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), autofocusing and phase recovery are needed, which are in general cumbersome and time-consuming to perform digitally. Here we demonstrate a convolutional neural network (CNN)-based approach that simultaneously performs autofocusing and phase recovery to significantly extend the depth of field (DOF) and the reconstruction speed in holographic imaging. For this, a CNN is trained by using pairs of randomly defocused 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.

Recent advances in deep learning have given rise to a new paradigm of holographic image reconstruction and phase recovery techniques with real-time performance. Through data-driven approaches, these emerging techniques have overcome some of the challenges associated with existing holographic image reconstruction methods while also minimizing the hardware requirements of holography. These recent advances open up a myriad of new opportunities for the use of coherent imaging systems in biomedical and engineering research and related applications.

Autofocusing is essential to digital holographic imaging. Previously used autofocusing criteria exhibit challenges when applied to, e.g., connected objects with different optical properties. Furthermore, in some of the earlier autofocusing criteria, the polarity, i.e., whether to search for the peak or the valley as a function of depth, changes for different types of samples, which creates another challenge. Here, we propose a robust and accurate autofocusing criterion that is based on the edge sparsity of the complex optical wavefront, which we termed the “sparsity of the gradient” (SoG). We demonstrated the success of SoG by imaging a wide range of objects, including resolution test targets, stained and unstained Papanicolaou smears, stained tissue sections, and blood smears.