 Research
 Open Access
A germgrain model applied to the morphological study of dual phase steel
 Alessandra Micheletti^{1}Email authorView ORCID ID profile,
 Junichi Nakagawa^{2},
 Alessio A Alessi^{3},
 Daniela Morale^{3} and
 Elena Villa^{3}
https://doi.org/10.1186/s1336201600335
© Micheletti et al. 2016
Received: 17 January 2016
Accepted: 8 November 2016
Published: 22 November 2016
Abstract
The mechanical properties of dual Phase steels (DP steels) are strictly related to the spatial distribution and the geometry of the two phases composing the steel, ferrite and martensite. Due to the high costs to obtain images of sections of steel samples, one important industrial problem is the reduction of the number of 2D sections needed to build and simulate a geometric model which may reproduce in a realistic way the 3D geometry of the material. In this context, the availability of suitable techniques of parameter estimation or identification is fundamental to solve the problem.
In this work we present a germgrain model which approximates the main geometric characteristics of the martensite, taking into account the inhomogeneities of the material. The parameters of the model are estimated on the basis of the morphological characteristics of the images of about 150 tomographic sections of a real sample, quantified by the Minkowski functionals. Here we replace the Mahalanobis distance, introduced in previous literature, with the \(\mathcal{N}\)distance, which provides computational advantages. In order to test if the estimated model is reproducing the distribution of the Minkowski functionals of the real material, both confidence bands from the simulated model are computed and compared with the real data and techniques for the detection of functional outliers are applied to quantify the accuracy of fit of the estimated model.
Keywords
 dual phase steel
 germ grain model
 Minkowski functionals
 mathematical morphology
1 Introduction
Dual Phase steels (DP steels) have shown high potential for many applications due to their remarkable combination of high strength and good formability.
Here we consider a sample of steel formed by martensite and ferrite. The relative position and geometric structure of the two phases are responsible for the mechanical properties of the material, thus it is particularly important to provide statistical models which may reproduce the main geometric characteristics of the two phases. Our results are based on images of about 150 tomographic sections taken from a lab sample of steel.
The formation of the two phases of the material starts after a cooling phase of the melted alloy of iron and carbon, during which austenite is formed, followed by a rolling phase, transforming slabs of steel into thin metal foils.
A further cooling phase follows the rolling; during this phase the formation of ferrite starts. Crystals of ferrite nucleate mainly from the interfaces of the rolled (and thus deformed) austenite, and grow up to impinge on other crystals of ferrite, driven by the evolving field of carbon concentration. After a fixed time interval the formation of ferrite is stopped by a sudden quenching, during which the material that is still not transformed into ferrite, becomes martensite. The final result is a dual phase steel formed by ferrite and martensite, having a stochastic geometric structure.
In order to define a dynamical model able to reproduce the complete geometric structure of the material, a stochastic birth and growth process coupled with the evolution of the carbon field, the temperature evolution and the mechanical stresses due to rolling should be used (see [1] for similar models applied to polymer crystallization). A first model which goes in this direction, though facing the problem at only a macroscopic scale, neglecting the microscopic geometry, has been studied in [2].
The problem of building a random geometric model, at the microscopic level, for DP steels has already been faced in [3, 4]. In these papers the results were based on measurements taken on a 3D reconstruction of the material, obtained by means of a large number of 2D tomographic sections of a real sample. Since taking sections is expensive, here we aim to build a 3D model using methods for parameters identification applicable also in case of a lower number of available sections, in order to reduce the industrial experimental costs.
Furthermore in [3, 4] the material has been considered homogeneous, even though anisotropies in the real sample are present in one direction. In order to take such anisotropies into account, we here propose a different germgrain model, with an higher number of unknown parameters, and based on more detailed measurements. In such situation the common Mahalanobis distance, used in the previous papers, is computationally expensive to be used for the parameters estimation, since the sample covariance matrix becomes singular in presence of small samples. Thus we here introduce a new metric, the \(\mathcal{N}\)distance, which is convenient in geometric problems, as mentioned in [5].
As from a confidentiality agreement with Nippon Steel & Sumitomo Metal, who provided the real data, the images of the real sample will not be shown.
2 Structure of the austenite phase
We first considered the geometric structure of the interfaces of austenite after rolling, since nucleation of ferrite happens mainly on such interfaces, so that the location of the final ferrite and martensite crystals depends on the location of such interfaces.
The shape of the crystals of austenite before rolling is quite close to a 3D Voronoi tessellation, but after the deformation due to rolling, the interfaces between different crystals can be approximated by parallel planes, with random levels, as discussed in [3, 4] and supported by previous studies [6].
3 Morphological analysis of the martensite
In order to set up a geometrical model which may realistically represent the material, we first described quantitatively the geometrical structure of the martensite via a morphological analysis of the real sample. The morphology of a random set may be characterized [7, 8] by the densities of the relevant Minkowski functionals (the interested reader might see [9] for the definition and [10–12] for properties and examples of applications). Since the about 150 sections of our real sample have been taken at a distance equal to the side of a pixel, the real sample can be regarded either as a collection of (correlated) 2D sections, each represented by an image composed by 2D pixels, or as a 3D sample of the material, represented by a parallelepiped composed by 3D voxels, obtained by piling the sections one over the other. Thus we can either consider the threedimensional densities of Minkowski functionals, that is densities of Hausdorff measures per unit volume at different dimensions (volume density or volume fraction \(V_{V}\), volumetric surface density \(S_{V}\), volumetric density of average breadth \(B_{V}\) and volumetric density of EulerPoincaré characteristics \(E_{V}\)) on the whole 3D sample, or we may consider the twodimensional Minkowski functionals, that is densities of Hausdorff measures per unit area (volume fraction \(V_{A}\), areal surface density \(S_{A}\), and areal density of EulerPoincaré characteristics \(E_{A}\)), when we consider every single 2D section.
In general all these functionals are constant in space only if the random set under study is isotropic and stationary, i.e. its distribution is invariant under rigid motions.
The functionals have been computed according to the estimators described in [13], using the Matlab codes which can be downloaded from [14], both in their 2D or 3D versions.
The advantage of performing a volumetric study of the real sample is that in this way we may understand if the material presents some anisotropy or non stationarity in specific directions, and take these features into account in the construction of the model. By the way, since the industrial aim is to reduce the number of sections needed to set up a reasonable model, in the parameter estimation phase we need to use the 2D Minkowski functionals. Indeed they can be computed also in the case of a smaller number of available sections, even not equally spaced, and taken at distances bigger than those of our real sample.
3.1 Study of 3D Minkowski functionals
The functionals show a very low variability along the directions Z and X, while they show a sort of periodicity along the direction Y. This fact is confirmed by a visual inspection of the real sample, which shows a ‘striped’ structure along the direction Y, which is reasonable since the direction of rolling is parallel to the XZ plane.
We will use the Minkowski functionals of the closed real image as a reference in the procedure of parameter identification of the geometrical model reproducing the material, described in the following sections.
3.2 Study of the covariance and autocorrelation functions
The covariance function of a random closed set is related to the second order properties of the set and gives information on the stationarity and isotropy of the object under study [10, 11].
Other techniques to investigate the presence of anisotropies in a material are present in literature (see e.g. [15]), that we don’t apply here, since we are investigating only the presence of main sources of anisotropy, to build a first (and maybe rough) geometric model, but with a few unknown parameters. Anyway such techniques could be used in the future to refine the model.
Definition
The plot shows that in direction Y there is a tendency to clustering, since the covariance function in direction Y is lower than those in the other directions and, at scales bigger than \(h=25\), \(C(h)\) is also below the prescribed sill value. This confirms the striped structure of the martensite that had been already noticed by a visual inspection of the sample. On the other side, the shape of the covariance function in the directions X and Z confirms that the material is stationary in such directions.
Again we observe the presence of a significant (negative) autocorrelation only in direction Y, at lag 3, of the first three Minkowski functionals, revealing thus a periodicity of the material only in direction Y, with half period equal to 3 lags, that is 30 voxels.
The results obtained with both the covariance and the autocorrelation functions are consistent with the experimental conditions, which are schematically depicted in Figure 2: the anisotropies in the material are due to the rolling phase and thus in order to avoid to loose relevant information on its structure, the sections must be taken in a direction parallel to the Y axis; for example, as in our case, in direction orthogonal to Z.
3.3 Study of 2D Minkowski functionals
Since in the parameter estimation procedure we will use the estimates of the 2D Minkowski functionals on the sections of the material, we report here a preliminary study of these functionals.
Also in this case, as expected, the mean of the Minkowski functionals over the sections is oscillating more in direction Y than in direction X. Furthermore we note an increased local variability of the functionals in direction Y with respect to direction X.
4 A germgrain model
In order to set up a statistical model able to reproduce the mean geometric structure of the real sample of steel, we propose a germgrain model with spherical grains, depending on a small set of unknown parameters. The model will reproduce the structure of the ferrite phase, neglecting the interfaces between different crystals, so that the martensite will be represented by the empty space between different grains of the model.
We modelled the point process of germs as a clustered point process of NeymanScott type, taking into account that ferrite nucleates in the surrounding of parallel planes, and also observing that martensite in the real sample exhibits a ‘striped’ structure.
The NeymanScott point process ([10], Section 5.3) is obtained by generating a spatial Poisson point process of parents having intensity \(\lambda_{p}(x)\) and then surrounding the parents by a random number of daughter points, scattered independently and identically distributed around the parents. The parents are then removed and the NeymanScott process is formed just by the daughter points.
Hence the germs are generated according to the following algorithm.
Algorithm 1
Input:
\(n_{\text{planes}} = \text{number of parallel planes}\);
\([z_{1},\ldots, z_{n_{\text{planes}}}]= \text{levels of the parallel planes}\);
\(\sigma_{\text{vert}}= \text{standard deviation of the daughters' distribution in the vertical direction}\);
\(\sigma_{\text{hor}}=\text{standard deviation of the daughters' distribution in the horizontal direction}\);

Step 1: locate parallel planes, from which ferrite nucleates, into positions \(z_{1},\ldots, z_{\text{planes}}\);

Step 2: generate the number \(N_{g}\sim \operatorname{Poisson}(\lambda )\) of germs to be located in the 3D space;

Step 3: fix the number of parent germs to \(0.03\cdot N_{g}\);

Step 4: distribute the parent germs uniformly on the parallel planes;

Step 5: distribute the \(N_{g}\) daughter germs around the parents according to a 3variate normal distribution having diagonal covariance matrix given by$$\varSigma =\operatorname{diag}\bigl(\sigma_{\text{hor}}^{2}, \sigma_{\text{hor}}^{2}, \sigma_{\text{vert}}^{2}\bigr). $$
We used the results of preliminary analyses of the model in order to reduce the number of parameters to be estimated: the proportion of parents, fixed to the 3% of the daughter germs, has been obtained by applying the optimisation technique described in [4] and by including the proportion of parents between the parameters to be estimated.
The number of parallel planes from which the germ process originates has been fixed to 7, since again this was the optimal estimate for this quantity in [4]. The seven levels of the planes \([z_{1},\ldots, z_{7}]\) are among the parameters to be estimated.
The grains have been modeled as independent spheres of random radius \(R=L\cdot\rho\), where \(L=25\) is a constant representing the maximum possible radius of the spheres (it has again been fixed to 25 because this was the optimal value for this parameter obtained in [4]) and ρ is a random variable distributed as a \(\operatorname{Beta}(3,b)\) distribution, where b is a parameter to be estimated. In previous works [3, 4] the random radii of the spheres had a distribution obtained by a mixture of two Beta’s, one favouring small radii and the other favouring big radii, but results of parameters estimation proved that the two estimated distributions were not much different. Thus, in this work we used only a single distribution in order to reduce the number of parameters to be estimated.
In order to avoid edge effects, the simulation of the model has been performed in a window of observation enlarged by L on each side, and, then, only the central portion of the window with dimensions equal to the real sample has been considered.
5 Parameters estimates
For each set of the parameters \(\underline{p}=( [z_{1},\ldots, z_{7}], \sigma_{\text{hor}}, \sigma_{\text{vert}}, b,\lambda)\) we performed 10 simulations of the germ grain model, each producing a 3D binary sample with the same dimensions as the real one. Each simulated sample was then cut orthogonally to the Z direction, obtaining the same number of sections available from the real sample. On each section we estimated the volume, surface, and Euler characteristic areal densities, i.e. the densities of 2D Minkowski functionals. The parameters can be estimated by minimising a suitable distance between the values of the densities estimated on the sections of the (closed) real sample and the densities estimated on the sections of the simulated germgrain model.
In [3, 4] the Mahalanobis distance has been used, but since here we want to take into account the inhomogeneities of the material in direction Y, we need to measure the Minkowski functionals more in detail in this direction. As will be clear in the following, this implies that our data will be random vectors of rather high dimension (at least 72), and the estimate of the corresponding covariance matrix, to be used in the Mahalanobis distance, would need a very high number of simulated experiments, increasing thus too much the computational costs. We then looked for other metrics which are more computationally efficient also in presence of high dimensional data.
5.1 The \(\mathcal{N}\)distance
A suitable distance to compare geometric descriptors of different random closed sets has been proven to be the \(\mathcal{N}\)distance (see [5, 17]).
Definition 1
Definition 2
For the discussion on the properties of the \(\mathcal{N}\)distance the interested reader may see [5, 17], while in [18] an example of application can be found.
We describe here shortly how the \(\mathcal{N}\)distance will be applied in the context of our application.
Assume that m geometrical quantities \(\underline{X}=[X_{1}, \ldots, X_{m}]'\) are measured for each available section of the simulated material. If k sections are available, the measurements will be collected in a data matrix \({\mathcal{X}}\) having dimension \(m\times k\). Note that in our case, if the sections are not sufficiently far apart, the measures of the geometrical quantities can not be considered independent, and thus the columns of \({\mathcal{X}}\) do not form an i.i.d. sample of size k of a multivariate random variable \(\underline{X} \in\mathbb{R}^{m}\). The independence of the columns of the data matrix unfortunately is essential to the theoretical proof of many properties of the \(\mathcal{N}\)distance. We will disregard this problem for a moment.
The same m geometrical quantities are also measured on the same number k of sections of the real sample material, forming thus a data matrix \({\mathcal{Y}}\), again having dimensions \(m\times k\).
In our application, we considered two approaches in the construction of the data matrices \({\mathcal{X}}\), \({\mathcal{Y}}\): the first one gives more emphasis to direction Y, which is the one with bigger variability, the second one gives equal emphasis both to directions Y and X.
Each section both of the real and the simulated material is represented by an image having dimension \(200\times240\) pixels in directions \(X\times Y\).
Case 1. Construction of \(\mathcal{Y}\): in order to ‘increase the independence’ of the elements of our sample, along Z direction we considered only one section out of two. As already mentioned, this choice does not guarantee that the theoretical properties of the \(\mathcal{N}\)distance hold, but enforces the possibility that they are anyway satisfied. The study of the autocorrelation functions of the Minkowski functionals in direction Z, reported in Figure 6, suggests that a good choice to obtain (almost) uncorrelated sections would be to consider one section out of 20, since the autocorrelations of the first three functionals are almost null at lag 2 (note that each lag corresponds to 10 sections), but in this way we would reduce too much our sample.

the area densities of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[V^{\text{real}}_{1},\ldots, V^{\text{real}}_{24} \bigr]_{\text{section } l}; $$

the perimeter densities of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[S^{\text{real}}_{1},\ldots, S^{\text{real}}_{24} \bigr]_{\text{section } l}; $$

the density of EulerPoincaré characteristics of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[E^{\text{real}}_{1},\ldots, E^{\text{real}}_{24} \bigr]_{\text{section } l}. $$

the area densities of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[V^{\text{sim}}_{1}(\underline{p},r),\ldots, V^{\text{sim}}_{24}(\underline{p},r)\bigr]_{\text{section } l}; $$

the perimeter densities of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[S^{\text{sim}}_{1}(\underline{p},r),\ldots, S^{\text{sim}}_{24}(\underline{p}, r)\bigr]_{\text{section } l}; $$

the density of EulerPoincaré characteristics of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[E^{\text{sim}}_{1}(\underline{p}, r),\ldots, E^{\text{sim}}_{24}(\underline{p},r)\bigr]_{\text{section } l}. $$

the area densities of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[V^{Y,\text{real}}_{1},\ldots, V^{Y,\text{real}}_{24} \bigr]_{\text{section } l}; $$

the perimeter densities of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[S^{Y,\text{real}}_{1},\ldots, S^{Y,\text{real}}_{24} \bigr]_{\text{section } l}; $$

the density of EulerPoincaré characteristics of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[E^{Y,\text{real}}_{1},\ldots, E^{Y,\text{real}}_{24} \bigr]_{\text{section } l}; $$

the area densities of martensite computed in each 2dimensional stripe of 10 pixels along direction X,$$\bigl[V^{X,\text{real}}_{1},\ldots, V^{X,\text{real}}_{20} \bigr]_{\text{section } l}; $$

the perimeter densities of martensite computed in each 2dimensional stripe of 10 pixels along direction X,$$\bigl[S^{X,\text{real}}_{1},\ldots, S^{X,\text{real}}_{20} \bigr]_{\text{section } l}; $$

the density of EulerPoincaré characteristics of martensite computed in each 2dimensional stripe of 10 pixels along direction X,$$\bigl[E^{X,\text{real}}_{1},\ldots, E^{X,\text{real}}_{20} \bigr]_{\text{section } l}. $$

the area densities of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[V^{Y,\text{sim}}_{1}(\underline{p},r),\ldots, V^{Y,\text{sim}}_{24}(\underline{p},r)\bigr]_{\text{section } l}; $$

the perimeter densities of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[S^{Y,\text{sim}}_{1}(\underline{p},r),\ldots, S^{Y,\text{sim}}_{24}(\underline{p}, r)\bigr]_{\text{section } l}; $$

the density of EulerPoincaré characteristics of martensite computed in each 2dimensional stripe of 10 pixels along direction Y,$$\bigl[E^{Y,\text{sim}}_{1}(\underline{p}, r),\ldots, E^{Y,\text{sim}}_{24}(\underline{p},r)\bigr]_{\text{section } l}; $$

the area densities of martensite computed in each 2dimensional stripe of 10 pixels along direction X,$$\bigl[V^{X,\text{sim}}_{1}(\underline{p},r),\ldots, V^{X,\text{sim}}_{20}(\underline{p},r)\bigr]_{\text{section } l}; $$

the perimeter densities of martensite computed in each 2dimensional stripe of 10 pixels along direction X,$$\bigl[S^{X,\text{sim}}_{1}(\underline{p},r),\ldots, S^{X,\text{sim}}_{20}(\underline{p}, r)\bigr]_{\text{section } l}; $$

the density of EulerPoincaré characteristics of martensite computed in each 2dimensional stripe of 10 pixels along direction X,$$\bigl[E^{X,\text{sim}}_{1}(\underline{p}, r),\ldots, E^{X,\text{sim}}_{20}(\underline{p},r)\bigr]_{\text{section } l}. $$
6 Numerical results
Since parameters \([z_{1},\ldots, z_{7}]\) are integers while the other parameters are real numbers, we needed to apply an optimisation algorithm to a function which is not expressed in algebraic form and depending upon mixed integer and real parameters, so that we decided to apply a genetic algorithm for the minimisation procedure stated in (2).
All the simulations and the optimisation procedure have been performed using Matlab R2015a. In order to speed up the execution of the genetic algorithm, the Parallel Computing Toolbox has been used to parallelise the algorithm on a parallel machine using up to 30 simultaneous workers. Anyway the computational time of the optimisation procedure is about 48 hours.
Optimal values of the parameters obtained with the procedure described in Case 1
Parameter  Optimal value 

\(z_{1}\)  15 
\(z_{2}\)  56 
\(z_{3}\)  100 
\(z_{4}\)  132 
\(z_{5}\)  180 
\(z_{6}\)  193 
\(z_{7}\)  244 
b  4.38 
\(\sigma_{\text{vert}}\)  9.55 
\(\sigma_{\text{hor}}\)  12.08 
λ  14,687 
Optimal values of the parameters obtained with the procedure described in Case 2
Parameter  Optimal value 

\(z_{1}\)  12 
\(z_{2}\)  53 
\(z_{3}\)  91 
\(z_{4}\)  132 
\(z_{5}\)  178 
\(z_{6}\)  201 
\(z_{7}\)  246 
b  4.74 
\(\sigma_{\text{vert}}\)  9.37 
\(\sigma_{\text{hor}}\)  13.14 
λ  14,379 
The numerical values of the estimated parameters seem to be quite similar, but techniques to validate and compare quantitatively the two models will be applied in the next section.
7 Models comparison and validation
7.1 Validation based on confidence bands
The results reveal that both models may capture some insights of the overall variability of the true sample, in terms of the location and amplitude of the oscillations of the functionals in direction Y. By comparing the relative deviations between the mean of the functionals computed on the simulations and the mean of the functionals computed on the real sample, we observe that the strategy stated in Case 2 produces a very small gain in precision of the agreement along direction X, but also a big loss along direction Y. This fact suggests that the strategy stated in Case 1 should be preferred.
We observe that the scatter in the estimates is reducing while the optimization procedure progresses, but our estimators are biased. The bias may be related to the small number of simulations that we perform at each iteration of the genetic algorithm, or to the correlation present between different sections of the sample, which does not guarantee the optimality properties of the \(\mathcal{N}\)distance. It must be further investigated how to reduce such bias.
Note that the confidence bands shown in Figures 12 and 13 are built using pointwise confidence intervals at each spatial lag; they thus provide information only on the ‘marginal’ distribution of the Minkowski functionals at each lag. We then introduced the functional boxplots which provide bands which take into account the ‘joint’ distribution of the Minkowski functionals at different lags, by regarding them as continuous functions.
7.2 Validation based on functional boxplots
Functional boxplots have been introduced in Functional Statistics as an instrument to detect and identify the presence of possible outliers in a dataset where each datum is represented by a function. Functional boxplots are based on the definition of band depth (BD) measures, or modified band depth (MBD) measure introduced in [20], where each function in the sample is ordered and ranked from the center outward and, thus, it is possible to define functional quantiles and the centrality or outlyingness of an observation. The construction of functional boxplots, as well as the associated outlier detection rule are described in [19]. In the same paper, a simulation study comparing the performance of functional boxplots with other techniques present in literature for functional outliers detection is also reported, showing that functional boxplots provide a very reliable instrument to detect outliers in a functional dataset.
Note also that the almost total absence of outliers in the functionals reveals that the distribution of the Minkowski functionals has a tendency to be concentrated around the (functional) median, without showing long or heavy tails. This is a remarkable property in univariate statistics, since statistics having distributions without heavy tails give usually origin to robust parameter estimators.
8 Conclusions
In this paper we faced the problem of building a realistic stochastic geometric model for ferritemartensite dual phase steel, based on the availability of about 150 sections of one sample of the material. After a morphological analysis of the given real sample, we proposed a germgrain model with germs scattered around some nucleation planes, and spherical random grains, depending on a set of 11 unknown parameters. Since the distribution of the random variables involved in the model are very difficult to be theoretically retrieved, classical statistical methods for parameters estimation, like maximum likelihood, could not be applied to this case. Thus we based our parameter estimation technique on the minimization of a suitable distance (the \(\mathcal{N}\)distance) between the Minkowski functionals of the real data and the mean of the simulated data. This choice was motivated by the fact that the Minkowski functionals characterize the morphology of a random set. Two different ways to estimate the Minkowski functionals have been tested and the best method (that is the one described in Case 1, in Section 5) has been selected, on the basis of a comparison between the results of the simulations of the model and the real data. The model validation has been performed by building confidence bands and functional boxplots for the Minkowski functionals.
Unfortunately a simulated test revealed that the parameter estimators are biased, but since the functionals computed on the real sample are always included in the confidence bands, and are not identified as outliers in the functional boxplots, we conclude that the proposed model is coherent with the overall variability of the real data, and can thus be used to reproduce the geometric characteristics of the real sample. Methods to correct the bias should anyway be studied to improve the results.
The advantage of our method is that the parameters of the model can be estimated starting from any number of available sections of the real material, thus they can be identified also when the industrial sampling costs need to be reduced. Obviously the reduction in the number of sections would lead to an increase of the variability of the estimated parameters. Such increase could be quantified for example by computing the mean increase in the \(\mathcal{N}\)distance between the Minkowski functionals of the real data and of the simulated optimal model, or by comparing the amplitude of the confidence bands reported in Figure 12 with the analogous ones obtained via the optimal model estimated reducing the number of sections, or via other evaluations of the fitting between the real data and the simulated optimal model.
Such comparisons would give indications to the industry on the minimum number of sections needed to avoid a dramatic increase in the uncertainty in the estimation procedure.
Declarations
Acknowledgements
AM, AA, DM, EV thank Nippon Steel & Sumitomo Metal Corporation for the project funding and for the constant support during all the project phases. Special thanks are due to prof. Vincenzo Capasso, ADAMSS Centre, Università degli Studi di Milano, for inspiring and fruitful discussions and for making this collaboration with Nippon Steel & Sumitomo Metal Corporation possible.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Capasso V, editor. Mathematical modelling for polymer processing: polymerization, crystallization, manufacturing. Mathematics in industry. vol. 2. Heidelberg: Springer; 2003. MATHGoogle Scholar
 Suwanpinij P, Togobytska N, Prahl U, Weiss W, Hömberg D, Bleck W. Numerical cooling strategy design for hot rolled dual phase steel. Steel Res Int. 2010;81:10019. View ArticleGoogle Scholar
 Micheletti A, Nakagawa J, Alessi AA, Capasso V, Grimaldi D, Morale D, Villa E. Mathematical morphology applied to the study of dual phase steel formation. In: Russo G, Capasso V, Nicosia G, Romano V, editors. Progress in industrial mathematics at ECMI 2014. Berlin: Springer; 2016. Google Scholar
 Micheletti A, Nakagawa J, Alessi AA, Capasso V, Morale D, Villa E. Mathematical morphology and uncertainty quantification applied to the study of dual phase steel formation. In: Papadrakakis M, Papadopoulos V, Stefanou G, editors. Proceedings of UNCECOMP 2015, 1st ECCOMAS thematic conference on uncertainty quantification in computational sciences and engineering. 2015. p. 71432. ISBN:9789609999496. Google Scholar
 Rachev ST, Klebanov LB, Stoyanov SV, Fabozzi FJ. The methods of distances in the theory of probability and statistics. Berlin: Springer; 2013. View ArticleMATHGoogle Scholar
 Villa E, Rios PR. Transformation kinetics for nucleation on random planes and lines. Image Anal Stereol. 2011;30:15365. MathSciNetView ArticleMATHGoogle Scholar
 Møller J, Stoyan D. Stochastic geometry and random tessellations. In: Tessellations in the sciences: virtues, techniques and applications of geometric tilings. Berlin: Springer; 2008. Google Scholar
 Matheron G. Random sets and integral geometry. New York: Wiley; 1975. MATHGoogle Scholar
 Capasso V, Villa E. On the geometric densities of random closed sets. Stoch Anal Appl. 2008;26:784808. MathSciNetView ArticleMATHGoogle Scholar
 Chiu SN, Stoyan D, Kendall WS, Mecke J. Stochastic geometry and its application. 3rd ed. New York: Wiley; 2013. View ArticleMATHGoogle Scholar
 Serra J. Image analysis and mathematical morphology. vol. 1. San Diego: Academic Press; 1982. MATHGoogle Scholar
 Benes V, Rataj J. Stochastic geometry: selected topics. Berlin: Springer; 2004. MATHGoogle Scholar
 Legland D, Kieu K, Devaux MF. Computation of Minkowski measures on 2D and 3D binary images. Image Anal Stereol. 2007;26:8392. http://www.iasiss.org/ojs/IAS/article/view/811. MathSciNetView ArticleMATHGoogle Scholar
 Legland D. Geometric measures in 2D/3D images. http://it.mathworks.com/matlabcentral/fileexchange/33690geometricmeasuresin2d3dimages.
 Gutkowski P, Jensen EBV, Kiderlen M. Directional analysis of digitized threedimensional images by configuration counts. J Microsc. 2004;216:17585. MathSciNetView ArticleGoogle Scholar
 Chatfield C. The analysis of time series: an introduction. 6th ed. London: Taylor & Francis; 2005. MATHGoogle Scholar
 Klebanov LB. NDistances and their applications. Prague: Karolinum Press; 2005. Google Scholar
 Benes V, Lechnerovà R, Klebanov L, Slàmovà M, Slàmab P. Statistical comparison of the geometry of secondphase particles. Mater Charact. 2009;60:107681. View ArticleGoogle Scholar
 Sun Y, Genton MG. Functional boxplots. J Comput Graph Stat. 2011;20:31634. MathSciNetView ArticleGoogle Scholar
 LòpezPintado S, Romo J. On the concept of depth for functional data. J Am Stat Assoc. 2009;104:71834. MathSciNetView ArticleMATHGoogle Scholar