  One of the outstanding analytical problems in X-ray single particle imaging
(SPI) is the classification of structural heterogeneity, which is especially
difficult given the low signal-to-noise ratios of individual patterns and that
even identical objects can yield patterns that vary greatly when orientation is
taken into consideration. We propose two methods which explicitly account for
this orientation-induced variation and can robustly determine the structural
landscape of a sample ensemble. The first, termed common-line principal
component analysis (PCA) provides a rough classification which is essentially
parameter-free and can be run automatically on any SPI dataset. The second
method, utilizing variation auto-encoders (VAEs) can generate 3D structures of
the objects at any point in the structural landscape. We implement both these
methods in combination with the noise-tolerant expand-maximize-compress (EMC)
algorithm and demonstrate its utility by applying it to an experimental dataset
from gold nanoparticles with only a few thousand photons per pattern and
recover both discrete structural classes as well as continuous deformations.
These developments diverge from previous approaches of extracting reproducible
subsets of patterns from a dataset and open up the possibility to move beyond
studying homogeneous sample sets and study open questions on topics such as
nanocrystal growth and dynamics as well as phase transitions which have not
been externally triggered.
