  The trade-off between throughput and image quality is an inherent challenge
in microscopy. To improve throughput, compressive imaging under-samples image
signals; the images are then computationally reconstructed by solving a
regularized inverse problem. Compared to traditional regularizers, Deep
Learning based methods have achieved greater success in compression and image
quality. However, the information loss in the acquisition process sets the
compression bounds. Further improvement in compression, without compromising
the reconstruction quality is thus a challenge. In this work, we propose
differentiable compressive fluorescence microscopy ($\partial \mu$) which
includes a realistic generalizable forward model with learnable-physical
parameters (e.g. illumination patterns), and a novel physics-inspired inverse
model. The cascaded model is end-to-end differentiable and can learn optimal
compressive sampling schemes through training data. With our model, we
performed thousands of numerical experiments on various compressive microscope
configurations. We show that learned sampling encodes important information
about the specimens in the illumination field of the microscope allowing higher
compression up to $\times 1024$. We further utilize our framework for Task
Aware Compression. The experimental results show superior performance on the
cell segmentation task.
