  Determining the phase of a wave from intensity measurements has many
applications in fields such as electron microscopy, visible light optics, and
medical imaging. Propagation based phase retrieval, where the phase is obtained
from defocused images, has shown significant promise. There are, however,
limitations in the accuracy of the retrieved phase arising from such methods.
Sources of error include shot noise, image misalignment, and diffraction
artifacts. We explore the use of artificial neural networks (ANNs) to improve
the accuracy of propagation based phase retrieval algorithms applied to
simulated intensity measurements. We employ a phase retrieval algorithm based
on the transport-of-intensity equation to obtain the phase from simulated
micrographs of procedurally generated specimens. We then train an ANN with
pairs of retrieved and exact phases, and use the trained ANN to process a test
set of retrieved phase maps. The total error in the phase is significantly
reduced using this method. We also discuss a variety of potential extensions to
this work.
