  Phase separation mechanisms can produce a variety of complicated and
intricate microstructures, which often can be difficult to characterize in a
quantitative way. In recent years, a number of novel topological metrics for
microstructures have been proposed, which measure essential connectivity
information and are based on techniques from algebraic topology. Such metrics
are inherently computable using computational homology, provided the
microstructures are discretized using a thresholding process. However, while in
many cases the thresholding is straightforward, noise and measurement errors
can lead to misleading metric values. In such situations, persistence
landscapes have been proposed as a natural topology metric. Common to all of
these approaches is the enormous data reduction, which passes from complicated
patterns to discrete information. It is therefore natural to wonder what type
of information is actually retained by the topology. In the present paper, we
demonstrate that averaged persistence landscapes can be used to recover central
system information in the Cahn-Hilliard theory of phase separation. More
precisely, we show that topological information of evolving microstructures
alone suffices to accurately detect both concentration information and the
actual decomposition stage of a data snapshot. Considering that persistent
homology only measures discrete connectivity information, regardless of the
size of the topological features, these results indicate that the system
parameters in a phase separation process affect the topology considerably more
than anticipated. We believe that the methods discussed in this paper could
provide a valuable tool for relating experimental data to model simulations.
