Ashish Chauniyal, Interdisciplinary Centre for Advanced Materials Simulation
(ICAMS)
Ruhr-Universität Bochum, Univeristätstraße 150, 44780 Bochum, Germany
<EMAIL_ADDRESS>NA

# Employing semi-supervised learning for high-resolution segmentation of
crystal defects

Ashish Chauniyal  Pascal Thome  Markus Stricker

###### Abstract

Materials characterization using electron backscatter diffraction (EBSD)
requires indexing the orientation of the measured region from Kikuchi
patterns. In the presence of defects the quality of the Kikuchi patterns can
degrade. Or if the measured region contains a grain boundary, two or more
orientations can cause pattern overlap which may lead to erroneous orientation
calculations. In this work we show that constrained non-negative matrix
factorization can be successfully employed to characterize such measurements
containing signals from multiple crystallographic orientations, particularly
if the misorientation is very small ($<1\degree$). Using simulated patterns,
we demonstrate the resolution limit of the method and subsequently showcase a
computationally efficient segmentation of a dendritic microstructure, which is
on par with high resolution EBSD. By analysis of the weight metrics of the
components across a grain boundary or dendrites, we demonstrate how very small
misorientations/low-angle grain boundaries can be resolved at a pixel level.
Our approach constitutes a versatile and robust tool complementing other fast
indexing methods for microstructure characterization.

###### keywords:

Kikuchi patterns, segmentation, semi-supervised learning, pattern overlap,
grain boundary, HR-EBSD, RVB-EBSD

##  1 Introduction

Electron backscatter diffraction (EBSD) has been the prime workhorse for the
analysis of crystal structures and orientation of crystalline materials for
many years [11, 15]. While the fundamental principles of diffraction and
recording of Kikuchi patterns have not changed over the years, there has been
continuous developments in novel ways of post-processing [27]. Starting from
Hough-based indexing [10, 1, 41, 20], which transforms the patterns to a Hough
space followed by indexation, to the more recent Dictionary Indexing (DI) [8,
33, 22] and pattern matching [28, 29] approaches, which directly compare the
patterns to a simulated library and find out the best match. Such indexing
methods process noisy patterns effectively and also provide good lateral and
angular resolutions. The angular resolutions of EBSD has further been improved
using cross-correlation methods that measure pattern shifts, along with the
ability to quantify strains [43, 37, 39, 5, 34, 40, 33]. The cross-
correlation-based methods thus provide a high resolution of angular variations
by directly identifying sub-pixel level pattern shifts and go by the acronym
high-resolution electron backscattered diffraction (HR-EBSD). Onwards from
here with the progress of machine learning in material science, several data-
based supervised machine learning (ML) methods have recently been employed for
indexation and segmentation of microstructures [17, 30, 19, 18, 9, 23]. These
methods circumvent the need to evaluate computationally expensive pattern
comparisons and instead train a ML model that is adapted to the intricacies of
the orientation space. While for routine tasks such methods are well suited,
their robustness remains in question, as machine conditions, sample
preparations, crystallography, and the need for large training datasets can
greatly limit the extent to which such methods can be quickly deployed. An
additional aspect is their transferability to unseen data. Due to the above
reasons, it is generally desirable to employ unsupervised ML methods.

Unsupervised methods promise a higher efficiency and robustness [25]. Several
such unsupervised data-based approaches have been presented [25, 4, 38] which
can discriminate between similar or dissimilar phases and improve the
effective spatial resolution of automated EBSD analysis, particularly
segmentation tasks. Denoising of electron backscattered diffraction patterns
(EBSPs) is possible using (unsupervised) dimensionality reduction strategies
[2]. For unsupervised methods an essential precept is that EBSPs are not
directly mapped to orientations or phases in the standard EBSD sense, but
instead undergo dimensionality reduction on raw pattern data followed by
clustering [38, 25, 4]. Herein, EBSP patterns are treated as linear
combinations of spatially simple components [4] which can then be used
efficiently for segmentation. While the above methods do not directly provide
the orientation information of the microstructure, they are still very useful
in conjunction with standard EBSD methods. The most important advantage in
this respect lies in the ability to segment and categorise data without prior
knowledge of the crystallography of the microstrucutre. This implies that such
methods are easily transferable to different types of microstructures without
the need for expensive supervised training, often involving human-created
masks or labels because these methods detect pattern existing in data without
human bias.

The simplest dimensionality reduction algorithms that are used for EBSD data
analysis, are principle component analysis (PCA) [4, 38, 2] and non-negative
matrix factorization (NMF) [25]. NMF is shown to perform especially well for
the segmentation of crystallographically similar phases, which extols the
sensitivity of the technique in detecting very small changes in an EBSPs [25].
So far, only a canonical NMF method has been implemented for segmentation
which is essentially a subset of PCA. However, the completely unsupervised
nature of NMF implies that the derived NMF coefficients may not be amenable to
physical interpretation. The derived NMF coefficients look similar to an EBSP
with only small variations in intensities [25], however from an EBSD
standpoint they cannot be directly associated with an orientation or a phase.
For a connection to physics, the coefficients in such a factorization should
be related to the either an orientation or a phase, as well, which is not
possible with canonical NMF. For this reason we need a factorization scheme
that can exploit the prior knowledge of orientation or phases. Such a
specialized factorization method is called constrained non-negative matrix
factorization (cNMF) which incorporates the prior knowledge of one or all
components during the factorization [24]. In the context of EBSD, this
translates to having a prior knowledge of certain EBSPs belonging to a
particular orientation or phase. These known EBSPs need to be supplied
externally during factorization and therefore the scheme can be classified as
semi-supervised. In this work we implement a cNMF algorithm to factorize EBSPs
using known EBSPs and calculate the weights of the components. These weights
thus provide a metric to estimate the fraction of known EBSPs in any pattern.
We show how this approach can be used to segment microstructures in great
detail, which is on par with HR-EBSD [34] methods, with much less
computational overhead.

Fixing components before factorization implies that we select certain EBSPs
from the microstructure and then resolve the rest in terms of their weights
with respect to these EBSPs. This approach of using weight-based metric to
resolve a microstructure, subsequently allows us to address the problem of
pattern overlaps at grain boundaries or other defects [42]. The subject of
pattern overlaps has been explored extensively in the past [3, 44, 42, 35, 32,
36, 21, 31, 6, 13] as it is well known that EBSD measurements nearby grain
boundaries can create uncertainties in indexation. Inherent to EBSD
measurements is the fact that the intensity contribution to the final pattern
from each grain, when measured at a grain boundary, is related to the
respective interaction volume of each grain. So far, the pattern overlap
problem has been treated using rigorous pattern matching approaches which
incorporate multiple crystal orientations [31, 35, 21], or by determining a
resolution limit to avoid overlaps in the first place [3, 44, 13]. Yet, these
approaches rely on mapping EBSPs to orientations. Even if these approaches
lead to increase in accuracy, the volume fraction contribution from each grain
may still remain unknown. Though the consideration of volume fraction has
recently been made by simulating the mixing of Kikuchi patterns and
subsequently matching patterns [21]. Still, there is a need for employing a
much less computationally expensive and more robust scheme.

In this regard, the work of [4] is most relevant to our work, as it employs a
very simple yet robust PCA method to segment grains. Not only does this method
deliver angular resolutions on par with standard EBSD, but can also
discriminate between crystallographically similar phases. Furthermore, the
authors foresaw the ability of PCA in deconvolving pattern overlaps, which can
exceed the capabilities of EBSD. At the core of such analysis is the shift to
a paradigm that EBSPs can be analyzed using an alternate metric, instead of
orientations alone. In this work we follow this new paradigm of using
alternate weight-based metric to characterize a microstructure [4, 38, 25]. We
employ an advanced data-based method - cNMF, which allows us to estimate the
contribution of individual EBSPs in an overlapped EBSP using the weight
metric. In this paper we briefly describe the cNMF method and explain its
implementation for EBSPs. Thereafter, we first test the limits of the method
on simulated patterns. We then present results pertaining to an experimental
data set of low angle grain boundaries to showcase the segmentation ability
for a limiting case of very small misorientations. This data set is specially
chosen as it is synonymous to crystallographically similar EBPSs. Finally, we
elaborate how to extract more information from the predicted values and
discuss the identification of grain boundaries.

##  2 Methods

###  2.1 Experimental Details

The specimen used for investigation is a cross-section of a cylindrical bar of
single-crystalline superalloy ERBO1/C which is directionally solidified using
a seeded Bridgeman technique [16]. This casting technique results in a single
crystal along $\langle 001\rangle$ direction. However, during solidification
individual dendrites frequently deviate by small misorientation angles, which
leads to a fine substructure [34]. Cross-sections are carefully polished using
a sequence of $6\mu$m, $3\mu$m and $1\,\mu$m diamond suspensions, followed by
a $3\,$h Vibrometer polishing with a colloidal silica suspension of
$0.25\,\mu$m size. Finally, the surface has been cleaned by ion milling with
$6\,$kV beam for $30\,$min at $40\,\degree$ inclination. The above steps are
essential to obtain good quality Kikuchi patterns.

EBSD measurements are performed using a FEI Quanta 650 FEG SEM at $30\,$kV
accelerating voltage, $60\,\mu$m aperture, $17\,$mm working distance WD and
$200\,$mm camera length (distance between detector and sample). Patterns are
recorded using an EDAX-TSL system with a Hikari-XP camera at a resolution of
$512\times 512$ pixels, with a step size of $1\,\mu$m.

###  2.2 Constrained Non-Negative Matrix Factorization

Constrained non-negative matrix factorization (cNMF) [24] is essentially a
version of the canonical non-negative matrix factorization (NMF) where the
former is semi-supervised and the latter is unsupervised. Starting with NMF
first, for a given non-negative matrix $\mathbf{X}$ of dimensions $m\times n$
we seek a factorization such that

$\mathbf{X}\approx\mathbf{WH},$ (1)

where $\mathbf{W}$ and $\mathbf{H}$ are non-negative matrices of dimensions
$m\times k$ and $k\times n$ respectively. Here $k$ is the number of components
which are provided by the user. The above then reduces to a minimization
problem which can be solved as

$\min_{\mathbf{W}\in\mathbb{R}^{m\times k},\mathbf{H}\in\mathbb{R}^{k\times
n}}D(\mathbf{X},\mathbf{WH}),$ (2)

such that $\mathbf{W}\geq 0,\mathbf{H}\geq 0$ and $D$ is a separable, positive
measure of fit. In the above, the $\beta-$divergence loss function is used for
minimization [12]. $\beta$ is the single shape parameter used in the
minimization and we employ $\beta=2$ or squared Euclidean distance given by

$D_{2}(\mathbf{X},\mathbf{X}^{\prime})=\frac{1}{2}\sum_{i=1}^{m}\sum_{j=1}^{n}(\mathbf{X}_{i,j}-\mathbf{X}^{\prime}_{i,j})^{2}$
(3)

An alternating non-negative least squares (ANLS) algorithm is used for
minimization that operates by alternately minimizing
$D_{\beta}(\mathbf{X},\mathbf{WH})$ with respect to $\mathbf{W}$ and
$\mathbf{H}$ such that all matrices are constrained to be non-negative. The
above algorithm has been implemented with PyTorch by Maffetone et. al. [24]
which is different from a canonically implemented NMF [25].

It is noted that the canonical NMF has some drawbacks [24] and from the
standpoint of scientific data analysis, a major one is that the output
components may not correspond to physically-interpretable entities. The above
drawback is circumvented by using cNMF where the initial values for
$\mathbf{W}$, $\mathbf{H}$ \- $\mathbf{W}_{0}$ and $\mathbf{H}_{0}$
respectively, are constrained using prior knowledge and preset before the
minimization as a constraint. Therefore, either the weights $\mathbf{W}_{0}$
or components $\mathbf{H}_{0}$ can be constrained at a time for factorization.
In the following work, we constrain the components $\mathbf{H}_{0}$ and solve
for the corresponding weights. In the next section we describe how this
mathematical framework is implemented for EBSP data.

###  2.3 Data Processing

Figure 1: Schematic of the matrix algebra used in constrained non-negative
matrix factorization. The data matrix $X$, the constrained components $H$
(preset EBSPs) and the unknown weights $W$.

Each raw EBSP is first subtracted by a static background and a dynamic
background but not processed further. Subsequently, the intensities for each
pattern are normalized in the range 0 to 1 and vectorized as a single
continuous vector. For the analysis of experimental data, the chosen region of
interest is shown in Figure 3. Component EBSPs are selected from grain
interiors marked by red markers in Figure 3. To obtain sharp component EBSPs
from the marked regions, a 3x3 pixel area is selected and averaged using a
Gaussian 3x3 kernel. These component patterns $C_{1-6}$ are also shown in
Figure 3.

The component patterns $C_{1-6}$ shown in Figure 3 are very close w.r.t. their
orientation. This can be rationalized based on the fact that the chosen
microstructure is essentially a single crystal with only small amounts of
misorientations ($<0.1\degree$), hence the patterns $C_{1-6}$ must look very
similar. Conventional Hough-based EBSD methods are unsuitable to analyze such
a microstructure and, therefore, we employ the Rotation Vector Base Line
(RVB)-EBSD method [34] for a baseline orientation mapping as reference. The
RVB-EBSD method is an HR-EBSD technique specifically designed to study crystal
mosaicity and fine microstructural details. Such microstructures are
associated with slightly deviating growth paths of dendrites during
directional solidification of Ni-based SX superalloys in a Bridgeman process
[16]. RVB-EBSD utilizes a normalized cross-correlation function along with a
rotational vector baseline function to determine pattern shifts caused by
changes in orientation relative to a selected reference pattern. This allows
to detect small angular deviations ($<0.1\degree$) clearly. Therefore,
consistently in this paper, we use the RVB-EBSD analyzed microstructure as a
reference benchmark to compare our results obtained using cNMF. For example in
Figure 3, the central figure (orientation map) and Figure 5 (Kernel Average
Misorientation). Additional details of the RVB-EBSD method and the applied
color coding schemes can be found in the relevant publication [34].

To prepare data for cNMF, we first take note that each EBSP with a resolution
of $\mathrm{n}_{\mathrm{px}}\times\mathrm{n}_{\mathrm{px}}$, is an image with
gray values from 0-255 and therefore by construction non-negative. As
described in Equation 1, $\mathbf{X}$ forms a matrix with dimensions $m\times
n$, where $m$ is the number of independent measurements or scanned points and
$n$ is the feature dimension of an EBSP or total pixels
$N=\mathrm{n}_{\mathrm{px}}\times\mathrm{n}_{\mathrm{px}}$. To analyze a grain
boundary region between two grains, cNMF is carried out using only 2 known
component patterns - $\mathbf{H_{1}}$, $\mathbf{H_{2}}$. These component
patterns are selected from the $C_{1-6}$ shown in Figure 3, depending on the
analyzed region of interest. The constrained factorization then returns a pair
of weights - $\mathbf{W}$ corresponding to each component $\mathbf{H_{1}}$,
$\mathbf{H_{2}}$. These 1-dimensional weights are then reshaped into
respective 2-dimensional spatial dimensions for visualization. In this way we
obtain 2 weight maps - $w_{1}$,$w_{2}$, corresponding to each constrained
component.

##  3 Results & Discussion

###  3.1 Mixing Simulated Patterns

As a first step, we test the theoretical angular resolution limits of the
method using artificially generated overlapping patterns through intensity
mixing [4] that can be interpreted as a simulation of the conditions near a
grain boundary [35]. An arbitrary orientation is chosen and a Kikuchi pattern
is extracted from a gnomonic projected space of a dynamically simulated master
pattern [7]. Similarly, another pattern is extracted with a relative
misorientation angle $\omega$ with respect to the first pattern. These two
patterns are linearly mixed into 20 intensity fractions, which includes the
two original pure patterns as the limiting cases. Further data processing is
the same as described in Section 2.3. Note however, that in this case the
predicted weights are 1-dimensional, which can be compared to a linear
behavior for verification. Figure 3 shows the resulting weights of the
factorization of patterns oriented along $[111]$ direction and misoriented by
$\omega$. We consider 4 cases in which $\omega$ is gradually decreased as
$\omega=5\degree,\,2.3\degree,\,1\degree,\,0.5\degree$. The displayed
overlapping patterns in Figure 2 correspond to the 10th index where there the
original patterns are mixed with equal weight intensities.

Figure 2: Constrained non-negative matrix factorization of a synthetic dataset
showing predicted weights (points: blue = $w_{1}$, orange = $w_{2}$), compared
to linear behavior (grey lines). The misorientation angle between the
component pattern used for synthetic EBSP generation is (a) $\omega=5\degree$
(b) $\omega=2.3\degree$ (c) $\omega=1\degree$ (d) $\omega=0.5\degree$. The
Kikuchi patterns in each subfigure show the 10th pattern in the index that has
equal ratio of mixing intensities.

Figure 2 (a) shows a superimposed Kikuchi pattern with visible overlap of two
patterns, a common case in grain boundary regions. The predicted weights
$w_{1}$ and $w_{2}$ for different mixing ratios follow the grey lines or the
linear interpolation in the same manner as they are mixed. Note that the
pattern overlap is clearly discernible in the image, due to a relatively large
misorientation angle $\omega=5\degree$. Besides the misorientation, the result
also highlights that very small changes in pattern intensities or weights are
captured using cNMF as 20 different weights, and therefore mixing ratios, are
successfully resolved. This is akin to crossing over a grain boundary in 20
discrete steps. In addition, each data point is independently predicted in
cNMF. In other words, each EBSP is individually factorized, without any
influence of the outcome of other data points.

With a decrease of $\omega$ to $2.3\degree$ in Figure 2 (b), it becomes
difficult to visibly discern the overlap of two patterns. Instead, only a blur
is visible, e.g. in the bottom left corner. Nevertheless, cNMF successfully
predicts the calculated weights as per the linear mixing.

Further decrease of $\omega$ to $1\degree$, shown in Figure 2 (c), leads to
some data points which begin to deviate from linearity. Even though there is
hardly any visible overlapping signature in the Kikuchi pattern itself, the
cNMF method performs well even in this case.

A misorientation of $\omega=0.5\degree$, as shown in Figure 2 (d) constitutes
a case beyond the resolution limit for factorization. At this low
misorientation angle, the weights cannot be determined accurately any more.
However, such a small misorientation is not necessarily the limit of the
method itself, but only selectively applies to the above dataset of 20
patterns. To attest to this, notice that even though the predicted weights
deviate from absolute linearity, they still follow the linear trend. This
implies that if the instead of 20 a lower number of patterns are factorized,
the method may well predict better results. This scenario is akin to
decreasing the number of steps when moving across the grain boundary. However,
in this work we are only introducing a new method, so we refrain from such an
analysis, which will be a part of a separate, future study.

Using the above analysis we are able to simultaneously test two parameters.
The first is the sensitivity of cNMF to misorientation and the second is
sensitivity to mixing steps. In practice the first is akin to resolving small
misorientations between grains while the second is related to step size. We
demonstrate that the cNMF method successfully predicts these variations.
Moreover, we conclude that it is possible to locate grain boundaries in a
microstructure using a weight based metric.

###  3.2 Experimental Patterns

Figure 3: Weight maps of cNMF factorization chosen from different regions of
the scanned area. The central figure shows the full scanned area and is
colored according to a pole figure generated using RVB-EBSD method with the
pole along a $[001]$ direction as a $2\degree$ pole. Within the pole figure,
specific regions are highlighted using red stars. They mark the locations
chosen for the constrained components. Note that there are 5 locations of
component patterns in total corresponding to 5 separate dendritic regions. All
weight maps show a low-angle grain boundary between two regions and,
correspondingly, two weight maps which show individual variations in weight
$w_{1}$ and $w_{2}$ of the chosen patterns as constraints.

The use of simulated patterns is an essential step to test the limits the
method but it is difficult to simulate real patterns containing noise,
background signal, beamshifts, etc., and which are affected by sample surface
quality, sample preparation artifacts, plastic deformation zones, surface
relief and contamination issues. Therefore, in this section we apply cNMF on
an experimental dataset of an ERBO1/C Ni-based single crystal superalloy
specimen. The details of specimen preparation are provided in 2.1. This
dataset is specifically selected due to the prevalence of low angle grain
boundaries (the grains are within $1.5\degree$ orientation spread). This
approach further tests the practical limits of the cNMF method by resolving
patterns that are very similar (also see $C_{1-6}$ in Figure 3). We compare
our results to a RVB-EBSD based high-resolution analysis [34], which is state-
of-the-art in analyzing such microstructures.

Figure 3 (R1-6) shows the reconstructed weight maps using cNMF. The analysis
is made using small sub-sections of the scanned area, such that each region of
interest contains a low-angle grain boundary between two dendritic regions.
The constrained components for each region are extracted from within the grain
interior marked C1-6 with red stars. For example, in region R1 the constrained
component EBSPs are C1 and C5. Similarly for region R5 they are C3 and C5. In
this manner, corresponding to each component combination, a weight map is
obtained, shown using $w_{1}$ and $w_{2}$. Note that the weight maps are shown
using both $w_{1}$ and $w_{2}$ as both together can be used as indicators to
demonstrate variations in each individual weight. Unlike the synthetic pattern
analysis consisting of superimposed patterns in section 3.1, in this case only
the grain boundary EBSPs are expected to exhibit pattern overlaps whereas a
single pattern (orientation) typically dominates in the grain interior. This
can be seen in Figure 3 region R1, where each weight map $w_{1}$, $w_{2}$ has
distinct bright and dark colored regions which invert symmetrically across the
grain boundary. That is, weights are inversely proportional which allows us to
demarcate the extent of each grain. Notice that in terms of misorientation,
each pair of two grains lie within $1\degree$ pole as shown in the RVB-EBSD
pole figure in the center bottom. In addition, also note that in the pole
figure the grain boundary pixels may not be indexed to distinct orientations
because of pattern overlap (the grain boundary pixels are white), which is not
the case with weight maps $w_{1}$, $w_{2}$. Thus, using weights, the extent of
the grain boundary is distinctly known. Furthermore, in the weight map $w_{1}$
the intensity of the pixels varies within the grain which is indicative of
small orientation changes, consistent with the RVB-EBSD pole figure.

In region R4, the grain boundary is clearly visible in the weight maps.
Interdentritic orientations, however, are not visible, consistent with RVB-
EBSD pole figure.

In region R5, two boundaries exist between intersecting dendritic regions
which consist of only 2 grains. The corresponding weight maps exhibit similar
pixel intensities that lie within the same grain which distinctly demarcates
the grain boundaries. Moreover, weight map $w_{2}$ shows a variation in
intensity within the respective grain (bottom left), that is consistent with
interdendritic orientation changes in the pole figure.

In region R6, weight maps demarcate the extent of the grain boundary as well
as local changes in orientation along the grain boundary in map $w_{2}$. The
above cases demonstrate that cNMF is able to detect orientation variations
which are on par with the RVB-EBSD reference (orientations changes are less
than $0.25\degree$). Note however, the variations in intensities in region R6
within a dendritic region implicitly arise due to sensitivity of the method as
the factorized EBSPs are dissimilar to the component EBSPs. This implies that
deviations from the component pattern lead to changes in weights, even though
they arise from pattern degradation and not alone from orientation changes.
Following this, it can also be argued that all EBSPs that deviate from the two
chosen component EBSPs may correspond to a 3rd orientation or grain.

This aspect is brought out more clearly in region R2 where, apart from a grain
boundary, a triple junction exists in the top right corner. In the weight maps
$w_{1}$, $w_{2}$, two dominant grains are distinctly visible. Herein, the
constrained components are C1, C4. Yet, the predicted weights in the 3rd
intersecting dendritic region (right corner), which is not used as component
in cNMF, are very similar in intensities for both $w_{1}$ and $w_{2}$ maps.
Thus, attesting to the argument that any orientation that is not used in cNMF
may still manifest itself in the form of weight intensity variation. This,
generally, allows the segmentation of more than two grains even if only two
components are used, albeit with some loss of accuracy. More discussion on
this aspect is provided within Section 3.4.

In Region R3, the component EBSPs are located at C3 and C4, where there is a
change in orientation within the left grain. In other words, the grain
boundary region is slightly misoriented to the orientation of the C4 location.
Furthermore, the misorientation angle between the two grains is lowest in this
region ($\approx 0.5\degree$) as shown in the central pole figure.
Consequently, the calculated weight maps exhibit a considerable amount of
noise even through the demarcation of dendritic and interdendritic grain
boundaries are successfully identified. Thus, this case forms the limiting
case of the cNMF method. Moreover, this demonstrates that the weight maps are
sensitive to the location of the selected component patterns in the scan space
with respect to the region of interest.

###  3.3 Resolving the grain boundaries

Figure 4: Profiles of individual weights $w_{1}$, $w_{2}$ across two grain
boundaries: (a) sharp boundary and (b) smeared boundary. The weight map
$w_{1}$ of region R6 is used to depict the microstructure. Along profile ‘a’,
the EBSPs at 3 points at the grain boundary from index 35-37 are shown. The
location of these EBSPs along the profile are indicated with red circles in
(a). The enlarged section of the EBSP at index 36 is shown where a faint
signature of pattern overlap is visible and indicated with a yellow arrow.

Using a weight based metric instead of orientation allows us to resolve grain
boundaries by leveraging the change in weights and, thereby, address the issue
of pattern overlaps. Consider Figure 4, which shows the weight profiles across
a grain boundary for two cases - (a) across where orientation appears to
transition sharply from one pixel to the other and (b) where the grain
boundary appears smeared out. Along profile (a), weights $w_{1}$ and $w_{2}$
exhibit a discrete step-like change across the grain boundary. Notice the
relative symmetric reversal of the weights across the grain boundary. This
implies that the grain boundary is sharp and patterns do not have a measurable
overlap within the grain interior but only at the interface. To better explore
this behavior we consider three distinct patterns which lie along the grain
boundary profile (a). Pattern indices 35,36,37 lie across profile ‘a’ at the
transition from left grain to the right. Their respective weights are marked
with red circles in the Figure 4(a). Pattern 36 exhibits an overlap of
patterns 35 and 37 which is, however, very hard to visually discern. An
enlarged section of pattern 36 shows a faint secondary band marked by a yellow
arrow. This indicates that there is some pattern overlap of patterns 35 and 37
which is successfully resolved. The above results thus depict a use case which
is comparable to the scenario tested using simulated patterns in Section 3.1.

On the other hand, along profile ‘b’, weights $w_{1}$ and $w_{2}$ exhibit a
gradual change across the grain boundary. Across the grain boundary along
profile ‘b’, the weight map shows a visibly smeared grain boundary which
indicates that the orientation changes gradually. Nevertheless, the position
of the grain boundary can be estimated at the point where the two weights
$w_{1}$, $w_{2}$ make a transition. This scenario is comparable to the
scenario where the grain boundary is crossed in several steps as discussed
previously (Sec. 3.1). Herein, the relative transition of weights $w_{1}$,
$w_{2}$ can also be exploited to estimate the location of the grain boundary.

To locate the grain boundary, we need to extract the information contained
implicitly in the weight metric. The transition between very similarly
oriented regions can be captured by calculating the absolute differences in
weights $abs|w_{1}-w_{2}|$ per pixel since pixels at the grain boundary have
similar or equal weights as a consequence of pattern mixing. Consequently, the
absolute difference in weight is lower at and near grain boundary region
compared to the grain interior (also consider the weight profile in Figure 4
for an intuition). To test this hypothesis, we show the absolute weight
difference maps of separate regions in Figure 5. For comparison, we
additionally present a Kernel Average Misorientation (KAM) map for the same
region as obtained using RVB-EBSD [34].

Figure 5: Absolute weight difference maps of different regions and comparison
to a KAM map. The KAM map is obtained after processing the pole figure map in
Figure 3. Enlarged versions of the regions are shown in their correct relative
size. The highlighted regions are the same as selected in Figure 3.

In Figure 5, region R1, the grain boundary is revealed as dark colored pixels,
where as the brighter pixels belong to the grain interior. This follows from
the hypothesis that the absolute difference in weights in the darker pixels
should be lowest while the grain interiors would show relatively larger
values. A comparison with the KAM sub-region shows that in the difference
weight map the grain boundary is much sharper and well defined. Other
interesting features are the apparent discontinuities in dark pixels at the
grain boundary which may be understood as a consequence of the step size and
orientation of the grain boundary with respect to the scan direction [36].
Darker pixels are clustered around regions where the grain boundary curvature
changes. Local orientation changes arising from sub-grain boundaries are also
visible in these maps as variations in intensity. Overall, it is possible to
resolve a grain boundary at the level of a pixel. This resolution would not be
possible using a KAM analysis as each pixel is assigned to a particular
orientation which then leads to an average misorientation with respect to the
surrounding pixels. In other words, depending upon the size of the kernel, a
grain boundary will always be a few pixels wide and also continuously
connected.

Our proposed approach is capable of detecting regions of pattern overlaps very
effectively and efficiently. In cases where minimal overlap exist, we can
nevertheless capture the relative variations from one grain to the other. For
example, in Figure 5, region R5, the grain boundary can be identified as
different pixel intensities in the two grains. This allows us to sharply
demarcate the grain boundary. A similar scenario unfolds in Figure 5, region
R4, where the grain extent is distinctly demarcated as a change in mean
intensity, along with some darker pixels at the grain boundary. That is, the
left grain is brighter than the right grain and separated by a dark grain
boundary region.

Even maps derived from noisy weight maps, such as Figure 5, region R2, allows
the detection of the grain boundary region very accurately. Another
interesting feature is notable in this region. The small portion of a third
grain (top right) manifests in the form of dark pixels. Understandably so,
because the absolute difference scheme entails low intensity pixels for that
portion of the map, which belongs to a 3rd grain orientation which is not part
of the constraints. This, in principle, allows to resolve triple junctions
around this region similar to the predictions of Brewer [4]. However, the
analysis of triple junctions does not lie within the scope of the present work
but can be subject of future studies.

Finally, Figure 5, region R6, reveals that a combination of the previously
described scenarios can be present. The grain boundary is sharp and denoted by
dark pixels is specific regions. A relative variation of mean intensity exists
between the two adjoining grains, thereby allowing us to demarcate the extent
of each grain. That is, the left grain is brighter than the right grain. In
addition, some areas around the grain boundary show substantially darker and
smeared-out transitions at the grain boundary. This can be rationalized as a
combined affect of the presence of sub-grain structures and because the
component EBSPs are slightly different from the sub-grain structure EBSPs.
These features, however, only show up as gradual variations in intensity.

###  3.4 General discussion

First and foremost, the cNMF method provides a novel method of segmentation as
compared to an orientation indexing approach. The use of a weight metric to
resolve the extent of each grain allows us to capture orientation changes
within the grain as well, which is on par with HR-EBSD methods. While the
weight maps shown in Figure 3 belong to a grain boundary between only two
grains, it is in principle also possible to resolve larger areas. For example
in Figure 6, a two-component segmentation with one weight map is enough to
demarcate the extent of dendrites and interdendritic features. In the figure
$w_{1}$ map shows the intensity of weights which are most similar to the
component EBSP at C3. Notice the level of detail within the large dendrite
that can be captured along with a clear demarcation of grain boundaries. To
further extend the same idea, we can also calculate multiple two-component
factorizations and then combine the results together. For example in Figure 7
four different instances of factorization are shown with component EBSPs
originating from the grain interiors. These four weight-maps are then combined
to form a 4 channel color composite image that shows the microstructure in
fine detail. Herein, similarity in colors signify similarity of orientation as
well.

Note that this is possible in our experimental data set because the patterns
belong to similarly-oriented grains or very similar EBSPs. In case of large
angle grain boundaries, the regions which deviate substantially from the grain
of interest exhibiting low intensities. This may be construed as a drawback of
the method. However, we propose our method as a companion tool useful in
special cases such as those presented above or in segmentation problems where
indexing crystallographically similar phases creates its own problems [25,
21].

The second key aspect of using cNMF is the ability to factorize overlapping
pattern into their constituents which follows a paradigm of employing a
weight-based metric. This is advantageous as the binary classification of a
pixel that is used in orientation imaging [35] is avoided. So far, the method
of reducing the resolution of the scan step is used to accurately resolving
grain boundaries [44]. However, a weight-driven approach can allow to clearly
locate a grain boundary without changing the resolution limits (accelerating
voltage) of the electron beam itself. A continuous variable in the form of
weights allows us to appreciate the smooth transitions across a grain boundary
and detect EBSPs with pattern overlaps distinctly. These transitions may arise
either due to inherent orientation changes or due to the step size during
measurement. Following the analysis in Sections 3.1 and 3.3, we can also argue
that the grain boundary can be resolved to a single pixel, irrespective of the
step size. That is, the most probable location of the grain boundary should be
at pixels with largest amount of overlaps. This aspect of cNMF can be
particularly useful in spatially resolving nanoscale phases where the
electron-matter interaction volume is of comparable size to the characteristic
length scale of the phase [6]. In which case resolving pattern overlaps and
cleaning data is of utmost importance for proper characterization. Thus cNMF
can prove to be a viable alternate.

Finally, a comment on the robustness and efficiency of cNMF. The method is
already in use for various scientific analysis [24] and allows for enormous
flexibility in the input data. We have implemented the scheme for EBSD data
analysis using EBSPs. In practice, however, a variety of signals can be used,
as the method is not affected by material/signal/dimensions and therefore can
work for different crystallographic orientations and phases. For example,
locating the boundaries of $\gamma$/$\gamma^{\prime}$ phases in superalloys
[14]. In addition, our proposed scheme would also work for a hybrid dataset
where energy dispersive X-ray spectrocopy (EDS) data can be combined with EBSP
signal [26], i.e. use a multimodal representation of a material or part of it.
With respect to efficiency, since each pattern/signal can be read and
factorized independently it is also possible to parallelize for GPUs. In terms
of speed, the method is less computationally expensive than the HR-EBSD
methods but like such methods, requires storing Kikuchi patterns. Some part of
this storage space can be minimized by only analyzing small grain boundary
regions on the fly and discarding after analysis. Furthermore, the pattern
overlap detection of cNMF can still be used efficiently within the the
standard indexing procedure, as an add-on. To close, we note that any existing
dataset where Kikuchi patterns are recorded can be re-analyzed, since our
proposed method does not depend on any specific measurement settings or
crystallographic information. Consequently more information regarding phases,
orientations and defects can be extracted from already existing datasets.

##  4 Conclusions

We implement a semi-supervised constrained non-negative factorization (cNMF)
method to segment and demarcate grain boundaries. Using linear superpositions
of synthetic patterns, we demonstrate the angular sensitivity and resolution
of this technique. The process simulates two scenarios simultaneously - (1)
the sensitivity to misorientation between patterns (2) sensitivity to mixing
steps. Our method performs well in resolving misorientations considerably
$<1\degree$. This proves that the method can work effectively, in particular
for low-angle grain boundaries.

To test the limiting case of grain boundary segmentation (LAGB), we
demonstrate the use of the method on a dataset of a technical single crystal
superalloy. The specimen comprises of dendrites which deviate by small
misorientation angles, $\omega<1\degree$. Using cNMF, the grain boundary
regions are segmented using weight maps. Inherent to the method is the
paradigm of segmenting a microstructure in the form of weights w.r.t. to user-
defined references instead of absolute orientations. The weight maps clearly
segment the grains and show features which are on par with RVB-EBSD method -
an HR-EBSD technique specifically designed for mapping small misorientations
during dendritic growth. Using the weight metric, we are able to consider
overlaps in patterns and show how more information can be extracted from the
data. The regions of highest pattern overlaps are regions where the absolute
difference in weights is lowest. Using this metric, the grain boundary can be
resolved even at a pixel level, albeit with some nuances, which are discussed
in detail.

The cNMF method is versatile and may find use in applications such as grain
and phase segmentation with similar EBSPs and small-scale phases with
extensive overlapping EBSPs. Furthermore, due to its robustness, hybrid
signals can be used which use a combination of both EBSP and EDS signals.
While the processing speed may still be slower than the standard indexing
speeds today, the method can be useful when used in conjunction with
orientation imaging methods or in localized regions.

## Appendix

Figure 6: A1: cNMF factorization of the entire region of interest using
weight-maps $w_{1}$ \- (b) and $w_{2}$ \- (c). The component EBSPs originate
from C1 and C3 which have also been marked. For comparison an RVB-EBSD
generated pole figure has also been provided - (a) Figure 7: A2: Constitution
of a 4 channel composite color image formed by combining 4 weight-maps of the
entire region of interest.

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## Data statement

The Kikuchi pattern data set can be downloaded from Zenodo
(https:doi.org/10.5281/zenodo.10705510 ). An up-to-data version of the
implementation of cNMF is available from a github respository
(https://gitlab.ruhr-uni-bochum.de/icams-mids/kikuchi_cnmf.git) .

## Conflict of interests

The authors declare no competing financial interests or personal relationships
that could have influenced this work.


# Characterisation of a new RPC prototype using conventional gas mixture

A. Sen<EMAIL_ADDRESS>Department of Physics, Bose Institute,
Kolkata-700091, INDIA  S. Chatterjee  S. Das  S. Biswas

###### Abstract

Resistive Plate Chamber (RPC) is a well-known gaseous detector in the field of
High Energy Physics (HEP) experiments for its good tracking capability, high
efficiency, good time resolution, and low cost of fabrication. The main issue
in RPC is its limitation in the rate handling capability. Several experimental
groups have developed sophisticated techniques to increase the particle rate
capability and reduce the noise rate of this detector. In this article, we
discussed a new method for linseed oil coating in case of bakelite RPC
detector to achieve good efficiency and the results obtained using a
conventional gas mixture.

###### keywords:

RPC, Gaseous detector , Cosmic ray , Efficiency , Noise rate

††journal: Journal of LaTeX Templates

##  1 Introduction

HEP experiments have been using RPCs for triggering and tracking for their
high efficiency and good time resolution. RPCs are also being used in several
cosmic ray experiments to cover large detection area [1, 2]. Future heavy-ion
experiments $e.g.$ CBM will use RPCs for muon detection [3]. RPC is a gaseous
detector made up of resistive electrode plates $e.g.$ bakelite, glass,
ceramic, etc [4, 5, 6].

In bakelite RPC linseed oil coating [7] is done to get rid of surface
roughness issue [8]. The linseed oil coating also helps to reduce the noise
rate of the detector, protects the electrode plate from Hydrofluoric Acid (HF)
corrosive effect and it also has photon quenching properties that reduce the
UV sensitivity of the electrode plates.

In convensional linseed oil coated bakelite RPC a serious problem was observed
in the BaBar experiment. The coated linseed oil formed stalagmite, that
subsequently forms the conducting paths through the gas gap, around the
spacers and the discharge permanently damages the detector. The formation of
stalagmite is due to the polymerization of uncured linseed oil droplets
present on the surface [9]. However, lot of R&D is performed and the solution
is found out for this issue [10].

A new technique of linseed oil coating [11] is introduced in the bakelite
plates in which one can check visually whether there is any uncured oil
present inside the gas gap. With this new technique one RPC prototype was
built and tested with 100% C2H2F4 gas. In this article the test results of the
RPC prototype with conventional C2H2F4 and i-C4H10 mixture is presented.

##  2 Detector description and experimental set-up

The detector is built with two 27 cm$\times$ 27 cm bakelite plates of
thickness 2 mm and having bulk resistivity of $\sim$ 3$\times$ 1010 $\Omega$
cm (at 22∘C). Four edge spacers, two gas nozzles, one button spacer made of
polycarbonate (resistivity $\sim$ 1015 $\Omega$ cm) and having thickness of 2
mm are used to make the gas gap. The surface resistivity of the outer graphite
surface of the electrode plates are measured to be $\sim$ 510 k$\Omega$/$\Box$
and 540 k$\Omega$/$\Box$.

The signal is read out from the copper strips of dimension 2.5 cm$\times$ 27
cm. The strips are covered with 100 $\mu$m thick mylar foils to isolate them
from the graphite layers. The details of the fabrication of the chamber is
described in [11].

Three scintillation detectors of dimensions 10 cm $\times$ 10 cm (SC1), 10 cm
$\times$ 2 cm (SC2), 20 cm $\times$ 20 cm (SC3) are used to generate the
trigger for the detector. All the scintillators are operated at +1550 V and
-15 mV threshold is applied to the leading edge discriminator (LED). The RPC
signal from the pick-up strip is first fed to a 10x fast amplifier and then
the output goes to the LED. Suitable thresholds are applied to the LEDs to
reduce the noise. From the LED, one output goes to the scalar to count the
number of the signal from the RPC which is known as the noise count or singles
count of the chamber. The other output from LED goes to the dual timer where
the discriminated RPC signal is stretched to avoid any double counting of the
pulses and also to apply the proper delay to match the signal with the
trigger. The output of the dual timer is taken in coincidence with the trigger
and this is defined as the 4-fold signal. The window of the cosmic ray test
set-up is of area 10 cm $\times$ 2 cm. The block diagram of the set-up is
shown in Figure 1.

Figure 1: Block diagram of the cosmic ray test set-up for characterization of
the detector [11].

##  3 Result

The chamber is tested with cosmic rays in avalanche mode. The detector is
purged with Tetrafluoroethane (C2H2F4) and Isobutane (i-C4H10) gas mixture in
90/10 volume ratio. The leakage current through the RPC module is measured as
a function of the applied high voltage (HV) and shown in Figure 2.

Figure 2: Leakage current as a function of the voltage for C2H2F4 and i-C4H10
gas mixture in 90/10 volume ratio (colour online).

The efficiency of the RPC module for the cosmic rays is defined as the ratio
of the 4-fold counts to the 3-fold coincidence trigger count of the plastic
scintillator telescope for a fixed duration and the noise rate of the RPC, is
defined as the number of counts per unit area of the strip per second. The
efficiency and noise rate are studied by varying the applied HV. The RPC is
tested with - 20 mV and - 25 mV threshold settings to the LED. An efficiency
of greater than 90% is achieved from 10 kV onwards for both the threshold
settings. The maximum noise rates are found to be 120 Hz/cm2 and 80 Hz/cm2 for
the - 20 mV and - 25 mV thresholds respectively.

Figure 3: (a) Efficiency as a function of the applied voltage, (b) Noise rate
as a function of the applied voltage for a gas mixture of C2H2F4 and i-C4H10
in the 90/10 volume ratio (colour online).

##  4 Summary

A linseed oil-coated RPC prototype of dimension 27 cm $\times$ 27 cm is
fabricated using indigenous resistive bakelite plates. The linseed oil coating
is done before making the gas gap. Before building the detector, electrode
plates are checked visually whether the oil is cured properly or if any
uncured oil is present on the surface.

The detector is tested with Tetrafluoroethane (C2H2F4) and Isobutane (i-C4H10)
in 90/10 volume ratio. Both the current and noise rate are very low for this
gas mixture compared to the 100% C2H2F4 used earlier for the same detector
[11]. An efficiency of greater than 90% is found from 10 kV onwards with a
maximum noise rate of 120 Hz/cm2 at - 20 mV threshold.

##  5 Acknowlegement

The authors would like to thank Ms. Rudrapriya Das for her help in running the
experiment and Prof. Sanjay K Ghosh for many valuable discussions and
suggestions during the course of study. The authors would also like to thank
Mr. Subrata Das for building the readout strips. This work is partially
supported by the CBM-MuCh project from BI-IFCC, DST, Govt. of India. A. Sen
acknowledges his Inspire Fellowship research grant [DST/INSPIRE
Fellowship/2018/IF180361].

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