appli-cations. Springer; 1999. p. 111 – 3.

Fig. 6 – Particles of the 80% deformed material projected onthe factorial planes of Fig. 4 .


3D complex shape characterization by statistical analysis

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