  Fourier ptychography is a recently developed imaging approach for large
field-of-view and high-resolution microscopy. Here we model the Fourier
ptychographic forward imaging process using a convolution neural network (CNN)
and recover the complex object information in the network training process. In
this approach, the input of the network is the point spread function in the
spatial domain or the coherent transfer function in the Fourier domain. The
object is treated as 2D learnable weights of a convolution or a multiplication
layer. The output of the network is modeled as the loss function we aim to
minimize. The batch size of the network corresponds to the number of captured
low-resolution images in one forward / backward pass. We use a popular
open-source machine learning library, TensorFlow, for setting up the network
and conducting the optimization process. We analyze the performance of
different learning rates, different solvers, and different batch sizes. It is
shown that a large batch size with the Adam optimizer achieves the best
performance in general. To accelerate the phase retrieval process, we also
discuss a strategy to implement Fourier-magnitude projection using a
multiplication neural network model. Since convolution and multiplication are
the two most-common operations in imaging modeling, the reported approach may
provide a new perspective to examine many coherent and incoherent systems. As a
demonstration, we discuss the extensions of the reported networks for modeling
single-pixel imaging and structured illumination microscopy (SIM). 4-frame
resolution doubling is demonstrated using a neural network for SIM. We have
made our implementation code open-source for the broad research community.
