Accurate distances measures and machine learning of the texture-property relation for crystallographic textures represented by one-point statistics

Grain orientations can be parameterized in different ways, see [29]. Most frequently used in experimentation are Euler angles [7, 30], which describe an orientation as three subsequently executed rotations around defined axes of a Cartesian coordinate system. However, due to singularities, the Euler space is distorted, in general. To overcome this issue, the axis-angle description [29] or Rodrigues vectors [31] can be used, which describe orientations as a rotation around a 3d-vector (the length of the vector yields the rotation). A general drawback of these descriptions, however, remains, which is that for doing calculus, signs and special conventions have to be respected and the amount of permutations that follows from it is relatively high. For this reason, unit quaternions, which are unique and well-defined, have been introduced for the description of orientations [32].

To describe the crystallographic texture of materials, typically, the ODF defined by

$$\begin{align} f\left(g\right)\mathrm{d}g = \frac{\mathrm{d} V}{V},\end{align} \tag{ 1 }$$

with g being an orientation in SO(3) and V(g) describing the volume in SO(3), is used. Although our study is not limited by certain symmetry conditions, in this work, we assume f(g) underlying cubic symmetry conditions as we focus particularly on steel materials. As a consequence of such symmetry conditions, certain subspaces of SO(3) are equivalent and, thus, SO(3) can be reduced to the so-called fundamental zone, see [29]. In the following, we present commonly used texture descriptions and methods to measure distances between textures.

For expressing an orientation g, several representations exist (see [29]). For this study, however, two typically used representations are of importance, which are the Euler angle representation following [7] and the representation using unit quaternions, see for example [31, 32]. The Euler angles $\varphi_1,\phi,\varphi_2$ define subsequent rotations about the axes X, Z, X of a Cartesian coordinate system, while quaternions $q_0,q_i,q_j,q_k$ form a vector in a four-dimensional vector space [32].

In material science the commonly used distance measure between two ODFs $f^{{\,}(1)}(g)$ and $f^{{\,}(2)}(g)$ is defined in [15], equation (49) as:

$$\begin{align} \mathscr{D}_{\mathrm{odf}}\left(f^{\left(1\right)} \left(g\right),f^{\left(2\right)} \left(g\right)\right) = \int_g \left(f^{\left(1\right)}\left(g\right) - f^{\left(2\right)}\left(g\right)\right)^2 \mathop{}\!\mathrm{d} g,\end{align} \tag{ 2 }$$

with g being an orientation in the orientation space SO(3). This distance measure represents the integral squared difference between the amplitudes of the density functions only. In equation (2) the ODFs are usually represented as truncated GSH series, but can also be approximated by orientation histograms. In [15], approximations of the ODF distance are calculated similiarly to equation (2), but on the basis on two-point correlation functions. In our work, we transfer this approach to ODFs, which capture one-point correlation functions. As already mentioned in the Introduction (see section 1), due to the property of equation (2), neither the distance nor the neighborhood relationships between the corresponding orientations are reflected by this distance measure. Instead, the Sinkhorn distance is proposed in the study.

2.2.1. GSH functions.

The ODF f(g) can be approximated by a series expansion on the basis of GSHs [7]

$$\begin{align} f_{\mathrm{gsh}}\left(g\right) = \sum_{l = 0}^{\infty} \sum_{m = -l}^{+l} \sum_{n = -l}^{+l} C_{l}^{mn} T_l^{{\,}mn}\left(g\right),\end{align} \tag{ 3 }$$

with the orthonormal basis functions

$$\begin{align} T_l^{{\,}mn}\left(\varphi_1,\phi,\varphi_2\right) = \textrm{e}^{\textrm{i}m\varphi_2} P_{l}^{mn} \left(\cos\left(\phi\right)\right) \textrm{e}^{\textrm{i}n\varphi_1},\end{align} \tag{ 4 }$$

in which g is expressed as Euler angles and $P_l^{mn}$ denotes generalizations of the associated Legendre functions. The coefficients of the series expansion $C_l^{mn}$ are often used as descriptors for crystallographic texture, as they encode the shape of the ODF, see for example [33–35]. In these works, however, symmetries of the ODF are taken into account, yielding a more compact form of equation (3) as certain functions $T_l^{{\,}mn}(g)$ can be transformed or eliminated, depending on the symmetry [7]. In this study, GSHs are generated using the software PyMKS [36].

The distance between two ODFs $f^{{\,}(1)} (g)$ and $f^{{\,}(2)} (g)$ can be expressed according to equation (2) as

$$\begin{align} \begin{aligned} \mathscr{D}_{\mathrm{odf}}\left(f^{\left(1\right)} \left(g\right),f^{\left(2\right)} \left(g\right)\right) & = \frac{1}{8 \pi^2} \int\limits_{0}^{2\pi} \int\limits_{0}^{\pi} \int\limits_{0}^{2\pi} \left(f^{\left(1\right)}\left(\varphi_1, \Phi, \varphi_2\right) - f^{\left(2\right)}\left(\varphi_1, \Phi, \varphi_2\right)\right)^2 \\ & \quad \times \textrm{d} \varphi_1 \sin\left(\Phi\right) \textrm{d} \Phi \textrm{d} \varphi_2. \end{aligned}\end{align} \tag{ 5 }$$

When the densities $f^{{\,}(1)} (g)$ and $f^{{\,}(2)} (g)$ are expanded in two series of GSH base functions

$$\begin{align} f_{\mathrm{gsh}}^{\left(1\right)}\left(g\right) = \sum_{l = 0}^{\infty} \sum_{m = -l}^{+l} \sum_{n = -l}^{+l} C_{l}^{mn} \textrm{e}^{\textrm{i}m\varphi_2} P_{l}^{mn} \left(\cos\left(\Phi\right)\right) \textrm{e}^{\textrm{i}n\varphi_1}\end{align} \tag{ 6 }$$

and

$$\begin{align} f_{\mathrm{gsh}}^{\left(2\right)}\left(g\right) = \sum_{l = 0}^{\infty} \sum_{m = -l}^{+l} \sum_{n = -l}^{+l} \tilde{C}_{l}^{mn} \textrm{e}^{\textrm{i}m\varphi_2} P_{l}^{mn} \left(\cos\left(\Phi\right)\right) \textrm{e}^{\textrm{i}n\varphi_1},\end{align} \tag{ 7 }$$

the distance measure of equation (5) becomes

$$\begin{align} \begin{aligned} \mathscr{D}_{\mathrm{gsh}}\left(f_{\mathrm{gsh}}^{\left(1\right)}\left(g\right),f_{\mathrm{gsh}}^{\left(2\right)}\left(g\right)\right) & = \frac{1}{8 \pi^2} \int\limits_{0}^{2\pi} \int\limits_{0}^{\pi} \int\limits_{0}^{2\pi}\\ & \quad \times \left( \sum_{l = 0}^{\infty} \sum_{m = -l}^{+l} \sum_{n = -l}^{+l} \left(C_{l}^{mn} - \tilde{C}_{l}^{mn}\right) \textrm{e}^{\textrm{i}m\varphi_2} P_{l}^{mn} \left(\cos\left(\Phi\right)\right) \textrm{e}^{\textrm{i}n\varphi_1} \right)^2\\ & \quad \times d \varphi_1 \sin\left(\Phi\right) \textrm{d} \Phi \textrm{d} \varphi_2. \end{aligned}\end{align} \tag{ 8 }$$

Considering the orthogonality of $P_l^{mn}$, the distance measure between orientation densities $f_{\mathrm{gsh}}^{(1)}(g)$ and $f_{\mathrm{gsh}}^{(2)}(g)$ reduces to

$$\begin{align} \mathscr{D}_{\mathrm{gsh}}\left(f_{\mathrm{gsh}}^{\left(1\right)}\left(g\right), f_{\mathrm{gsh}}^{\left(2\right)}\left(g\right)\right) = \sum_{l = 0}^{\infty} \sum_{m = -l}^{+l} \sum_{n = -l}^{+l} \frac{1}{2l + 1} \left(C_{l}^{mn} - \tilde{C}_{l}^{mn}\right)^2.\end{align} \tag{ 9 }$$

For the sake of completeness, we included the full derivation of equation (9) in appendix B.

2.2.2. Orientation histograms.

Orientation histograms have been introduced in [2] to facilitate formulating statistical distance measures. The basis of the approach is the quaternion distance $\Phi(q_1,q_2)$ between two unit quaternions q₁ and q₂ as given in [26]:

$$\begin{align} \Phi\left(q_1,q_2\right) = \min\left(|q_1-q_2|, |q_1+q_2|\right).\end{align} \tag{ 10 }$$

To construct an orientation histogram, a set O of j nearly uniformly distributed orientations o_j in SO(3) (or the fundamental zone, if symmetry conditions hold) is needed. The set O forms the bin centers of the orientation histograms and is derived using the algorithm introduced in [37], as is implemented in the software neper [38].

For a set of orientations G that describes the crystallographic texture of a material, assignment vectors w _g are calculated for each element $g \in G$ by

$$\begin{align} w_\mathrm{g}^{\left(j\right)} = \left\{\begin{array}{ll} \frac{\Phi\left(g,o_j\right)}{\sum_{o_i \in N_l} \Phi\left(g,o_i\right)}, & \text{if} ~~ o_j \in N_l \\ 0, & \text{else} \end{array}\right. ,\end{align} \tag{ 11 }$$

with $w_\mathrm{g}^{(j)}$ denoting the jth component of w _g . Furthermore, N_l is a set of l nearest neighbor orientations of g in terms of the orientation distance equation (10). This so-called soft-assignment approach (with soft assignment factor l) guarantees that orientations that do not lie perfectly in the bin center are assigned proportionally to the neighbor bins $o_i \in N_l$. Finally, a volume average over the assignment vectors of the individual orientations w _g yields the orientation histogram

$$\begin{align} f_{\boldsymbol{\mathrm{b}}}\left(g\right) = \frac{1}{V}\sum_{g \in G} V\left(g\right)\boldsymbol{w}_{g}.\end{align} \tag{ 12 }$$

On the basis of the orientation histograms, texture distances can be defined using any histogram-based distance measure. The Chi-Squared distance and the EMD, as well as a faster implementation of the EMD, called the Sinkhorn distance [39], are introduced in the following.

2.2.2.1. Chi-Squared distance.

In [2], the Chi-Squared distance measure [25] was shown to be suitable for this purpose. The Chi-Squared distance between two orientation histograms $f_{\boldsymbol{\mathrm{b}}}^{(1)} (g)$ and $f_{\boldsymbol{\mathrm{b}}}^{(2)} (g)$ is defined as follows [25]:

$$\begin{align} \mathscr{D}_{\mathrm{\chi^2}}(f_{\boldsymbol{\mathrm{b}}}^{(1)} (g), f_{\boldsymbol{\mathrm{b}}}^{(2)} (g))) = \sum_{j = 1}^J \frac{(f_{\mathrm{b}_j}^{(1)} (g) - f_{\mathrm{b}_j}^{(2)} (g))^2}{f_{\mathrm{b}_j}^{(1)} (g) + f_{\mathrm{b}_j}^{(2)} (g)}.\end{align} \tag{ 13 }$$

However, for comparing sparsely populated histograms, the applicability of the Chi-Squared distance measure is limited. In such cases, the Chi-Squared distance measure is very likely to show very high values (close to the maximum) and is therefore not sensitive for changes in the histograms. This is why we propose to use the EMD [27].

2.2.2.2. Earth movers distance.

The EMD [27, 28], going back to the works [40] and [41], is based on a solution to the well-known transportation problem [42]. It is typically used for comparing distributions over a discrete region (e.g. the bins of a histogram as being used in image retrieval [28]). EMD measures the minimum amount of work needed to transport the probability mass from one distribution to the other to make them equal. The work of transfer between two bins is the transported probability mass times the distance between the representatives of the bins. In our case we consider the minimum amount of work needed to transform one orientation histogram into the other. Thereby, orientation distances between bins play an important role, compared to the Chi-Squared distance.

The computation of the EMD can be formalized as a linear programming problem with constraint conditions as described in [27], which is used in our approach. The principle procedure of this approach is as follows: Given two sets of tuples: $\boldsymbol{t}^{(1)} = \{(q^{(1)}_1, f_{\mathrm{b}_1}^{(1)} (g)), (q^{(1)}_2, f_{\mathrm{b}_2}^{(1)} (g)), {\ldots}, (q^{(1)}_m, f_{\mathrm{b}_m}^{(1)} (g)) \}$ and $\boldsymbol{t}^{(2)} = \{(q^{(2)}_1, f_{\mathrm{b}_1}^{(2)} (g)), (q^{(2)}_2, f_{\mathrm{b}_2}^{(2)} (g)), {\ldots}, (q^{(2)}_n, f_{\mathrm{b}_n}^{(2)} (g))\}$ where $q^{(1)}_i$, $q^{(2)}_j$ are the histogram bin centers represented as unit quaternions and $f_{\mathrm{b}_i}^{(1)}(g)$, $f_{\mathrm{b}_j}^{(2)} (g)$ are the entries (weights) of the corresponding histogram. Respecting the crystal symmetry, the ground distance matrix $D = [d_{ij}]$ is calculated, where d_ij is the distance between the quaternion representations $q^{(1)}_i$ and $q^{(2)}_j$. The quaternion distance [26] is used as the distance function:

$$\begin{align} d_{ij} = \arccos\left(|q^{\left(1\right)}_i \cdot q^{\left(2\right)}_j|\right),\end{align} \tag{ 14 }$$

where · denotes the product of the vector elements (not the quaternion multiplication, which results in another quaternion). The aim of solving the transportation problem is to find a flow matrix $F = [f_{ij}]$, where f_ij is the flow between $f_{\mathrm{b}_i}^{(1)} (g)$ and $f_{\mathrm{b}_j}^{(2)} (g)$, that minimizes the overall work

$$\begin{align} \textrm{WORK}\left(\boldsymbol{t}^{\left(1\right)}, \boldsymbol{t}^{\left(2\right)}, F\right) = \sum_{i = 1}^m \sum_{j = 1}^n f_{ij} d_{ij}.\end{align} \tag{ 15 }$$

Once the transportation problem is solved, and thus the optimal flow F is found, the EMD is defined as the normalized distance between the sets $\boldsymbol{t}^{(1)}$ and $\boldsymbol{t}^{(2)}$:

$$\begin{align} \mathscr{D}_{\mathrm{emd}} \left(\boldsymbol{t}^{\left(1\right)}, \boldsymbol{t}^{\left(2\right)}\right) = \frac{\sum_{i = 1}^m \sum_{j = 1}^n f_{ij} d_{ij} } {\sum_{i = 1}^m \sum_{j = 1}^n f_{ij}}.\end{align} \tag{ 16 }$$

2.2.2.3. Sinkhorn distance.

The solution of the transport problem as discussed above suffers from the increase in complexity with increasing number of histogram bins, which is also the dimensionality δ of the histogram representation space. If two histograms of dimension δ are compared, the costs for the calculation of the optimal transport distances scale at least in $O({\delta}^3 \textrm{log}(\delta))$ [39, 43]. The transportation problem therefore becomes intractable when δ exceeds around 100. To overcome this issue, in [39] the Sinkhorn algorithm is proposed instead. In this work, an additional constrain is introduced, in order to ensure that the the KL divergence between the flow matrix F and π_i $\zeta_j^T$ is smaller to the pre-defined parameter α:

$$\begin{align} \textrm{KL} \left(F, \pi_i \zeta_j ^T \right) \unicode{x2A7D} \alpha \quad \textrm{with} \quad \zeta_j = \sum_i f_{i,j} \quad \textrm{and} \quad \pi_i = \sum_j f_{i,j}.\end{align} \tag{ 17 }$$

This results in the requirement that s(F) should be as large as possible,

$$\begin{align} s\left(F\right) = \max \left( \sum_{i,j} f_{i,j} \log{\;} f_{i,j} \right).\end{align} \tag{ 18 }$$

The resulting (dual) Sinkhorn distance is

$$\begin{align} \mathscr{D}_{\mathrm{sh}} = \min \left(\sum_{i,j} f_{i,j} \cdot d_{i,j} - \frac{1}{\lambda} s\left(F\right) \right),\end{align} \tag{ 19 }$$

which is again a convex function. By forming the Lagrangian and searching for its minimum with respect to $f_{i,j}$, a strictly convex problem is obtained yielding a unique optimal solution. It is shown that the resulting optimum is also a distance. Since the Sinkhorn algorithm relies on matrix-vector products, the distance can be computed through Sinkhorn's matrix scaling algorithm, whereby it is several orders of magnitude faster than transport solvers.

In our approach, we calculate the histogram distances using the highly efficient Sinkhorn algorithm ([39], algorithm 1) with 100 iterations and setting the value of parameter λ to 0.01. Furthermore the algorithm can be integrated as a fully differentiable function in optimization frameworks such as being used for neural networks and implemented on Graphics Processing Units.

2.2.3. Pole figures.

A pole figure is a graphical representation used in materials science and crystallography to display the orientation of polycrystals. It is based on a stereographic projection that maps three-dimensional information on a two-dimensional plane [6]. When using pole figures, the neighborhood relations of the orientations are represented by the neighboring pixels in the image. In this study, pole figures are generated using the software mtex [44]. Distances between textures are measured by comparing two pole figures $f_{\mathrm{pf}}^{(1)} (g)$ and $f_{\mathrm{pf}}^{(2)} (g)$ pixel-wise by means of the absolute or l₁ loss:

$$\begin{align} \mathscr{D}_{\mathrm{pf}} \left(f_{\mathrm{pf}}^{\left(1\right)} \left(g\right), f_{\mathrm{pf}}^{\left(2\right)} \left(g\right)\right) = \frac{1}{M} \frac{1}{N} \sum_{m = 1}^M \sum_{i = 1}^N | f_{\mathrm{pf}_{m,i}}^{\left(1\right)} \left(g\right) - f_{\mathrm{pf}_{m,i}}^{\left(2\right)} \left(g\right) |,\end{align} \tag{ 20 }$$

where N denotes the number of pixels and M denotes the number of pole figures.

When texture-property relations are learned, the textures are transformed into a latent space, which arranges the texture representations according to their properties. Looking at machine learning in materials design, we want to retrieve texture, which correspond to given properties. For this purpose it is very important that the latent space representation also reflects the distance in the original texture space. This can be enforced by using e.g. siamese neural networks [3]. It still remains crucial that the original texture representation contains the essential information about the properties.

In order to investigate the ability of the discussed texture representations being suitable for carrying property information, we develop neural network-based models for learning the relation between textures and their corresponding properties, based on the texture-property data set described in section 2.4.2. This forward mapping from textures to properties is modeled as a regression problem. The regression loss is given by the mean squared error between the predicted properties $\boldsymbol{\hat{p}}$ and the true properties p :

$$\begin{align} \mathscr{L}_{\mathrm{regr}} \left(\boldsymbol{p}, \boldsymbol{\hat{p}}\right) = \frac{1}{N} \sum_{i = 1}^N \left({p_i} - {\hat{p}_i} \right)^2 + \lambda R\left(\boldsymbol{\theta}\right),\end{align} \tag{ 21 }$$

where θ are the trainable parameters and $R(\boldsymbol{\theta})$ is a regularization term that is used to prevent overfitting with the hyperparameter λ defining the strength of the regularization (also known as weight decay, see [45, 46]). We use E₁₁, E₂₂, E₃₃, $\tilde{R}_{11}$, $\tilde{R}_{22}$, $\tilde{R}_{33}$ to describe the elastic and plastic of material properties. For learning the relation between textures and corresponding properties, we represent the textures in three different ways:

• as orientation histograms having the resolution J = 512 and l $\in$ {1, 3}.
• as pole figures for the Miller's indices (001), (110), (111), where the polar coordinates are transformed into Cartesian coordinates with a resolution of $64 \times 64$ pixels. The resulting pole figure representation for one exemplary ODF is depicted in figure 1.
• as GSH textures for degree L = 16 and L = 10. For the mapping, we use only a subset of the GSH coefficients, as the first coefficient equals one and half of the remaining coefficients are complex conjugates of the other half [47].

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Fully connected neural networks are used for orientation histograms and GSH representations, whereas convolutional neural networks (CNN) are used for pole figure representations [48, 49]. CNNs are well-studied deep learning methods for image data [50–52] and are particularly powerful for processing images, where the underlying neighborhood relations between pixels are exploited, as is the case with pole figures. The topologies of the neural network models are listed in table 7 (appendix D), visualized in figure 9 (appendix D) and are explained in more detail in the following:

2.3.1. Orientations histograms.

2.3.2. GSH.

2.3.3. Pole figures.

2.3.4. Implementation details.

2.4.1. Texture evolution and properties calculation.

The basis for the investigations in the present paper is a simulated metal forming process, which is described in [2] and for which a simulation framework has been published [58]. The simulation is based on the Taylor-type crystal plasticity model introduced in [59], which is modified in terms of hardening as described in [60]. For a detailed description of the material model, we refer to appendix C.

Basically, the simulation framework allows for applying sequences of tension and compression operations on a point within a workpiece of the material under consideration (which we call material point in the following). These operations are defined by applying the deformation gradient

$$\begin{align} \widetilde{\boldsymbol{F}} = \widetilde{F}_{11} \boldsymbol{e}_1 \otimes \boldsymbol{e}_1 + \widetilde{F}_{2} \boldsymbol{e}_2 \otimes \boldsymbol{e}_2 + \widetilde{F}_{33} \boldsymbol{e}_3 \otimes \boldsymbol{e}_3,\end{align} \tag{ 22 }$$

with orthogonal basis vectors e _i and the outer product denoted by the operator ⊗. Expressed as a matrix, the deformation gradient writes

$$\begin{align} \widetilde{\boldsymbol{F}} = \left[ \begin{array}{ccc} \widetilde{F}_{11} & 0 & 0 \\ 0 & \widetilde{F}_{22} & 0 \\ 0 & 0 & \widetilde{F}_{33} \\ \end{array} \right].\end{align} \tag{ 23 }$$

$\widetilde{F}_{11}$ is defined by the applied tensile load increment and both, $\widetilde{F}_{22}$ and $\widetilde{F}_{33}$, are determined such that the stress boundary conditions are fulfilled, see [2]. Each of the individual tensile load increments $\widetilde{\boldsymbol{F}}$ can be oriented arbitrarily to the material point yielding $\hat{\boldsymbol{F}}$, which is a rotated form of the deformation gradient, following

$$\begin{align} \hat{\boldsymbol{F}} = \boldsymbol{R}\widetilde{\boldsymbol{F}}\boldsymbol{R}^\top,\end{align} \tag{ 24 }$$

with the rotation matrix R . We use the framework to simulate texture evolution and generate different crystallographic textures for our analysis. It is well known that Taylor-type mean-field models typically overestimate texture evolution [6]. However, for the purpose of our study the used mean-field model is sufficient as we prioritize execution speed over predicting highly realistic material behavior.

In addition, the simulation framework is used to calculate material properties. The properties we focus on are the orientation dependent Young's moduli and anisotropy similar to R-values, as in [61]. R-values, however, are originally developed to express plastic anisotropy in sheet metals. On this basis, we use an R-values-like measure, $\tilde{R}$, to described the plastic anisotropy at the material point in three space directions. To do so, tension tests are conducted in each space direction using the simulation framework. The $\tilde{R}_i$-values are given by the relation between tensile and lateral strain. On the basis of the simulated tension tests, also, the Young's modulus E_i is calculated in three space directions.

2.4.2. Crystallographic texture data generation.

The distances of cristallographic textures are usually measured in materials science by means of equation (2) with the texture ODFs represented as truncated GSH expansions. A more recent approach uses orientation histograms as discrete representations of the ODFs, while the distance is still calculated according to equation (2) in its discretized form (Chi-Squared distance $\mathscr{D}_{\mathrm{\chi^2}}$ (equation (13))). Our alternative approaches of using the Sinkhorn distance $\mathscr{D}_{\mathrm{sh}}$ (equation (16)) between orientation histograms and of representing ODFs as pole figures to calculate the pole figure distance $\mathscr{D}_{\mathrm{pf}}$ (equation (20)) capture orientation neighborhoods of and correlations between orientations. We first compare the non-GSH approaches among each other, and then compare the best of them, the Sinkhorn distance, to the classical GSH approach.

The effect of the various discussed ODF representations on the corresponding ODF distance measures are shown in the following. For this purpose, we use the distance measurement evaluation set $\{f^{{\,}(0)}(g), f^{{\,}(1)}(g), \dots , f^{{\,}(19)}(g)\}$ described in section 2.4.2, resulting from the forming process, which consists of 20 tension steps. We determine the distances from the initial texture $f^{{\,}(0)}(g)$ to each subsequent texture $f^{{\,}(i)}(g), i = 1, \dots , 19$: $\mathscr{D}(f^{{\,}(0)}(g), f^{{\,}(1)}(g))$,...,$\mathscr{D}(f^{{\,}(0)} (g), f^{{\,}(19)} (g))$, where $\mathscr{D}$ represents the distance function. Such sequences of distance measure values are then compared using different texture representations. An overview of the methods of data generation and distance measurements is depicetd in figure 10 in appendix E.

3.1.1. Orientation histogram and pole figure representations.

In the following, we compare the distances, which result from using

• 1.  
  the Chi-Squared distance $\mathscr{D}_{\mathrm{\chi^2}}$ (equation (13)),
• 2.  
  the Sinkhorn distance $\mathscr{D}_{\mathrm{sh}}$ (equation (16)) and
• 3.  
  the pole figure distance $\mathscr{D}_{\mathrm{pf}}$ (equation (20)).

These measures are calculated for the textures resulting from the forming process starting at the three different initial textures. For each texture set, the orientation histograms and pole figures (depicted in figures 2–4) are created as follows: The textures are represented by orientation histograms of different resolution, where the number of histogram bins J $\in$ {$128, 256, 512, 1024, 2048$}. The histogram of each resolution is smoothed with three different soft-assignment values l $\in$ {$1, 3, 5$}. We generate pole figures for the Miller's indices $(001), (110), (111)$ and transform the polar coordinates into Cartesian coordinates with a resolution of $64 \times 64$ pixels.

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To make the distances comparable independent of the measure under consideration, we normalize the distances by dividing the raw distance by the maximum possible distance in each case. The maximum possible distance is the distance between two single crystals $f_{\mathrm{sc}}^{(0)}(g)$ and $f_{\mathrm{sc}}^{(1)}(g)$, which have the maximum distance in their orientations:

$$\begin{align} \mathscr{D} = \mathscr{D}\left(f^{\left(0\right)}\left(g\right), f^{\left(k\right)}\left(g\right)\right) / \mathscr{D}\left(f_{\mathrm{sc}}^{\left(0\right)}\left(g\right), f_{\mathrm{sc}}^{\left(1\right)}\left(g\right)\right).\end{align} \tag{ 25 }$$

The sequences of these normalized distances are calculated for our three different texture sets and plotted as distance curves over process step number (depicted in figure 5) They are discussed in the following.

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3.1.1.1. Initial Texture 1 ($n_{\mathrm{oris}} = 512$).

The distance curves of the Chi-Squared and Sinkhorn distance measures for the (J, l)-settings of the orientation histograms as well the distance curves of the l₁-distance measure for the pole figures are depicted in figures 5(a)–(c). It can be seen that the Chi-Squared and Sinkhorn distance values for J = 512 and l = 1 do not increase until step 2 despite the expected texture evolution due to increasing load (figure 5(a)). This is because in this special case, all texture orientations are located initially in the bin centers. Our process then needs two steps to force the movement of the orientations into another bins. On the other hand, when applying smoothing by soft-assignment with l = 3 and l = 5, a small process-enforced movement from the bin center is enough to change the histogram and the distance value. Furthermore, the Chi-Squared distance curve is not smooth for l = 1 (figure 5(a)) and erroneously not monotonic for l = 1 and J = 2048. At higher soft-assignment values l = 3 (figure 5(b)) and l = 5 (figure 5(c)) the Chi-Squared distance curve shows a monotonic behavior and is smoother. Another effect of the discretization parameters of the histograms on their distance curves is the variance of the distance values at a process step. One can see from figures 5(a)–(c) that this variance is much higher for the Chi-Squared than for the Sinkhorn distance measure and therefore the Chi-Squared measure is much more sensitive against the choice of histogram parameter values. The Sinkhorn distance as well as the pole figure distance result in monotonic and smoother curves, while the Sinkhorn distance is less sensitive to the number of bins J and soft-assignment l.

3.1.1.2. Initial texture 2 ($n_{\mathrm{oris}} = 100$).

The results of the investigated distances for the orientation histograms and pole figures are depicted in figures 5(d)–(f). The general behavior of the curves is similar to the case $n_{\mathrm{oris}} = 512$, but the curves are rougher for the Chi-Squared measure where also the maximum value is much higher compared to $n_{\mathrm{oris}} = 512$ than for the Sinkhorn distance, the curves of which still remain smooth. The distance curve resulting for the pole figure measure remains almost unaffected.

3.1.1.3. Initial texture 3 ($n_{\mathrm{oris}} = 1$).

The results of the investigated distances for textures with only one orientation are shown in figures 5(g)–(i). The simulated process delivers a sequence of single crystals where the texture is determined by their orientation quaternions. The quaternion distance equation (14) can be used in this case as the ground truth for the texture distance. The quaternion distance curve is strictly monotonic and almost linear. The Chi-Squared distance measure can not cope with this case because it changes its value only if the single crystal orientation moves into a different bin and then immediately takes on the maximum possible value (saturation). Higher-order soft-assignments smoothen this step function behavior and let the curves saturate slower. The discontinuity problem of the Chi-Squared curves is less prominent for Sinkhorn distance curves because the quaternion distances between bin centers are also taken into account. This also forces the Sinkhorn distance curves to be closer to distance curve and also prevents saturation. The pole figure distance curve is monotonic and smooth, but deviates from the linear ground truth curve by forming a half sigmoid function which saturates smoothly with increasing step number.

3.1.2. GSHs.

After having learned that the Sinkhorn distance delivers the best results related to histogram distance measures at textures composed of 512 orientations, we now compare these results to the results from using the distance measure $\mathscr{D}_{\mathrm{gsh}}$ (equation (9)) based on GSH texture representations of various GSH degrees. For this part of our study, we use the Initial Texture 1 set ($n_{\mathrm{oris}} = 512$) from the distance measurement evaluation set described in section 2.4.2. The ODF is approximated by a GSH series expansion ([7]) of the degrees L $\in$ {4, 6, 8, 10, 16} [7]. For comparison, we show the corresponding Sinkhorn distance.

The value ranges of the different analyzed distances differ due to different representations. To compensate this effect the distances are normalized to the value of the distance at step 19 (maximum distance value):

$$\begin{align} \mathscr{D} = \mathscr{D}\left(f^{\left(0\right)}\left(g\right), f^{\left(k\right)}\left(g\right)\right) / \mathscr{D}\left(f^{\left(0\right)}\left(g\right), f^{\left(19\right)}\left(g\right)\right).\end{align} \tag{ 26 }$$

The results for the GSH distance are shown over process step numbers in figure 6. It can be seen that the Sinkhorn distance increases stronger in the first steps and then tends to saturate at later steps. In contrast, the GSH distance shows a small increase at the beginning and increases stronger after certain process steps, resulting in an approximately linear increase from step 8 onwards. In general, the GSH distance is strictly monotonic and smooth. The curves show that the GSH distance measure is less sensitive to small textures distances than the Sinkhorn distance measure, while the latter tends to saturate at higher distances.

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3.1.3. Investigation of experimentally measured textures.

Finally, we evaluate the distance measures at experimentally measured textures of a DC04 steel sheet at different processing steps. The data was obtained from EBSD measurements conducted after (1) hot rolling, (2) cold rolling and (3) heat treatment in order to study the texture evolution [62]. The resulting textures are shown as pole figures and orientation histograms in figure 7. After hot rolling, a nearly gray texture can be observed, while cold rolling and heat treatment result in pronounced textures typical for these processing steps. In figure 8, we additionally provide visualizations of each individual orientation distribution as $\varphi_2 = 45^\circ$ sections through the Euler space. Therein, the typically pronounced α and γ fiber of cold rolling textures can be clearly seen. The α fiber disappears after heat treatment, while the γ fiber gets more pronounced [63].

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The distances of the ODFs are calculated between textures after (1) hot rolling and (2) cold rolling, (2) cold rolling and (3) heat treatment, and (1) hot rolling and (3) heat treatment. For our experimental data it is not possible to compare GSH-based distance measures to histogram-based and pole-figure-based distance measures, because the prerequisites of normalization as in the simulated process are not given: With only three textures, it makes no sense to use one distance for normalization of the others as we did it by using the distance of the last process step in the simulated 20-step process. To compare distances between the three experimental textures, we were normalizing the distances of the histogram-based and pole-figure-based measures by the distance between two single crystals (maximum possible), but which can not be calculated for GSH representations.

The observed qualitative behaviour of the textures is reflected by all distance measures, which are shown in table 1. The difference between the behaviours of the different distance measures is small, because all the experimental textures are not very sharp (sparse), which would induce distinct distance measure behaviours.

Table 1. Results of the measured distances between each of the experimentally measured textures. The results are shown for the orientation histograms (OH) with a fixed number of bins J = 512 and varying soft-assignments l, as well as for the pole figures.

Texture	Distance	Hot rolled vs. Cold rolled	Cold rolled vs. Heat treated	Hot rolled vs. Heat treated
OH for l = 1	$\mathscr{D}_{\mathrm{\chi^2}}$	0.421	0.296	0.453
OH for l = 3	$\mathscr{D}_{\mathrm{\chi^2}}$	0.368	0.209	0.340
OH for l = 5	$\mathscr{D}_{\mathrm{\chi^2}}$	0.343	0.184	0.305
OH for l = 1	$\mathscr{D}_{\mathrm{sh}}$	0.206	0.180	0.207
OH for l = 3	$\mathscr{D}_{\mathrm{sh}}$	0.195	0.166	0.192
OH for l = 5	$\mathscr{D}_{\mathrm{sh}}$	0.189	0.163	0.186
Pole figures	$\mathscr{D}_{\mathrm{pf}}$	0.241	0.172	0.228

The results of the texture-property relations are summarized in table 2. The coefficient of determination R² is used for comparison, as well as the regression errors MAE$_\mathrm{E}$ and MAE$_{\tilde{\mathrm{R}}}$, which denote the mean absolute error (MAE) between the true and predicted Young's moduli and $\tilde{R}$-values. An MAE of Young's moduli $\unicode{x2A7D} 1000$ and of $\tilde{R}$ of $\unicode{x2A7D} 0.1$ is sufficient in most applications. Additionally, the numbers of input features of the used texture representations are noted. It is shown that texture-property relations with a sufficient prediction quality can be achieved for all texture representations. By considering the orientation histograms, better results are obtained with the soft-assignment l = 3. Comparing all texture representations, the estimator quality for GSH for L = 16 is best considering R² while pole figures achieves the best result considering MAE. The distribution of the property values of the learning data set is shown in table 3.

Table 2. Results of the investigated textures for learning the texture-property relations. Regression error MAE$_\mathrm{E}$ is given in $[$GPa$]$, Regression error MAE$_{\tilde{\mathrm{R}}}$ in $[$-$]$ and Regression score R² in $[$%$]$. The best results for the metrics are noted in bold.

Texture	Number of features	R2	MAE$_\mathrm{E}$	MAE$_{\tilde{\mathrm{R}}}$
Orientation histograms for $J = 512, l = 1$	512	98.2	0.329	0.022
Orientation histograms for $J = 512, l = 3$	512	99.5	0.125	0.012
GSH for L = 10	80	99.3	0.152	0.015
GSH for L = 16	280	99.7	0.087	0.011
Pole figures	$64 \times 64 \times 3$	99.6	0.080	0.011

Table 3. Distribution of the property space for the underlying data set. The E-Values are given in $[$GPa$]$ and the $\tilde{R}$-Values are given in $[$-$]$.

Property	Min. Value	Max. Value	Range	Mean	Std.
E11	204	231	27	216	3.14
E22	206	234	28	218	3.42
E33	204	229	25	217	3.27
$\tilde{R}_{11}$	0.373	4.024	3.651	1.084	0.284
$\tilde{R}_{22}$	0.342	2.882	2.540	0.964	0.167
$\tilde{R}_{33}$	0.239	2.210	1.971	0.931	0.194

Pole figures are the richest representation of the investigated textures in this study, as they are represented as image data. In this representation, the neighborhood relations of the orientations are represented by the neighboring pixels in the image. Moreover, the visualization of the ODF by pole figures is interpretable by domain experts, which in turn allows them to interpret texture-property relations represented by neural regression networks. In contrast, neighborhood relations are not reflected by orientation histograms, since the order of bins does not reflect the order of orientations, cf section 2.2.2. For a comparison of the ODF represented as pole figures and orientation histograms see figures 2–4. Orientation relationships are only represented by orientation histograms when the orientations of the bin centers are included. The representations of ODFs by GSH coefficients does not contain any information about orientation relationships. When using GSHs, neighborhood relations between orientations are implicitly represented by the base functions, which belong to the GSH coefficients. When expanding an ODF using GSH with only low order coefficients, sharp textures can not be represented very well. Besides pole figure images, plots showing sections through the Euler space are also often used to visualize the orientation distribution. However, due to the distortion of the Euler space, this representations is disadvantageous for representing and processing crystallographic texture data.

Distances between textures using the distance measurement evaluation set described in section 2.4.2 are evaluated by means of a stepwise process, which pushes the texture further apart from the initial texture at each step. When constantly applying tensile load increments on a material with initial texture $f^{{\,}(0)}(g)$ as it is done in the example process (cf section 2.4.1), the distance between texture $f^{{\,}(0)}(g)$ and $f^{{\,}(k)}(g)$ is expected to be smoothly and monotonically increasing with step number k. The GSH-based distance measure as well as the distance measures based on pole figures show this behavior. The monotonic requirement of the distance measure is also preserved by the Sinkhorn distance of the histogram representation, while it is violated by the Chi-Squared distance measure at soft-assignment l = 1 of the histogram representation. The Chi-Squared distance measure is often not smooth (see figure 5), independent of the soft-assignment parameter, while the Sinkhorn distance is smooth in all of the presented studies (except for the single crystal example shown in figure 5(g)).

Another disadvantage of the Chi-Squared distance is its high sensitivity with respect to a shift of orientations from one bin to another one. This is less pronounced for dense histograms (i.e. where there is a non-zero number in each bin) than for sparse histograms. This is best seen by looking at extreme cases of three sparse histograms: Histogram $f_{\boldsymbol{\mathrm{b}}}^{(1)}(g)$ has orientations only in one single bin o₁, otherwise all entries are zero. Histogram $f_{\boldsymbol{\mathrm{b}}}^{(2)}(g)$ has orientations only in an immediate neighbor bin of o₁. Histogram $f_{\boldsymbol{\mathrm{b}}}^{(3)}(g)$ has orientations only in any other bin. The Chi-Squared distances in both cases $\mathscr{D}_{\mathrm{\chi^2}}(f_{\boldsymbol{\mathrm{b}}}^{(1)} (g), f_{\boldsymbol{\mathrm{b}}}^{(2)} (g)) = \mathscr{D}_{\mathrm{\chi^2}}(f_{\boldsymbol{\mathrm{b}}}^{(1)} (g), f_{\boldsymbol{\mathrm{b}}}^{(3)} (g)) = 2$ and thus maximal (following equation (13)), no matter how close the unit quaternions of the bins are.

In general, it can be seen that the soft-assignment parameter has a strong effect on the Chi-Squared distance measure, as it incorporates local orientation distances by distributing histogram weights to several bins. This is artificially countering the disadvantage of sensibility against the shift of orientations. The Sinkhorn distance, in contrast, is not affected much by the soft-assignment parameter as it incorporates also the distances between the orientations of the bin representatives (cf Eq(14)). This reflects the neighborhood relations of the orientation space over which the ODF is spanned. Nevertheless, larger values of soft-assignment l yield smoother distance curves. In the case of Chi-Squared distance the usage of l > 1 is mandatory for the above mentioned reasons.

In addition, the number of histogram bins of the orientation histogram has a significant effect on the Chi-Squared distance measure, which can be seen in figures 5(a)–(c) and (d)–(e): the higher the number of histogram bins, the lower the distance is. For higher number of histogram bins (e.g. J = 2048), the Chi-Squared distance even tends to converge towards the Sinkhorn distance. The Sinkhorn distance in-turn is much less sensitive to the number of histogram bins.

Regarding the single crystal texture study (Initial Texture 3 ($n_{\mathrm{oris}}$ = 1)), the orientation histogram-based distance measures do not show a smooth behavior, because the crystal rotation after loading cannot be tracked adequately by the orientation histograms. Therefore, the Chi-Squared and the Sinkhorn distance jumps at certain load steps, independently of the number of histogram bins and soft-assignment parameter. However, because of the bin-wise comparison, the Chi-Squared distance reaches quickly it is maximum value. In contrast, the Sinkhorn distance does not reach the maximum value at all and converges almost to the same value for different number of histogram bins and soft-assignments. Moreover, the Sinkhorn distance is closer to the quaternion distance, which can be considered to be the ground truth in this case.

In contrast to the orientation histogram-based distance measures, the pole figure distance shows a rather smooth behavior over the loading steps, as the rotation of the single crystal is tracked by the pole figure. This is due to the generation of pole figures by using a Gaussian kernel which spreads the orientations over several pixels. The distance curves resulting from pole figures are monotonic and smooth. Also the GSH distance shows a smooth behavior in all cases, as it is based on a smooth function approximation (cf equation (9)). However, in figure 6, it can be seen that the GSH distance behaves contrary to the Sinkhorn distance, as it has a slight increase at the beginning of the loading steps and a rather strong increase later. This characteristic cannot be influenced by the degree L of the GSH distance and does not seem to have a strong effect in this example. It can be seen from figure 6 that the GSH distance measure has lower resolution power at small texture distance than the Sinkhorn distance measure while it is higher at larger texture distances than Sinkhorn.

When applying the distance measures on experimentally measured textures, naturally, the texture distances do not behave linearly in the chain hot rolling, cold rolling and heat treatment. Heat treated and cold rolled textures are similarly far from the hot rolled texture, what in the sense of the Sinkhorn distance means that the work to change a grey texture to one of these textures is similar.

Besides evaluating different texture representations for texture distance measurement their suitability for modeling texture-property relations was investigated in section 2.3. The results show that sufficient prediction quality can be achieved for all texture representations (see table 2), whereas the best results are obtained with the textures represented as pole figures (considering MAE) and GSH with L = 16 (considering R²).

The pole figure representation allows the neural network models to exploit the pixel-represented neighborhood relationships within the textures through convolutional operations, which explains the superior prediction quality. However, the pole figure image data must be transformed by a backbone to a vectorized latent space by means of convolutional and pooling layers, before the neural network head can perform the regression to the property values. The additional backbone layers make processing of pole figures more complex than using histograms as input, which results in higher model training efforts.

In contrast, the orientation histograms can be directly used as input for the neural regression network, reducing the overall network complexity. Investigations were performed for the number of histogram bins J = 512 and soft-assignment l = 1 and l = 3, where the better result is achieved with l = 3. The improvement with l = 3 can be explained by the approximate consideration of neighborhood relationships by higher order smoothing via soft assignment. The number of bins of J = 512 was chosen as compromise between computational cost and representation quality.

The GSH representations can be directly used as input as well. The investigation was performed using GSH representations of the degrees L = 16 and L = 10, which comprise of 280 and 80 features respectively. A better prediction quality is achieved with L = 16, but with L = 10 sufficient results are already achieved with its lower number of features, requiring a less complex neural network model. Better prediction results than with orientation histograms can be achieved by GSH-based models with lower number of features due to the implicit representation of orientation neighborhood relations in the base functions of the GSHs.