  Phase retrieval is the inverse problem of recovering a signal from
magnitude-only Fourier measurements, and underlies numerous imaging modalities,
such as Coherent Diffraction Imaging (CDI). A variant of this setup, known as
holography, includes a reference object that is placed adjacent to the specimen
of interest before measurements are collected. The resulting inverse problem,
known as holographic phase retrieval, is well-known to have improved problem
conditioning relative to the original. This innovation, i.e. Holographic CDI,
becomes crucial at the nanoscale, where imaging specimens such as viruses,
proteins, and crystals require low-photon measurements. This data is highly
corrupted by Poisson shot noise, and often lacks low-frequency content as well.
In this work, we introduce a dataset-free deep learning framework for
holographic phase retrieval adapted to these challenges. The key ingredients of
our approach are the explicit and flexible incorporation of the physical
forward model into an automatic differentiation procedure, the Poisson
log-likelihood objective function, and an optional untrained deep image prior.
We perform extensive evaluation under realistic conditions. Compared to
competing classical methods, our method recovers signal from higher noise
levels and is more resilient to suboptimal reference design, as well as to
large missing regions of low frequencies in the observations. Finally, we show
that these properties carry over to experimental data acquired on optical
wavelengths. To the best of our knowledge, this is the first work to consider a
dataset-free machine learning approach for holographic phase retrieval.
