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Hansen et al. Microstructures 2023;3:2023029 https://dx.doi.org/10.20517/microstructures.2023.17 Page 5 of 17

a value of 1 indicates that the two compared images are identical.

[33]
In the third crystallographic variant mapping method, the k-means method was used to cluster the
diffraction patterns into different groups, with each group representing one crystallographic variant. The
algorithm uses the VGG16 Keras model (a pre-trained deep learning model that consists of 16 convolutional
layers and three fully connected layers used for feature extraction) to reduce all diffraction patterns in a
[34]
dataset into feature vectors . The feature vectors capture key characteristics of each diffraction pattern,
including structure and orientation. Next, the feature vectors are reduced from 4,096 to 100 components,
each using kernel principal component analysis (kernel PCA) to lower the amount of data processed in the
next step while retaining key information. Kernel PCA is a dimensionality reduction technique that uses
kernel methods to transform nonlinear data . These vectors are input into a k-means++ algorithm (an
[35]
[36]
extension of conventional k-means but with much better with centroid initialization) . The algorithm then
sorts the vectors into k clusters, which are mapped to create a similarity map. For each k-value, the inertia
(the sum of the squared distances of samples to their closest cluster centroid) is also calculated and shown in
an “elbow graph” that can be used to manually determine an optimal k-value.

RESULTS AND DISCUSSION
Conventional PED characterization of the model systems
Figure 1 shows the virtual bright-field (VBF), index, reliability, and orientation maps of the SMA and VO
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on sapphire, which are the model systems of this study. The VBF images are formed using the intensity of
the direct beam in diffraction patterns and are similar to the conventional bright-field TEM micrographs
but with less dynamical effect due to the beam precession. The index map is similar to the band contrast
map in EBSD, in which brighter pixels indicate better matches between the experimentally acquired
diffraction pattern and the simulated pattern. The reliability map is obtained by calculating the ratio of
similarities between the best and second-best matches obtained through template matching. Brighter pixels
in the reliability map indicate a higher level of confidence and reduced ambiguity between the best and
second-best matches in orientation indexing. The concept of “reliability” will serve as a metric to assess the
quality of our crystallographic mapping in subsequent sections of this work.

The SMA sample (B19’, monoclinic crystal structure) shows a typical martensitic microstructure where the
martensite grains are plate-like [Figure 1A]. The martensite plate thickness varies from tens to hundreds of
nanometers. The corresponding index, reliability, and orientation maps [Figure 1B-D] show that multiple
martensite variants exist in the sample, but the result is noisy. A VO (monoclinic crystal structure at room
2
temperature) thin film grows epitaxially on c-cut sapphire , which exhibits three crystallographic
[37]
[38]
variants . The VBF in Figure 1E shows that the VO film is approximately 50-100 nm thick. The
2
corresponding index, reliability, and orientation maps [Figure 1F-H] are extremely noisy, and the variants
of VO cannot be identified at all. The poor indexing of the martensitic SMA and VO may be caused by the
2
2
low symmetry nature of their monoclinic crystal structure, where the spacing and angles of different lattice
planes are close to each other, leading to confusion when the software does the diffraction pattern indexing.
Consequently, there is a demand for new methods to better illustrate the distribution of crystallographic
variants in this wide group of materials.

Method 1: user-selecting-reference-pattern approach
In this method, the user will first peruse the dataset to identify how many crystallographic variants are
present in the dataset and then select a reference pattern from each variant. We observed no apparent
variation in the diffraction patterns among pixels within each crystallographic variant. This allows high
flexibility in selecting the reference pattern for each variant by the user. Next, the diffraction pattern from