# Identification of recrystallized grains in EBSD maps: Automated adaptive thresholding using a Gaussian mixture approach

Hugo Latuner, Julien Favre, A. Helstroffer, G. Inacio da Rosa, P. Joly, Claire Maurice, E. Plancher, M. Roth, Christophe Desrayaud, Guillaume Kermouche

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## ► To cite this version:

Hugo Latuner, Julien Favre, A. Helstroffer, G. Inacio da Rosa, P. Joly, et al., Identification of recrystallized grains in EBSD maps: Automated adaptive thresholding using a Gaussian mixture approach. Materials Characterization, 2026, 231, pp.115920. 10.1016/j.matchar.2025.115920. emse-05483396

HAL Id: emse-05483396

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Submitted on 29 Jan 2026

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# Identification of recrystallized grains in EBSD maps: Automated adaptive thresholding using a Gaussian mixture approach

H. Latuner<sup>a,b,*</sup>, J. Favre<sup>a</sup>, A. Helstroffer<sup>c</sup>, G. Inacio Da Rosa<sup>b</sup>, P. Joly<sup>b</sup>, C. Maurice<sup>a</sup>, E. Plancher<sup>b</sup>, M. Roth<sup>c</sup>, C. Desrayaud<sup>a</sup>, G. Kermouche<sup>a</sup>

## ARTICLE INFO

### Keywords:

Dynamic recrystallization

EBSD

Hot deformation

Principal component analysis

Gaussian mixture model

Adaptive thresholding

Uncertainty band

## ABSTRACT

Modeling dynamic recrystallization requires a clear knowledge of how grain populations evolve during hot deformation. Electron backscatter diffraction (EBSD) is a common technique for discriminating between populations of new recrystallized grains and strain-hardened grains. One conventional method for identifying grain populations using EBSD data is to apply a predetermined threshold to grain orientation spread (GOS) maps. For a given alloy, the threshold must be manually adjusted, as it depends on the thermo-mechanical conditions investigated, making this conventional method user-dependent. This paper introduces a novel approach that aims to provide a more robust and reproducible method for identifying grain populations from EBSD data. This method uses principal component analysis (PCA) combined with Gaussian mixture modeling (GMM) to achieve automatic adaptive thresholding of grain populations. It incorporates an integrated uncertainty control for errors caused by threshold selection. This approach was used to monitor microstructural changes during the dynamic recrystallization of a 304 L austenitic stainless steel. Adaptive thresholding is expected to provide an alternative to conventional methods for recrystallization tracking.

## 1. Introduction

Hot processing plays a crucial role in the production of metal components. This process shapes the part as intended while also controlling the resulting microstructure and mechanical properties. During this stage, various microstructural mechanisms take place, such as work hardening, recovery, dynamic recrystallization, post-dynamic recrystallization, and grain growth. Quantifying these mechanisms is crucial for predicting microstructural changes during hot processing [1–6]. The distinction between recrystallized strain-free grains and deformed grains is critical for accurately modeling the mechanisms of nucleation, grain growth, and recovery that take place during hot deformation [7]. The simultaneous occurrence of these mechanisms in the dynamic recrystallization regime complicates the process of grain separation even further, as newly formed grains may undergo additional deformation [8].

Thanks to its continuous development over the past three decades, EBSD has become a powerful and widely used technique for

characterizing microstructural changes during hot deformation [9]. EBSD provides crystallographic information at the tenth-of-a-micron scale. This enables the reconstruction of grains and subgrains, measurement of size distribution, and estimation of orientation-based descriptors such as kernel average misorientation (KAM) and grain orientation spread (GOS). These descriptors are particularly useful for discriminating between deformed and strain-free recrystallized grains due to their sensitivity to internal lattice distortions. As a result, EBSD has emerged as the principal method for investigating recrystallization mechanisms. Yet, the challenge lies in establishing robust criteria for the classification of grains into recrystallized and deformed populations.

There is no consensus on the optimal method for distinguishing grain populations from EBSD data [10,11]. The most widely adopted approach involves a fixed GOS threshold. GOS is defined as the average misorientation between the pixels of a grain and the mean orientation of the grain.

However, the threshold varies considerably from study to study and is highly user-dependent, which can lead to arbitrary classification and

<footnote><sup>a</sup> Mines Saint-Etienne, CNRS, UMR 5307 LGF, Centre SMS, 158 Cours Fauriel, 42023 Saint-Etienne, France</footnote>

<footnote><sup>b</sup> FRAMATOME, Mechanical Engineering Division - Materials and Technology Department La Défense, France</footnote>

<footnote><sup>c</sup> FRAMATOME, Le Creusot - Forge shop - R&D, Le Creusot, France</footnote>

<footnote>* Corresponding author at: Mines Saint-Etienne, CNRS, UMR 5307 LGF, Centre SMS, 158 Cours Fauriel, 42023 Saint-Etienne, France. E-mail address: hugo.latuner@emse.fr (H. Latuner).</footnote>
inaccurate identification. For austenitic stainless steels, the GOS thresholds used to detect recrystallized grains vary over a wide range. Pinto et al. [12] used a fixed GOS threshold of 1° on EBSD maps of AISI 316 L steel. Kulakov et al. [13] employed 1.5° for REX734 austenitic stainless steel. For the same material, Lu et al. [14] used a fixed GOS threshold of 2.6° to detect dynamically recrystallized grains. Another method that uses fixed thresholds is based on a combination of descriptors. Ayad et al. [5] used a fixed GOS threshold of 2.5° on IF steel and a second GOS threshold normalized by grain diameter, GOS/D, of 1°/µm.

Such a dispersion clearly underlines the absence of a robust and transferable criterion. In particular, under dynamic conditions, where recrystallized grains may rapidly undergo secondary work hardening, visual separation of the two populations becomes complex and is open to interpretation [8]. To overcome these limitations, advanced statistical methods can be applied to EBSD data, thereby avoiding the biases inherent to manual thresholding. There are almost no referenced methods in metallurgy that address this issue. One of the few methods proposed in the literature has been developed to study dynamic recrystallization in the geology of the Earth's crust. Yeo et al. [15] employed a Gaussian mixture model (GMM) fitted to GOS distributions to detect and quantify recrystallized grain populations. GMM enables a dataset to be represented as a distribution through a combination of multivariate Gaussians. It can be used to determine an adaptive threshold, which avoids the subjectivity and the induced inaccuracies of a manually fixed threshold. Nevertheless, this approach presents some limitations. It relies on the GOS rather than combining multiple microstructural descriptors, which could improve the discrimination between grain populations. Additionally, in the context of modeling recrystallization mechanisms, the number of grain populations should not be treated as a variable determined *a posteriori*. Since the physical process involves two distinct states — recrystallized and non-recrystallized grains — it is essential to work with a fixed number of two populations rather than allowing a data-driven number of components.

This paper proposes a method to differentiate recrystallized grains from deformed grains in the context of discontinuous dynamic recrystallization (DDRX) of hot-deformed 304 L austenitic stainless steel. First, the experimental methodology is presented, including a description of the samples used and the process of obtaining reliable EBSD data. Next, the statistical methods used to develop the PCA-GMM method are presented. Then, the limitations of using a fixed GOS threshold are highlighted to emphasize the need for an adaptive threshold. Principal component analysis (PCA) and a Pearson correlation study were applied to eight microstructural descriptors derived from EBSD data. This approach identified the two most relevant descriptors for monitoring recrystallization in 304 L steel. Based on these parameters, a grain identification method combining PCA projection with a GMM fit is presented in detail. Compared to conventional fixed-threshold methods, this PCA-GMM framework reduces user dependency by automatically determining an adaptive threshold from the statistical distribution of the EBSD data. Additionally, an uncertainty band is introduced around the threshold, enabling ambiguous grains lying between the two populations to be flagged rather than misidentified, thus ensuring greater accuracy and reproducibility of the identification. Finally, the results obtained with this method are presented and discussed in relation to the dataset.

## 2. Experimental methodology

To develop and validate the proposed method, EBSD datasets that are representative of partially recrystallized microstructures are necessary. These datasets are obtained by processing 304 L austenitic stainless steel under different thermomechanical conditions. EBSD maps displaying both recrystallized and deformed grains are acquired after uniaxial compression tests. Several compression tests were conducted at varying strain levels and strain rates to obtain diverse conditions for

recrystallization progression.

### 2.1. Initial material and sample preparation

The material is an austenitic stainless steel X2 CrNi19.10 with controlled nitrogen content, similar to AISI grade 304 L. A residual quantity of ferrite is present in the material, around 3 vol%, but this study primarily examines the austenitic phase.

The as-received material underwent industrial hot forging followed by slow cooling at room temperature. Cylindrical compression samples of 15 × 9.4 mm² were machined along the longitudinal direction of the initial forged part (Fig. 1).

The samples were lubricated with boron nitride and subjected to high-temperature compression using a Schenck-Instron servo-hydraulic uniaxial compression device. An infrared heating furnace coupled with two servo-controlled thermocouples was used to heat and maintain a constant temperature throughout the deformation stage. The maximum temperature difference in the furnace was less than 2.5 °C [16]. Compression tests were performed at 1100 °C over strain levels ranging from $\bar{e} = 0.1$ to $\bar{e} = 0.8$ at a strain rate of 0.05 s$^{-1}$ in order to reproduce the thermo-mechanical conditions encountered during forging. At the end of the compression test, samples were quenched in water. The time between the end of the test and quenching was measured at less than one second.

Samples suitable for EBSD measurements were prepared using a standard metallographic preparation, involving mechanical polishing to a final granulometry of 0.25 µm. This step was followed by electro-polishing with a 6 % HClO₄ and 94 % ethanol solution for 30 s at 20 V at 6 °C.

### 2.2. EBSD measurement

EBSD measurements were performed in a ZEISS Supra 55 VP FEG SEM equipped with an *Oxford Instruments Symmetry 2* CMOS EBSD camera. The SEM-EBSD conditions used throughout the study are listed in Table 1.

Data post-processing was performed with *Matlab R2022b* using the *MTEX* library version 5.11.2 [17]. To avoid losing relevant substructural information, the EBSD step size should be comparable to the size of the typical substructure. If the step size is too large in relation to the subgrains or dislocation cells, the substructure will not be fully resolved. Conversely, as the step size decreases, the influence of angular resolution on orientation measurements becomes increasingly significant [18]. In our material, the typical size of smaller objects of interest, such as subgrains or dislocation cells, in the substructure is approximately 1–5 µm. The EBSD step size was set at 1 µm. In order to generate a dataset, the EBSD step size must remain constant since the microstructural descriptors depend on it. The angular misorientation between two adjacent pixels across a low-angle grain boundary (LAGB) is set between 0.5° and 10°. The criterion for high-angle grain boundaries (HAGBs) is set at 10°; this choice is justified in **Appendix C**. In the grain reconstruction process, objects with fewer than 10 pixels and less than 50 % indexation are systematically eliminated from the dataset. A total of 11,619 grains were reconstructed from EBSD data for this study. No

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Schematic representation of the samples before and after hot compression, as well as the area observed by EBSD.</figcaption></figure>
<table><caption>**Table 1**

EBSD conditions used for the microstructural characterization.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Step size</td><td>1 µm</td></tr><tr><td>Integration time</td><td>10 ms</td></tr><tr><td>Hough resolution</td><td>100</td></tr><tr><td>Number of bands</td><td>12</td></tr><tr><td>Mapping area</td><td>2.67 mm²</td></tr><tr><td>Indexation rate</td><td>Over 99 %</td></tr><tr><td>Indexation method</td><td>Optimized band detection</td></tr></tbody></table>

denoising filter has been employed in the reconstruction of grains to avoid bias in local misorientation data. This choice implicitly assumes that the noise level is homogeneous across all grains, thereby preventing any artificial smoothing that could affect the calculation of GOS and KAM. Such an assumption is not expected to represent a first-order effect in the present data treatment.

## 2.3. EBSD microstructural descriptors

Several microstructural descriptors can be extracted from EBSD data. In total, eight are analyzed in this study, as defined in Table 2. Some of these descriptors quantify local, intragranular misorientations. They can be used to visualize strain levels within grains and help distinguish recrystallized from deformed ones [19]. Among them, GOS and the grain average KAM (GAKAM) are widely used.

The GOS value is calculated for each grain as the mean misorientation between the pixels of the grain and the mean orientation of the grain. Thus, each grain has only one GOS value. It is defined as [20]:

$$GOS_i = \frac{1}{N} \sum_{j=1}^{N} \omega(g_j, <g_i>) \quad (1)$$

where N is the number of pixels of the grain i and $g_j$ is the orientation of the pixel j, and $<g_i>$ is the mean orientation of the grain i. $\omega(g_j, <g_i>)$ is the misorientation angle between the pixel j of the grain i.

The other microstructural descriptor, GAKAM, is the average misorientation of pixels relative to their neighbors. The GAKAM value of a grain i is defined as [19]:

<table><caption>**Table 2**

Description of the selected microstructural descriptors.</caption><thead><tr><th>Microstructural<br>descriptor</th><th>Description</th><th>Information</th></tr></thead><tbody><tr><td>Grain orientation<br>spread (GOS)</td><td>Mean misorientation between<br>the pixels of a grain and the<br>mean orientation of the grain</td><td>Local misorientations</td></tr><tr><td>Grain average kernel<br>average<br>misorientation<br>(GAKAM)</td><td>Grain average of the average<br>misorientations of the pixels<br>relative to their neighbors<br>(KAM)</td><td>Local misorientations</td></tr><tr><td>Grain median kernel<br>average<br>misorientation<br>(GMKAM)</td><td>Grain median of the KAM</td><td>Local misorientations</td></tr><tr><td>Standard deviation of<br>the KAM | stdev<br>(KAM)</td><td>Grain standard deviation of<br>the KAM</td><td>Local misorientations</td></tr><tr><td>Mean angular deviation<br>(MAD)</td><td>Mean angular deviation<br>between the simulated and the<br>measured Kikuchi bands</td><td>Raw EBSD data:<br>electron backscatter<br>pattern quality</td></tr><tr><td>Grain-averaged band<br>slope (BS)</td><td>Describes the intensity<br>gradient at the edge of Kikuchi<br>bands</td><td>Raw EBSD data:<br>electron backscatter<br>pattern quality</td></tr><tr><td>Grain-averaged aspect<br>ratio (AR)</td><td>Grain ratio of its maximum<br>length to its maximum width</td><td>Geometric: Grain<br>morphology</td></tr><tr><td>Grain diameter (D)</td><td>Diameter of the equivalent<br>circle of a grain</td><td>Geometric: Grain<br>morphology</td></tr></tbody></table>

$$GAKAM_i = \frac{1}{N} \sum_{j=1}^{N} \left( \frac{1}{K} \sum_{k=1}^{K} \omega(g_j, g_k) \right) = \frac{1}{N} \sum_{j=1}^{N} KAM_j \quad (2)$$

where K is the number of neighboring pixels.

When estimating the average KAM for a grain in lognormal or otherwise asymmetric distributions, it is more appropriate to use the median rather than the mean, as the median is less sensitive to extreme values and skewness. In such distributions, the mean can be strongly influenced by a few high or low values, potentially misrepresenting the typical local misorientation, whereas the median provides a more robust measure of the central tendency [21]. The median of the KAM over a grain is then defined as:

$$GMKAM_i = median_{j=1...N}(KAM_j) \quad (3)$$

GOS and KAM provide complementary information on the local misorientation state within grains. GOS reflects the overall spread of orientations across an entire grain, making it more sensitive to long-range orientation gradients and accumulated lattice distortions. In contrast, KAM evaluates the average misorientation between a pixel and its immediate neighbors, capturing short-range variations and local heterogeneities in the dislocation structure [22]. Together, these metrics allow a more complete characterization of both continuous and localized plastic deformation within the microstructure.

## 3. Statistical methods

Statistical processing of EBSD data enables a detailed and unsupervised identification of recrystallized grains [15]. The two data processing methods used in this study — PCA and GMM — are presented below.

### 3.1. Principal component analysis

Principal component analysis is a statistical method used to reduce the dimensionality of a dataset while preserving relevant information. PCA finds a new set of orthogonal basis vectors, called principal directions, that maximize the variance of the projected data. This is achieved by solving the eigenvalue problem of the covariance matrix S [23–27]:

$$SV = V\Lambda \quad (4)$$

where V is the eigenvectors matrix and Λ is the eigenvalues diagonal matrix.

Projection of the original data onto these directions yields new coordinates, known as principal components. To do so, the normalized dataset is projected on the principal directions:

$$Z = X_s V \quad (5)$$

where Z is the projected data or principal component, $X_s$ the standardized dataset and V the eigenvector matrix. More details on the PCA theory can be found in Appendix B.

Once the dataset is projected onto principal components, it becomes possible to compare the contribution of each descriptor to the total variance of the dataset. This can be done with the scaled loadings. The scaled loadings quantify the correlation of each descriptor to a principal component. These values range from 0 to 1, with higher values indicating a stronger contribution to the principal component and highlighting the descriptor's importance in capturing variance. The scaled loadings of PC1 are defined as [25]:

$$L_{PC1} = |V_{PC1}| \cdot \sqrt{\lambda_{PC1}} \quad (6)$$

where $L_{PC1}$ are the scaled loadings of the descriptors, $V_{PC1}$ the eigenvector of the PC1 and $\lambda_{PC1}$ the eigenvalue or explained variance of PC1.
## 3.2. Pearson correlation

The Pearson correlation coefficient $r_{XY}$ is a statistical measure that quantifies the strength and direction of the linear relationship between two descriptors, X and Y. It is defined as the normalized covariance of the two variables [23]:

$$r_{XY} = \frac{\operatorname{cov}(X, Y)}{\sigma_X \sigma_Y} = \frac{\sum_{i=1}^{N} (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum_{i=1}^{N} (X_i - \bar{X})^2 \sum_{i=1}^{N} (Y_i - \bar{Y})^2}} \quad (7)$$

The Pearson correlation coefficient ranges from -1 to +1. Values close to +1 indicate a strong positive linear correlation, values near -1 a strong negative correlation, and values around zero suggest no linear relationship.

## 3.3. Gaussian mixture model

Gaussian mixture models are probabilistic modeling methods for representing complex data distributions as a combination of K normal distributions such as:

$$p(X|\theta) = \sum_{k=1}^{K} \pi_k N(X|\mu_k, \Sigma_k) \quad (8)$$

where $p(X|\theta)$ is the probability density function of the Gaussian mixture for the data X in the model $\theta = \{\pi_k, \mu_k, \Sigma_k\}_{k=1}^K$, $\pi_k$ is the weight of each multivariate normal distribution $N(X|\mu_k, \Sigma_k)$ defined by its mean $\mu_k$ and its covariance matrix $\Sigma_k$ [28,29].

Based on the maximum likelihood principle, the estimation is typically performed using the Expectation-Maximization (EM) algorithm, which iteratively optimizes the Gaussian component parameters to maximize the fit to the data [30].

# 4. Development of the method

First, the limitations of conventional approaches involving a fixed GOS threshold are presented to highlight the benefits of the adaptive threshold method introduced in this paper. Next, the proposed method is described in detail. This method combines two complementary approaches. On the one hand, PCA is used to project the EBSD descriptor data onto the most relevant direction. On the other hand, a GMM provides an adaptive threshold to separate recrystallized and deformed grain populations.

## 4.1. Limits of fixed GOS thresholds to separate recrystallized grains

Microstructural descriptors that quantify local misorientations frequently follow log-normal distributions [31]. The semi-logarithmic GOS distributions obtained from EBSD analyses of 304 L stainless steel are shown in Fig. 2. Bimodal distributions are characteristic of partially recrystallized microstructures where recrystallized (ReX) grains appear as a distinct population with low GOS values. The higher GOS values correspond to deformed, work-hardened grains.

In Fig. 2, two thermo-mechanical conditions are compared. It is clear that the GOS threshold value separating the recrystallized and the deformed grain populations is dependent on the working conditions. This variability highlights the limitations of using a fixed GOS threshold to distinguish between recrystallized and deformed grains. Because the optimal separation depends on processing conditions, a single threshold may misclassify grains, directly affecting the estimated recrystallized fraction. Such inaccuracies compromise the interpretation of microstructural mechanisms and introduce uncertainty in model calibration. To overcome this issue, the threshold should be adaptively determined based on the data distributions.

In cases of dynamic recrystallization, recrystallized grains can be deformed again. Under these conditions, the two modes may overlap

<figure><img src="image_5.png" /><figcaption>Fig. 2. Semi-logarithmic GOS distributions from 304 L EBSD data under two different hot working thermo-mechanical conditions.</figcaption></figure>

further, making it more difficult to identify their separation. This increases the uncertainty in identifying grain populations. This also underscores the necessity of developing a new grain separation method that incorporates both an adaptive threshold and an uncertainty band.

## 4.2. Selection of the most relevant microstructural descriptors to monitor recrystallization

As previously mentioned, GOS and KAM are often used to track recrystallization. These local misorientation descriptors are the most relevant because recrystallized grains typically exhibit minimal work hardening [1]. Consequently, partially recrystallized microstructures show significant local misorientation contrast that can be used to identify grain populations.

However, other descriptors derived from EBSD data may also provide information on recrystallization. Recrystallized grains may be smaller and have a more rounded morphology than deformed grains [15]. Thus, these geometrical properties are tested by considering the grain diameter and aspect ratio to track recrystallization. Since recrystallized grains are less strain-hardened than deformed grains [1], Kikuchi bands tend to be of higher quality. This difference can be quantified directly from EBSD data using the mean angular deviation (MAD) or assessed from the electron backscatter diffraction patterns (EBSP) using the band slope. These two EBSP descriptors may also be relevant for tracking recrystallization. Thus, eight descriptors were selected for tracking recrystallization. They are presented briefly in Table 2.

Since not all descriptors are equally relevant, identifying the most significant ones for tracking recrystallization is essential. This selection is based on two criteria. The first criterion involves the scaled loadings. The method aims to separate non-recrystallized and recrystallized grains along a principal direction to form two clusters. Consequently, relevant descriptors are expected to contribute little to explain the variance at low strain levels, where only deformed grains exist, resulting in a single cluster. Once recrystallization initiates, however, the contribution to the principal component should increase as the grain population splits into two distinct groups: recrystallized and non-recrystallized. The second criterion considers Pearson correlations between descriptors. Highly correlated descriptors are disregarded since they convey essentially the same information. In such cases, one of the two descriptors can be omitted without loss of interpretive value.

A Python script based on the Scikit-learn 1.5.1 library [32] was developed to apply the selection criteria to the 304 L EBSD dataset. The script computes the scaled loadings of the eight microstructural descriptors on the first principal component (PC1), which contains most of
the dataset's variance. This analysis was applied to EBSD data from five equivalent plastic strain levels ranging from the undeformed state to $\bar{e} = 0.8$ at 1100 °C and $\dot{\bar{e}} = 0.05$ s⁻¹. Each descriptor was evaluated grainwise, ensuring an identical number of data points across all variables. A total of 11,619 grains were analyzed across the five conditions. Fig. 3 shows the contribution of each descriptor to the PC1 direction, represented as a heat map. Each value corresponds to the scaled loading of a descriptor at a specific strain level. The blue line between a strain of $\bar{e} = 0.1$ and $\dot{\bar{e}} = 0.2$ symbolizes the onset of recrystallization. The first four descriptors, which correspond to local misorientation descriptors, have their scaled loadings increasing suddenly as soon as recrystallization begins and remain significant throughout deformation. Therefore, these descriptors are particularly relevant for tracking recrystallization because they account for the sudden increase in variance along PC1, which likely reflects the emergence of two distinct clusters: recrystallized and deformed regions. Conversely, the grain diameter (D) ceases to correlate with PC1 as soon as recrystallization begins. The diameter makes a major contribution to variance when misorientation gradients are small but has almost no influence when misorientation gradients are large. Consequently, this descriptor provides little value in describing recrystallization compared to the misorientation descriptors and is effectively overshadowed by their strong contributions. The other descriptors, in purple shades, have little influence throughout the recrystallization progression and are also discarded.

As expected, the four most relevant descriptors describe local misorientations, and they are retained for the next stage of the selection. However, it is necessary to also consider possible correlations between the selected descriptors. Since these descriptors will be used to separate the populations of recrystallized and deformed grains, it is important to avoid redundancy with low Pearson correlation values.

GOS is retained, as it consistently exhibits the highest scaled loadings when recrystallization occurs. This confirms its dominant role in capturing the variance associated with recrystallization. Among the remaining descriptors, three are KAM-derived and show strong mutual correlations (Fig. 4), indicating redundant information. GMKAM is selected as the most representative since it shows the lowest correlation with GOS and a high scaled loading on PC1. Together, GOS and GMKAM provide a complementary and robust basis for monitoring recrystallization.

It is important to note that this study is material-dependent, meaning the retained microstructural descriptors may vary from one material to another. Determining the most relevant descriptors using PC1 and

Pearson correlation remains the most reliable way to ensure effective recrystallization monitoring.

### 4.3. Adaptive threshold with PCA and GMM

Following the selection process, GOS and GMKAM were identified as the most relevant descriptors for monitoring recrystallization in 304 L. Their distributions are subsequently used to differentiate the two grain populations during recrystallization. In the case of 304 L, the average grain size can exceed 200 µm, making it challenging to acquire EBSD maps large enough to ensure a statistically representative number of grains. Consequently, the number of analyzed grains per map remains limited (below 500 for the coarsest microstructures), which restricts the robustness of statistical analyses. To overcome this limitation, three EBSD datasets corresponding to different strain levels ($\bar{e} = 0.2, 0.4$, and 0.8) but acquired under identical temperature (1100 °C) and a strain rate of $\dot{\bar{e}} = 0.05$ s⁻¹ were combined. It is assumed that these three cases exhibit similar GOS and GMKAM distributions, as they originate from comparable thermo-mechanical conditions. Ideally, the input data would consist of a single EBSD map obtained under a given thermo-mechanical condition and containing a sufficiently large number of grains to ensure statistical representativeness.

To illustrate the principle of GMM, Fig. 5 presents an example showing how a GMM separates arbitrary 2D data into distinct clusters. Each cluster is represented by a multivariate Gaussian distribution. The isoprobability contours, shown at one, two, and three standard deviations, depict the dispersion of the data around their respective

<figure><img src="image_6.png" /><figcaption>Fig. 4. Pearson correlation heatmap between the four microstructural descriptors selected.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 3. Heatmap showing the evolution of scaled loadings $L_{PC1}$ (Eq. (6)) during deformation.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 5. Visualization of the Gaussian mixture model on two-dimensional arbitrary data.</figcaption></figure>

centroids. This illustrative example, processed using Python and the Scikit-learn library [32], provides a visual foundation for understanding how GMM identifies distinct grain populations in the subsequent analysis. An ideal clustering would exhibit no overlap between the Gaussian components, but such a situation is seldom encountered when dealing with experimental data.

Fig. 6 shows the application of GMM to the combined dataset of the three previously described EBSD maps to identify two clusters, labeled A and B. Each point represents a grain, characterized by its GOS and GMKAM values. The isoprobability contours at one standard deviation are shown for each detected cluster, illustrating their respective dispersion. Along the plot borders, the one-dimensional GOS and

GMKAM distributions are aligned with the 2D projection. The minima of these 1D distributions are marked with red dots, and the corresponding orthogonal lines indicate the separation thresholds on the 2D map.

Using only GOS or GMKAM individually would lead to significant misclassification of grains between the two clusters, emphasizing the need to combine both descriptors as complementary information. The resulting separation between clusters forms a curved boundary. This boundary remains uncertain: the clusters are continuously connected, without a sharp division, and alternative boundaries could be equally valid. This reflects the underlying physics: recrystallized grains can lose their work-hardening structure and deform again, gradually merging with the deformed-grain population. Therefore, the boundary proposed by the 2D GMM clustering is not always optimal, and two main issues arise. First, the quality of the separation cannot be reliably assessed visually when clusters overlap in two dimensions, making it difficult to evaluate potential misclassifications. Second, grains located in the transitional region should be treated separately. These grains, of ambiguous identity, may correspond to slightly deformed initial grains or to recrystallized grains that underwent plastic deformation later on. Since their history cannot be clearly established, they should be considered a third, intermediate group. This zone of uncertainty is represented by a green band in Fig. 6.

We propose an alternative method that robustly defines the two main clusters while introducing an uncertainty band between them (detailed in Eq. (10)).

To obtain a bimodal distribution that can be visually assessed while retaining the maximum amount of information from both descriptors, the data is projected onto the first principal component, PC1 (Fig. 7). Additionally, projecting the data from two dimensions to one makes it convenient to add an uncertainty band between the two grain populations.

As in Eq. (5), the data is projected using a linear combination of the principal directions (eigenvectors) and the values of each descriptor for a grain i. This is expressed as follows:

$$PC1_i = v_{PC1\_GOS} \cdot \log_{10}(GOS_i) + v_{PC1\_GAKAM} \cdot \log_{10}(GMKAM_i) \quad (9)$$

where PC1_i is the projected value of the two descriptors for the grain i. v_{PC1\_GOS} is the eigenvector of the GOS associated with the first component PC1. GOS_i is the GOS value of the grain i.

After projection onto the single dimension PC1, Fig. 7 reveals a bimodal distribution of the grain populations, allowing for a clearer and more comprehensive separation between the populations. The two Gaussians fitted with GMM are consistent with the distributions, and the separation threshold in dashed black corresponds to where the

<figure><img src="image_9.png" /><figcaption>Fig. 6. Two-dimensional GMM on GOS and GMKAM for three EBSD files with recrystallization. The projected distributions of the descriptors are shown on the border.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. One-dimensional GMM clustering on GOS and GMKAM after PC1 projection.</figcaption></figure>
Gaussians intersect.

Although introducing a third population in the GMM fitting could provide a slightly better statistical description of the data with the GMM, this approach was deliberately avoided. The objective here is not to achieve the best mathematical fit but rather to identify two populations of grains with high confidence. The goal of identifying these grains is to model a recrystallization mechanism that inherently involves two distinct populations of grains. Maintaining a two-population framework ensures consistency with the physical interpretation of the process. To compensate for potential overlap between the two distributions, the uncertainty band helps ensure a reliable classification of grains. This methodological choice differs from the approach proposed by Yeo et al. [15], who relied on GMM Bayesian criteria to determine the number of grain populations.

### 4.4. Uncertainty band: interquartile range

To account for the uncertainty around the population separation, an asymmetric uncertainty band corresponding to the interquartile range (IQR) of each Gaussian has been included. The IQR of the left Gaussian is defined as follows:

$$ IQR_{left} = Q3_{left} - Q1_{left} \qquad (10) $$

where $Q1_{left}$ and $Q3_{left}$ are, respectively, the first and third quartiles of

the left Gaussian.

The IQR characterizes the spread of the distributions. The uncertainty band is therefore wider for a spread-out distribution, such as the population of deformed grains in Fig. 7. The uncertainty band constitutes a third group of grains. It helps to filter out ambiguous cases within a separate group, such as incorrectly reconstructed grains during the EBSD data post-processing stage or grains that underwent dynamic recrystallization right before being slightly strained again.

Approximately 12% (1383 grains) of the 11,619 grains in the study dataset fall within the uncertainty band. This proportion varies from one dataset to another because it depends on the distance between the separation threshold and the Gaussian modes. A strong overlap between the Gaussians increases the proportion of uncertain grains because the uncertainty band covers a larger portion of the distributions. Conversely, when the Gaussians are well separated, the proportion of uncertain grains decreases significantly. The proportion of uncertain grains also depends on the standard deviation of each Gaussian distribution. A larger standard deviation results in a more spread-out distribution and a wider uncertainty band.

In grain population analysis, this third group can usually be regarded to retain only well-identified grains or, if relevant, merged with one of the two main groups, depending on the purpose of the analysis. The point here is to have a robust and reproducible process to identify these grains.

<figure><img src="image_11.png" /><figcaption>Fig. 8. Results of the PCA-GMM method for three strain levels (a): $\bar{\epsilon} = 0.2$, (b): $\bar{\epsilon} = 0.4$ and (c): $\bar{\epsilon} = 0.8$.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 9. Reconstruction issue from a zoom in Fig. 8(c). (a): Band slope and LAGB and (b): PCA-GMM identification.</figcaption></figure>

# 5. Application and assessment of the PCA-GMM method

## 5.1. Application of the PCA-GMM method on 304 L austenitic stainless steel

From the previous grain separation of Fig. 7, the corresponding grains can be found with MTEX and displayed on an EBSD grain boundary map. Fig. 8 shows the results obtained using the PCA-GMM method for the same three EBSD files (Fig. 6). On the left are the gray-scale band slope maps, with the low-angle grain boundaries (LAGBs) marked in red. The band slope illustrates the intensity gradients of the edges of the Kikuchi bands. The brightest areas are grains with little deformation and few impurities, while the darkest areas are the most deformed grains. LAGBs enable observation of the material's substructure and determination of a grain's internal level of deformation. Combining the two indicators provides a representation of deformed areas and helps assess the results of the PCA-GMM method. The light areas without substructure scattered throughout the microstructure are recrystallized, strain-free grains.

The maps on the right are EBSD grain boundary maps with PCA-GMM identification results. Deformed grains are shown in red, and recrystallized grains, considered strain-free, are shown in blue. Green grains correspond to those in the uncertainty band. White areas indicate the minor ferritic phase, which was not considered in this study.

In Fig. 8(a), the first recrystallized grains are detected even if they are not numerous compared to the deformed portion. Figs. 8(b) and (c) show dynamically recrystallized grains that underwent subsequent deformation at higher levels of deformation, in addition to the recrystallized and deformed grain populations. At the highest level of deformation (ε̄ = 0.8), a significant proportion of these recrystallized grains

that underwent subsequent deformation is observed. The grain population identification aligns well with the observed microstructural reality of the band slope, even with a low proportion of recrystallized grains. The following section provides more details on the assessment of the PCA-GMM method.

A close inspection of the top-right area of Fig. 8(c) reveals that a grain exhibits reconstruction issues. Fig. 9(a) shows a recrystallized grain adjacent to a deformed grain. After grain reconstruction with MTEX, the two grains are erroneously merged into a single reconstructed unit. This occurs because the misorientation at their boundary, represented with a dashed black line in Fig. 9(b), slightly falls below the reconstruction criterion. The presence of the deformed grain greatly increases the GOS and GMKAM of the reconstructed grain, causing it to be incorrectly identified as deformed. This example highlights the sensitivity of the results to the reconstruction criterion. In this study, the reconstruction threshold was fixed at a 10° misorientation (see Appendix C for more details). However, a few reconstruction artifacts are inevitable. These errors also affect conventional fixed-threshold approaches based on GOS because they rely on the same reconstructed grain dataset.

## 5.2. A case of dynamic recrystallization

On a finer scale, a complex case of dynamic recrystallization can be seen in Fig. 10. By zooming in on the previous map Fig. 8(b), for a strain of ε̄ = 0.4, one can observe an area where all three grain types are present.

The blue grain identified as recrystallized in Fig. 10(a) has a KAM close to zero in Fig. 10(b)-(d) and no substructure in Fig. 10(c). Similarly, the red grain identified as deformed has a large KAM and a forming

<figure><img src="image_13.png" /><figcaption>Fig. 10. Zoom in on an area containing all three grain types for ε̄ = 0.4, with the (a): PCA-GMM result map, (b): KAM map, (c): band slope and LAGB and (d): GOS map.</figcaption></figure>
substructure. This is likely a dynamically recrystallized grain that has undergone additional deformation. In a more complex case, the green grain, detected as uncertain, falls within the uncertainty band between the two grain populations. It is difficult to categorize it in one population rather than another because its KAM indicates slight deformation, but it does not have a pronounced substructure.

The proposed PCA-GMM method correctly categorized all three grains, including the more complex case involving slight deformation of a recrystallized grain, which is typical in dynamic recrystallization. To capture all recrystallized grains, including dynamic ones, merging the recrystallized and uncertain populations may be appropriate, although there is a risk of accidentally including some deformed grains. However, if a strict focus on recrystallized grains is required, limiting the analysis to the recrystallized population is preferable, even if this excludes part of the dynamically recrystallized grains.

## 6. Conclusions

This work demonstrates that using a fixed GOS threshold to separate recrystallized grains from deformed ones can lead to inaccurate grain identification. A method was developed to identify recrystallized grains using EBSD data from hot-compressed 304 L austenitic stainless steel. Preliminary PCA and Pearson correlation analyses indicate that GOS and GMKAM are the most relevant microstructural descriptors for monitoring 304 L recrystallization progress.

The PCA-GMM approach presented in this work has several advantages:

(1) PCA projection merges complementary information from multiple descriptors into one-dimensional distributions.

(2) An adaptive separation threshold with low user dependency results from fitting a Gaussian mixture model to the GOS and GMKAM PC1 projected distributions.

(3) The integration of an uncertainty band defined by the interquartile range avoids misidentification of grains, particularly in complex cases such as dynamic recrystallization. This gives control over grains that are difficult to categorize.

(4) The identification results align well with the observed band slope and LAGB maps, even in cases with a low proportion of recrystallized grains or complex dynamic recrystallization.

The PCA-GMM method can be quickly and easily implemented in Python using the publicly available code at the GitHub link in Appendix A.

### CRediT authorship contribution statement

H. Latuner: Writing – review & editing, Writing – original draft,

Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **J. Favre:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. **A. Helstroffer:** Supervision, Project administration, Methodology, Investigation, Funding acquisition. **G. Inacio Da Rosa:** Writing – review & editing, Validation, Supervision, Methodology, Investigation. **P. Joly:** Writing – review & editing, Validation, Supervision, Project administration, Funding acquisition. **C. Maurice:** Writing – review & editing, Visualization, Validation, Methodology, Investigation, Formal analysis, Conceptualization. **E. Plancher:** Writing – review & editing, Writing – original draft, Visualization, Validation, Project administration, Methodology, Investigation, Formal analysis, Conceptualization. **M. Roth:** Writing – review & editing, Validation, Supervision. **C. Desrayaud:** Supervision, Project administration, Funding acquisition. **G. Ker-mouche:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.

### Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work, the authors used OpenAI GPT-40 a few times in order to structure ideas into paragraphs. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

The authors gratefully acknowledge Framatome (France) for its financial support of and active contribution to this study. Thanks are also extended to S. Girard-Insardi for her strong participation in the hot compression tests, to G. Blanc for helping to establish the metallographic preparation method, and to M. Mondon and S. Sao-Joao for their help with the SEM-EBSD analyses. Special thanks to M. Lemetais and L. Hennocque for their valuable advice in developing the grain identification method.

## Appendix A. Supplementary data

The code developed and used for this method is publicly available on the GitHub Jupyter repository accessible via the following link: https://github.com/h-latuner/PCA_GMM_ReX

## Appendix B. Principal Component Analysis theory

It is possible to define the main stages of the PCA thanks to the works of Abdi et al. [26], Jolliffe et al. [27] and Li [35].

From a given dataset $X \in R^{n \times p}$ with $n$ the number of lines or grains and $p$ the number of descriptors.

The dataset is standardized to have a comparable dataset regardless of descriptors values:

$$ \forall i \in \{1, ..., n\}, \forall j \in \{1, ..., p\}; \quad (X_s)_{ij} = \frac{x_{ij} - \bar{x}_j}{\sigma_j} \qquad (B.1) $$

where $X_s$ is the standardized dataset, $x_{ij}$ is the (i, j)-th value of X, $\bar{x}_j$ is the mean value of a descriptor j and $\sigma_j$ is the standard deviation of a descriptor j.
Then, the covariance matrix $S \in \mathbb{R}^{p \times p}$ is calculated from the dataset:

$$S = \frac{1}{n-1} X_s^T X_s \qquad (B.2)$$

Diagonalization of the covariance matrix to decompose in eigenvalues:

$$SV = V\Lambda \qquad (B.3)$$

where $V = \begin{bmatrix} v_{PC-1,desc-1} & \cdots & v_{PC-1,desc-p} \\ \vdots & \ddots & \vdots \\ v_{PC-p,desc-1} & \cdots & v_{PC-p,desc-p} \end{bmatrix} \in \mathbb{R}^{p \times p}$ is the normalized eigenvector matrix or principal direction.

And $\Lambda = \begin{bmatrix} \lambda_{PC-1} & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \lambda_{PC-p} \end{bmatrix} \in \mathbb{R}^{p \times p}$ is the diagonal eigenvalue matrix or explained variance.

Finally, the dataset is projected onto the principal directions:

$$Z = X_s V \qquad (B.4)$$

where $Z \in \mathbb{R}^{n \times p}$ is the projected dataset onto the principal directions.

For this study, the PC1 is the only used component, Eq. (B.4) can be written:

$$\forall i \in \{1, \dots, n\}, \quad Z_{PC1,i} = X_{s,i} V_{PC1} = \sum_{j=1}^{p} (x_s)_{ij} (v_{PC1})_j \qquad (B.5)$$

where $Z_{PC1,i} \in \mathbb{R}$ is the projected dataset onto PC1 for the i-th value or grain, $(x_s)_{ij}$ is the (i,j)-th value of $X_s$.

And $(v_{PC1})_j$ is the j-th value of $V_{PC1} = \begin{bmatrix} v_{PC1,desc-1} \\ \vdots \\ v_{PC1,desc-p} \end{bmatrix} \in \mathbb{R}^{p \times 1}$

## Appendix C. Grain reconstruction threshold

The selection of the misorientation for grain reconstruction is a critical challenge in the EBSD community. Varying this value would significantly affect the grain-averaged descriptors and the final PCA-GMM identification, just as a fixed GOS threshold would affect grain separation.

In this study, the reconstruction threshold was fixed at a 10° misorientation. This threshold was initially chosen based on existing literature concerning 304 L austenitic stainless steels. A threshold between 10° and 15° is typically applied for this material. Specifically, Ruiz Sarrazola et al. [33] and Liang et al. [34] used a 10° threshold for 304 L recrystallization modeling. In a more applied context, the choice of using a 10° threshold in our study was also based on empirical validation. In our case, employing a higher threshold, such as 15°, resulted in several reconstruction issues, primarily because some grain boundaries were erroneously considered subgrain boundaries. This effect can be observed by comparing the grain reconstruction of the same EBSD data using 10° and 15° in Fig. C.1, which includes GOS and subgrain boundary plots. All reconstruction issues are highlighted with black arrows in Fig. C.1(b).

<figure><img src="image_14.png" /><figcaption>Fig. C.1. Comparison of the grain reconstruction with a threshold of (a): 10° and (b): 15°.</figcaption></figure>

Many grains are poorly reconstructed, which introduces issues in grain identification and subsequent data analysis. Conversely, using a lower reconstruction threshold leads to numerous subgrains being erroneously considered as part of larger grains. As a result, subgrains are integrated into these larger grains, as shown in Fig. C.2(b) when employing a 5° threshold. In this scenario, some subgrain boundaries were misoriented by more than 5° and were thus treated as separate grains within the reconstructed microstructure.
<figure><img src="image_15.png" /><figcaption>Fig. C.2. Introduction of subgrain in deformed grains with a (a): 10° and (b): 5° reconstruction threshold.</figcaption></figure>

To avoid subgrain reconstruction with a lower threshold and to limit excessive grain merging at higher thresholds, a 10° misorientation was thus selected as a good balance for recrystallization analysis. The uncertainty band associated with the identification is also helpful, as poorly reconstructed grains are flagged as uncertain and can be removed from further analysis, as illustrated in Fig. 9.

## Data availability

Data will be made available on request.

## GitHub

PCA-GMM Python code and dataset (Original data)

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