  Holography encodes the three dimensional (3D) information of a sample in the
form of an intensity-only recording. However, to decode the original sample
image from its hologram(s), auto-focusing and phase-recovery are needed, which
are in general cumbersome and time-consuming to digitally perform. Here we
demonstrate a convolutional neural network (CNN) based approach that
simultaneously performs auto-focusing and phase-recovery to significantly
extend the depth-of-field (DOF) in holographic image reconstruction. For this,
a CNN is trained by using pairs of randomly de-focused back-propagated
holograms and their corresponding in-focus phase-recovered images. After this
training phase, the CNN takes a single back-propagated hologram of a 3D sample
as input to rapidly achieve phase-recovery and reconstruct an in focus image of
the sample over a significantly extended DOF. This deep learning based DOF
extension method is non-iterative, and significantly improves the algorithm
time-complexity of holographic image reconstruction from O(nm) to O(1), where n
refers to the number of individual object points or particles within the sample
volume, and m represents the focusing search space within which each object
point or particle needs to be individually focused. These results highlight
some of the unique opportunities created by data-enabled statistical image
reconstruction methods powered by machine learning, and we believe that the
presented approach can be broadly applicable to computationally extend the DOF
of other imaging modalities.

Computer Vision and Pattern Recognition;Machine Learning;Optics