  Orientation mapping is a widely used technique for revealing the
microstructure of a polycrystalline sample. The crystalline orientation at each
point in the sample is determined by analysis of the diffraction pattern, a
process known as pattern indexing. A recent development in pattern indexing is
the use of a brute-force approach, whereby diffraction patterns are simulated
for a large number of crystalline orientations, and compared against the
experimentally observed diffraction pattern in order to determine the most
likely orientation. Whilst this method can robust identify orientations in the
presence of noise, it has very high computational requirements. In this
article, the computational burden is reduced by developing a method for
nearly-optimal sampling of orientations. By using the quaternion representation
of orientations, it is shown that the optimal sampling problem is equivalent to
that of optimally distributing points on a four-dimensional sphere. In doing
so, the number of orientation samples needed to achieve a indexing desired
accuracy is significantly reduced. Orientation sets at a range of sizes are
generated in this way for all Laue groups, and are made available online for
easy use.
