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

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Materials Characterization xx (2007) xxx–xxx

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Short communication

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

Received 9 August 2006; accepted 18 January 2007

EDPRAbstract

The goal of this paper is to describe a methodology for characterizing 3D complex shapes using 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 intermetallic particles of aluminium alloys investigatedusing X-ray microtomography. In that case, the analysis provides a description of shapes with a limited number of parameters, witha morphological interpretation for each of them. We finally demonstrate the practical interest of our work by comparing twopopulations extracted from the same aluminium sample at two deformation stages of a hot rolling process.© 2007 Elsevier Inc. All rights reserved.

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CORR1. Introduction

The microstructure of a material determines its phy-sical properties. Having an understanding of the micro-structure formation is a key tool for material scientists topredict the mechanical properties of the material and todevelop products with desired properties.

X-ray microtomography can now provide a 3D rep-resentation of the microstructure of materials with highresolution in a non destructive way. Image processing is

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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

Please cite this article as: Denis EP et al. 3D complex shape characterizaCharact (2007), doi:10.1016/j.matchar.2007.01.012

then essential to extract the relevant microstructuralcomponents and to perform 3D measurements to char-acterize quantitatively the material's microstructure ofinterest. These microstructural components often exhibitcomplex shapes, which makes their analysis difficult.

Many 3D shape analysis algorithms exist in theliterature [2,4]. However, most of the time, they onlyapply to simple 3D shapes or star-shaped objects. Hence,the analysis of 3D complex shapes like those encoun-tered in material studies required the development ofnew approaches.

In this paper, we propose a methodology to carry out3D complex shape analysis using morphologicalfeatures. This methodology is illustrated with theanalysis of intermetallic particles of aluminium alloys.It provides a description of shapes with a limited number

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63Fig. 1. Theoretical graph of λ2 versus λ1.

Fig. 2. Pairwise corre

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Please cite this article as: Denis EP et al. 3D complex shape characterizaCharact (2007), doi:10.1016/j.matchar.2007.01.012

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of parameters, with a morphological interpretation foreach of them. 2D or 3D plots can then be used to studythe shape variability of populations.

In the case of aluminium alloys, such analysis isuseful to reveal morphological differences between par-ticles and to track the deformation of the particles whenhot-rolling is applied to the studied alloy. During thisprocess used to transform aluminium slabs into sheets,the material undergoes important stress and strain andintermetallic particles then break up.

The processing was made on 3D images of an alu-minium sample of 1 mm2×1 cm at different rollingprocess stages of the material. X-ray microtomographywas performed at the European Synchrotron RadiationFacility (Grenoble, France). These images have aresolution of 0.7 μm3 and contain thousands of inter-metallic particles with volume ranging from 9 μm3 up to24.000 μm3.

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Fig. 3. PCA correlation circles (A) factorial axes 1–2 (B) factorial axes2–3.

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The following paper is organized as follows. First,we provide some Morphological features for 3D com-plex shape characterization. Second, we perform a sta-tistical multivariate analysis to select a set of parametersadapted to the morphology of particles. Third, results attwo stages of the hot rolling process are proposed todemonstrate the practical interest of the methodology.Finally, a Conclusion is given.

2. Morphological features

In this section, we provide a set of morphologicalparameters for characterizing 3D complex shapes. Theparameters can be divided into four categories: basicmeasures, shape indexes, geodesic measures, and massdistribution parameters.

2.1. Basic measures

They include volume and surface area:

- the volume (V) is calculated as the number ofvoxels that form the object.- the surface area (S) is estimated by the stereologicalmethod of Crofton [1].

2.2. Shape indexes

Shape indexes compare a studied shape to a referenceone [7]:

- the index of sphericity is defined as: Is=36πV2 /S2.

A sphere will have an index of sphericity equal to 1.- the index of compacity is Ic=6Is /π: It compares theshape to a cube instead of a sphere, a cube will havean index of compacity equal to 1.

2.3. Geodesic measures

Parameters of this category are based on the geodesicdistance [5], which is an important geometric measurefor understanding complex shapes of objects. Thegeodesic distance between two points x1 and x2belonging to a given shape X is equal to the length ofthe shortest path connecting x1 to x2, remaining includedin X. It allows to determine:

- the geodesic radius (Rmin) of a shape X whichcorresponds to the smallest ball included in X. Itprovides information about the size of the coreof the shape. It is normalized to correspond to ashape index which compares the volume of the

Please cite this article as: Denis EP et al. 3D complex shape characterizaCharact (2007), doi:10.1016/j.matchar.2007.01.012

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minimum ball included in the shape to the one ofthe object.- the geodesic elongation index (IGg), is a novelindex which we propose as an extension to 3D of thewell known geodesic stretching index in 2D [5]. Itcharacterizes the object elongation and is defined as:IGg=πLg

3 /6V, where Lg is the geodesic length. Lg isthe maximum length of a path which can be drawnwithin the shape.

2.4. Mass distribution parameters

Moments of inertia of an object depend on its shapeand characterize the distribution of mass within theshape. They correspond to the eigen values of the inertiamatrix of the shape [6] (computed from the center ofmass of X, under the assumption that the mass is uni-formly distributed and that the elementary volume is the

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voxel). They are normalized in order to be independentof the volume. If I1, I2, I3 denote the moments of iner-tia, the normalized moments λ1, λ2, λ3 are defined as:λi = Ii / I1+ I2+ I3, i=1,2,3. Their sum is equal to 1 andthey are ordered: λ1≥λ2≥λ3. From those equationsend the definition of inertia moments, the two followinginequalities can be deduced:

8i; ki V 0:5 and k2 z 0:5d 1� k1ð Þ

Plotting all these equations leads to a triangle havingoriginal properties to describe shapes (Fig. 1). At thetriangle vertices, we can distinguish 3 types of massdistribution within 3D objects: spherical, flat and needle.Between these extremities, shapes vary continuously.Along the triangle edges, shapes are prolate ellipsoid-typed, oblate ellipsoid-typed or flat ellipse-typed.

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Fig. 4. Data cloud in PC space and shape trends (A) Plane 1–2 (B) Plane 2

Please cite this article as: Denis EP et al. 3D complex shape characterizaCharact (2007), doi:10.1016/j.matchar.2007.01.012

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3. Shape statistical analysis

In the case of the analysis of a complex particlespopulation, a statistical study must be performed. Thegoal is to provide a description of shapeswith aminimumnumber of parameters. For that purpose, we propose inthis section to use the principal component analysis [3](PCA). This analysis and the way we select parameters isillustrated with a population of intermetallic particles.

3.1. Data matrix

Measurements of the previous morphological para-meters were made on 3500 intermetallic particles fora 10% deformed aluminium alloy. A pairwise correla-tion analysis between parameters presented in Fig. 2suggests to remove some parameters before applying

–3 (C) Plane 1–2 with shape trends (D) Plane 2–3 with shape trends.

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PCA. Indeed, as we can see on Fig. 2, the volume V andsurface S parameters are linearly correlated, as well as Isand the geodesic radius Rmin. Consequently, we chooseto disregard S and Rmin, as they do not bring any furtherinformation.

Finally, the considered parameters are: volume,sphericity index, geodesic elongation index and thenormalized eigen values of the inertia matrix (λ1, λ2).

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Fig. 5. 5 different types o

Please cite this article as: Denis EP et al. 3D complex shape characterizaCharact (2007), doi:10.1016/j.matchar.2007.01.012

The data matrix is therefore composed of 3500particles and 5 parameters describing each particle.Different types of PCA exist, varying in the waythe data are presented (centered/not centered–reduced/not reduced). In our case, it is appropriateto center and reduce the data because data pointsproject uniformly on all axes and they are measuredin different units.

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f AA5182 particles.

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Fig. 6. Particles of the 80% deformed material projected on thefactorial planes of Fig. 4.

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3.2. Principal components analysis

The computation of the PCA of our data matrixreturns the following percentages of the variability foreach eigen values: e1=45,5%, e2=28,5%, e3=13,8%,e4=8,0% and e5=4.2%. For the present study, weonly keep the first three axes since they represent 87.8%of the variability.

To understand the role of the initial variables in theformation of the principle axes, it is usual to projectthem onto the new axes leading to the correlation circlemaps (Fig. 3).

On the correlation circles, we observe that the geo-desic elongation index IGg is strongly negatively corre-lated with axis 1 and slightly correlated with axis 3. Theindex of sphericity Is is strongly positively correlatedwith axis 1 and close to IGg on axis 3. The volume isnegatively related with axis 1 and positively related withaxis 2. λ1 and λ2 are strongly negatively related withaxis 2 and they are mixed. They split according to axis 3where λ2 is positive and λ1 is negative.

The opposite correlation of IGg, V and Is with axis 1reflects that axis 1 characterizes elongation changes andthat the more elongated an object is, the larger it is. Axis2 suggests that the larger a particle, the smaller λ1 andλ2, which means, according to paragraph 2, that a largeparticles tend to have a spherical mass distribution,while small ones tend to have a flat or cylindrical massdistribution. Axis 3 allows to distinguish objects with aflat mass distribution from ones having a needle massdistribution. The interpretation of correlation circles ispresented with arrows on Fig. 4(C) and (D).

3.3. Analysis of morphological differences of alumi-nium particles

The previous interpretations of the principal axes areuseful to analyze the cloud of our data points in order toidentify some groups of particle shapes. The maps of theparticles on the two first factorial planes are given onFig. 4. It is clear that the shape of particles variescontinuously. No group of particles stands out.

3.3.1. Analysis of trends on plane 1–2From Fig. 4A, we can infer that small particles

(quadrant 2–3–4) are more numerous than large ones. Inquadrant 1, we observe a pointed distribution of objectsin the opposed direction of λ1 and λ2. It expresses thatthe higher Is, the more compact the object and naturally,the more spherical its mass distribution (particle B onFig. 5). Quadrant 2 corresponds to particles with a largevolume. Along the vertical axis, their mass distribution

Please cite this article as: Denis EP et al. 3D complex shape characterizaCharact (2007), doi:10.1016/j.matchar.2007.01.012

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gets more and more spherical (particle E). Along thehorizontal axes, objects gets more elongated (particle A).In quadrant 3, we find elongated particles (particle C).Quadrant 4 contains small particles having any possiblemass distribution.

3.3.2. Analysis of trends on plane 1–2The data cloud on the second factorial plane (Fig. 4B)

is characterized by a triangular structure. As a matter offact, axes 2 and 3 are mainly correlated with the λ1 andλ2 variables. The observed triangle corresponds to thetheoretical triangle explained in paragraph 2, up to ascale factor. Shape trends explained in Sections 3.3.1 and3.3.2 are reported on Fig. 4(D). This plane allows tocharacterize particles according to the type of their mass

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distribution. The 3 types are illustrated with particles C,D and E.

4. Application to the comparison of two deformationstages

In mechanical studies, it is fundamental to understandthe break-up of intermetallic particles in aluminiumalloys. We propose here to use our morphological shapeanalysis method to compare the two particle populationsof an aluminium sample at two deformation stages of anhot rolling process. The first population is the onestudied in Section 3. It corresponds to the beginning ofthe process (10% deformation). The second one corre-sponds to a more advanced stage of deformation (80%).

The comparison between two particle populations ismade possible by projecting one of them in the PCArepresentation space of the other. As the deformationprocess progresses, the number of particles increases.Fig. 6 plots the 80% data points in the 10% factorialplanes.

It is obvious that there are fewer particles in quadrant2 of graph 6-a than in graph 4-a. This quadrant corre-sponds to large particles. As expected, large particlestend to disappear, while needle-liked and flat ones tendto appear, which we have checked on a 2D histogram ofplane 2–3. Large particles are indeed the most brittle.As the deformation process goes along, they break.Their pieces become new smaller particles with simplershapes.

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5. Conclusion

In this paper, we have presented a set of morpholog-ical parameters adapted to the characterization of 3Dcomplex shapes. We have shown that applying PCA onthe measured parameters was efficient to characterizemorphological differences inside a large population ofintermetallic particles and to compare populations of asame sample at two stages of deformation. We now planto use the results for clustering particles into differentclasses. Our final goal is to model the microstructureevolution during hot-rolling process.

References

[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 Math ImagingVis 2002;17:131–7.

[5] Lantuejoul C, Maisonneuve F. Geodesic methods in quantitativeimage 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 Applica-tions. Springer; 1999. p. 111–3.

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