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Clustering misorientations#
In this tutorial we will cluster Ti crystal misorientations using data obtained from a highly deformed specimen, using EBSD, as presented in [Johnstone et al., 2020]. The data can be downloaded to your local cache via the orix.data module.
Import orix classes and various dependencies
[1]:
# Exchange "inline" for "notebook" (or "qt5" from pyqt) for interactive plotting
%matplotlib inline
from matplotlib.colors import to_rgb
from matplotlib.lines import Line2D
import matplotlib.pyplot as plt
import numpy as np
from skimage.color import label2rgb
from sklearn.cluster import DBSCAN
from orix.plot import register_projections, IPFColorKeyTSL
from orix import data
from orix.quaternion import Misorientation, Rotation
from orix.vector import Vector3d
plt.rcParams.update({"font.size": 20, "figure.figsize": (10, 10)})
register_projections()
Import data#
Load Ti orientations with the point group symmetry 622 (D6). We have to explicitly allow downloading from an external source.
[2]:
ori = data.ti_orientations(allow_download=True)
ori
/home/docs/checkouts/readthedocs.org/user_builds/orix/conda/latest/lib/python3.14/site-packages/tqdm/auto.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html
from .autonotebook import tqdm as notebook_tqdm
[2]:
Orientation (193167,) 622
[[ 0.3027 0.0869 -0.5083 0.8015]
[ 0.3088 0.0868 -0.5016 0.8034]
[ 0.3057 0.0818 -0.4995 0.8065]
...
[ 0.4925 -0.1633 -0.668 0.5334]
[ 0.4946 -0.1592 -0.6696 0.5307]
[ 0.4946 -0.1592 -0.6696 0.5307]]
Extract the symmetry
[3]:
sym = ori.symmetry
Reshape the orientation mapping data to the correct spatial dimension for the scan and select a subset of the orientations with a suitable size for this demonstration (bottom left corner)
[4]:
ori = ori.reshape(381, 507)
ori = ori[-100:, :200]
Plot orientation maps
[5]:
ckey = IPFColorKeyTSL(sym)
directions = [(1, 0, 0), (0, 1, 0)]
titles = ["X", "Y"]
fig, axes = plt.subplots(ncols=2, figsize=(15, 10))
for i, ax in enumerate(axes):
ckey.direction = Vector3d(directions[i])
ax.imshow(ckey.orientation2color(ori))
ax.set_title(f"IPF-{titles[i]}")
ax.axis("off")
# Add color key
ax_ipfkey = fig.add_axes(
[0.932, 0.37, 0.1, 0.1], # (Left, bottom, width, height)
projection="ipf",
symmetry=ori.symmetry.laue,
)
ax_ipfkey.plot_ipf_color_key()
ax_ipfkey.set_title("")
fig.subplots_adjust(wspace=0.01)
Map the orientations into the Rodrigues fundamental zone (find symmetrically equivalent orientations with the smallest angle of rotation) of 622
[6]:
ori = ori.reduce()
Compute misorientations \(m_{AB} = g_B \cdot g_A^{-1}\) (in the horizontal direction)
[7]:
mori_all = Misorientation(ori[:, :-1] * ~ori[:, 1:], symmetry=(sym, sym))
Keep only misorientations with a disorientation angle higher than 7\(^{\circ}\), assumed to represent grain boundaries
[8]:
boundary_mask = mori_all.angle > np.deg2rad(7)
mori = mori_all[boundary_mask]
Map the misorientations into the fundamental zone of 622-622
[9]:
mori = mori.reduce()
Compute distance matrix#
[10]:
D = mori.get_distance_matrix(lazy=False)
Clustering#
Apply mask to remove small misorientations associated with grain orientation spread
[11]:
small_mask = mori.angle < np.deg2rad(7)
D = D[~small_mask][:, ~small_mask]
mori = mori[~small_mask]
For parameter explanations of the DBSCAN algorithm (Density-Based Spatial Clustering for Applications with Noise), see the scikit-learn documentation.
[12]:
# Compute clusters
D = D.astype("float32")
dbscan = DBSCAN(eps=0.05, min_samples=10, metric="precomputed").fit(D)
unique_labels, all_cluster_sizes = np.unique(
dbscan.labels_, return_counts=True
)
print("Labels:", unique_labels)
n_clusters = unique_labels.size - 1
print("Number of clusters:", n_clusters)
Labels: [-1 0 1 2 3]
Number of clusters: 4
Calculate the mean misorientation associated with each cluster
[13]:
unique_cluster_labels = unique_labels[1:] # Drop "no-cluster" label
cluster_sizes = all_cluster_sizes[1:]
rc = Rotation.from_axes_angles((0, 0, 1), 15, degrees=True)
mori_mean = []
for label in unique_cluster_labels:
# Rotate
mori_i = rc * mori[dbscan.labels_ == label]
# Map into the fundamental zone
mori_i.symmetry = (sym, sym)
mori_i = mori_i.reduce()
# Get the cluster mean
mori_i = mori_i.mean()
# Rotate back and add to list
cluster_mean_local = ~rc * mori_i
mori_mean.append(cluster_mean_local)
cluster_means = Misorientation.stack(mori_mean).flatten()
# Map into the fundamental zone
cluster_means.symmetry = (sym, sym)
cluster_means = cluster_means.reduce()
cluster_means
[13]:
Misorientation (4,) 622, 622
[[ 0.8467 0.2664 0.4606 0.0037]
[-0.7858 -0.3094 -0.5355 -0.0044]
[-0.9515 -0.3075 -0.0015 -0.007 ]
[ 0.8656 0.4338 0.2501 0.0055]]
Inspect misorientations in the axis-angle representation
[14]:
cluster_means.axis
[14]:
Vector3d (4,)
[[0.5006 0.8656 0.007 ]
[0.5003 0.8658 0.0072]
[0.9997 0.0048 0.0228]
[0.8663 0.4994 0.0109]]
[15]:
np.rad2deg(cluster_means.angle)
[15]:
array([64.29349881, 76.40971082, 35.82903611, 60.10115054])
Define reference misorientations associated with twinning orientation relationships
[16]:
# From Krakow et al.
twin_theory = Rotation.from_axes_angles(
axes=[
(1, 0, 0), # sigma7a
(1, 0, 0), # sigma11a
(2, 1, 0), # sigma11b
(1, 0, 0), # sigma13a
(2, 1, 0), # sigma13b
],
angles=[64.40, 34.96, 85.03, 76.89, 57.22],
degrees=True,
)
Calculate difference, defined as minimum rotation angle, between measured and theoretical values. This procedure accounts for the edges of the fundamental zone.
[17]:
mori2 = twin_theory.outer(cluster_means)
sym_ops = sym.outer(sym).unique()
mori2_equiv = (
sym.outer(twin_theory).outer(sym_ops).outer(cluster_means).outer(sym)
)
D2 = mori2_equiv.angle.min(axis=(0, 2, 4))
[18]:
np.rad2deg(D2)
[18]:
array([[ 0.44196598, 12.01907556, 28.59166542, 31.00602591],
[29.33528603, 41.45127188, 1.18969475, 34.15185331],
[37.89471722, 35.23189781, 54.83306878, 25.25351347],
[12.60503699, 0.70196626, 41.07796156, 37.24075935],
[27.53333357, 34.58980591, 29.35942936, 4.44471339]])
We see that the first, second, third, and fourth clusters are within \(4.5^{\circ}\) of \(\Sigma7\)a, \(\Sigma13\)a, \(\Sigma11\)a, and \(\Sigma13\)b, respectively.
Visualisation#
Associate colours with clusters for plotting
[19]:
Inspect misorientation axes of clusters in an inverse pole figure
[20]:
cluster_sizes = all_cluster_sizes[1:]
cluster_sizes_scaled = 1000 * cluster_sizes / cluster_sizes.max()
fig = plt.figure(figsize=(6, 6))
ax = fig.add_subplot(projection="ipf", symmetry=sym)
ax.scatter(
cluster_means.axis, c=colors, s=cluster_sizes_scaled, alpha=0.5, ec="k"
)
Plot a top view of the misorientation clusters within the fundamental zone for the (D6, D6) bicrystal symmetry
[21]:
wireframe_kwargs = dict(
color="black", linewidth=0.5, alpha=0.1, rcount=361, ccount=361
)
fig = mori.scatter(
projection="axangle",
wireframe_kwargs=wireframe_kwargs,
c=labels_rgb.reshape(-1, 3),
s=4,
alpha=0.5,
return_figure=True,
)
ax = fig.axes[0]
ax.view_init(elev=90, azim=-60)
handle_kwds = dict(marker="o", color="none", markersize=10)
handles = []
for i in range(n_clusters):
line = Line2D(
[0], [0], label=i + 1, markerfacecolor=colors[i], **handle_kwds
)
handles.append(line)
ax.legend(
handles=handles,
loc="upper left",
numpoints=1,
labelspacing=0.15,
columnspacing=0.15,
handletextpad=0.05,
);
Plot side view of misorientation clusters in the fundamental zone for the (D6, D6) bicrystal symmetry
[22]:
fig2 = mori.scatter(
return_figure=True,
wireframe_kwargs=wireframe_kwargs,
c=labels_rgb.reshape(-1, 3),
s=4,
)
ax2 = fig2.axes[0]
ax2.view_init(elev=0, azim=-60)
Generate map of boundaries colored according to cluster membership
[23]:
Plot map of boundaries colored according to cluster membership
[24]:
fig3, ax3 = plt.subplots(figsize=(15, 10))
ax3.imshow(mapping)
ax3.set(xticks=[], yticks=[]);
