  We propose a flexible algorithm for feature detection and hypothesis testing
in images with ultra low signal-to-noise ratio using cubical persistent
homology. Our main application is in the identification of atomic columns and
other features in transmission electron microscopy (TEM). Cubical persistent
homology is used to identify local minima and their size in subregions in the
frames of nanoparticle videos, which are hypothesized to correspond to relevant
atomic features. We compare the performance of our algorithm to other employed
methods for the detection of columns and their intensity. Additionally, Monte
Carlo goodness-of-fit testing using real valued summaries of persistence
diagrams derived from smoothed images (generated from pixels residing in the
vacuum region of an image) is developed and employed to identify whether or not
the proposed atomic features generated by our algorithm are due to noise. Using
these summaries derived from the generated persistence diagrams, one can
produce univariate time series for the nanoparticle videos, thus providing a
means for assessing fluxional behavior. A guarantee on the false discovery rate
for multiple Monte Carlo testing of identical hypotheses is also established.
