Clustering directions and orientations

A cluster is a group of observations that lie close together in the space of the data. Clustering assigns each observation to a group without requiring the groups to be named in advance.

MTEX can cluster directions and orientations while respecting their geometry and symmetry. This is different from density estimation, which constructs a continuous density from discrete observations. Here the result is one integer label per observation and one representative center per cluster.

Cluster directional data

We start with directions sampled from a squared smiley function. Squaring the function concentrates the sample around its eyes and mouth while retaining points between those features.

plottingConvention.default('y↑→x');
rng default
S2F = S2Fun.smiley.^2;
v = S2F.discreteSample(10000);

scatter(v,'MarkerAlpha',0.2,'MarkerSize',2)

Read the cluster result

For directions, calcCluster uses CLASSIX by default. The vector cInd has one integer cluster label for every input direction. The entries of center are the corresponding mean directions.

Cluster labels are identifiers, not ranks: cluster 1 is not intrinsically more important than cluster 2. We use the labels only to assign colors and then mark each returned center.

[cInd,center] = calcCluster(v);

plot(v,ind2color(cInd),'MarkerAlpha',0.2,'MarkerSize',2)
annotate(center)

Tune the grouping

The colored regions follow the separated strokes of the smiley, while the centers sit inside the densest parts of those regions. CLASSIX first forms local groups and then merges nearby groups into clusters.

Two options control how readily this happens. Increasing 'radius' makes the local groups coarser and usually produces fewer clusters. The 'minPoints' option sets the size below which tiny groups are merged into neighboring clusters. If the automatic result is too fragmented, increase 'radius'; if isolated specks survive, increase 'minPoints'. This example supplies both values so that estimating the radius does not require the Statistics and Machine Learning Toolbox.

Respect antipodal symmetry

Antipodal directions identify a direction v with its opposite -v. Setting the antipodal property tells MTEX to measure proximity with this equivalence in mind.

The smiley sample on its own cannot show this. No feature of it lies opposite another, so identifying v with -v has nothing to merge. Add the opposite of every sampled direction, and each feature gains one.

vSym = [v(:); -v(:)];

[cInd,center] = calcCluster(vSym);

plot(vSym,ind2color(cInd))
annotate(center)

The two discs are the upper and the lower hemisphere. Read as ordinary directions, the sample falls into eight clusters. Every feature of the smiley and every one of their opposites carries a color of its own.

vSym.antipodal = true;
[cInd,center] = calcCluster(vSym);

plot(vSym,ind2color(cInd),'MarkerAlpha',0.2,'MarkerSize',2)
annotate(center)

One disc is left, and four clusters, each exactly twice the size of before. Every feature has been merged with the one opposite it. The two eyes stay apart, because one eye is not the opposite of the other.

The blue cluster reaches the rim of the disc at the top and at the bottom. Those are opposite directions, and they now share a label. This is a change in the meaning of a direction, not merely a change in how the same labels are drawn.

Build an orientation example

The same idea applies in orientation space. We construct an orientation distribution function (ODF) with a broad fibre component and a compact unimodal component. Kernel selection explains how the halfwidth controls smoothing when an ODF is estimated from observations.

cs = crystalSymmetry('432');
odf = 0.7*fibreODF(fibre.gamma(cs),'halfwidth',10*degree) + ...
  0.3*unimodalODF(orientation.byEuler(30*degree,10*degree,60*degree,cs));

ori = odf.discreteSample(10000);

Compare the model with its sample

Filled contours show sections through the model ODF. The points are the sampled orientations drawn over every section.

plotSection(odf,'contourf')
mtexColorMap white2black
plot(ori,'add2all','MarkerSize',3,'MarkerAlpha',0.25,'all')

Notice the narrow concentration of points around the compact peak and the elongated concentration along the fibre. These are the two structures the clustering step should distinguish.

Cluster the orientations

Orientation clustering does not use CLASSIX unless it is selected explicitly. The embedding used by this method accounts for crystal symmetry, so symmetrically equivalent representations of an orientation receive the same cluster label.

[cId,center] = calcCluster(ori,'method','classix');

plotSection(ori,ind2color(cId),'markerSize',3, ...
  'MarkerAlpha',0.25,'all')
annotate(center)

The dominant color follows the fibre and another color collects the compact unimodal portion. The annotated centers summarize the returned groups; they do not replace the full within-cluster spread visible here.

References