  Automated cell detection and localization from microscopy images are
significant tasks in biomedical research and clinical practice. In this paper,
we design a new cell detection and localization algorithm that combines deep
convolutional neural network (CNN) and compressed sensing (CS) or sparse coding
(SC) for end-to-end training. We also derive, for the first time, a
backpropagation rule, which is applicable to train any algorithm that
implements a sparse code recovery layer. The key observation behind our
algorithm is that cell detection task is a point object detection task in
computer vision, where the cell centers (i.e., point objects) occupy only a
tiny fraction of the total number of pixels in an image. Thus, we can apply
compressed sensing (or, equivalently sparse coding) to compactly represent a
variable number of cells in a projected space. Then, CNN regresses this
compressed vector from the input microscopy image. Thanks to the SC/CS recovery
algorithm (L1 optimization) that can recover sparse cell locations from the
output of CNN. We train this entire processing pipeline end-to-end and
demonstrate that end-to-end training provides accuracy improvements over a
training paradigm that treats CNN and CS-recovery layers separately. Our
algorithm design also takes into account a form of ensemble average of trained
models naturally to further boost accuracy of cell detection. We have validated
our algorithm on benchmark datasets and achieved excellent performances.
