Machine learning with sub-diffraction resolution in the photon-counting regime

The resolution of optical imaging is classically limited by the width of the point-spread function, which in turn is determined by the Rayleigh length. Recently, spatial-mode demultiplexing (SPADE) has been proposed as a method to achieve sub-Rayleigh estimation and discrimination of natural, incoherent sources. Here we show that SPADE yields sub-diffraction resolution in the broader context of image classification. To achieve this goal, we outline a hybrid machine learning algorithm for image classification that includes a physical part and a computational part. The physical part implements a physical pre-processing of the optical field that cannot be simulated without essentially reducing the signal-to-noise ratio. In detail, a spatial-mode demultiplexer is used to sort the transverse field, followed by mode-wise photon detection. In the computational part, the collected data are fed into an artificial neural network for training and classification. As a case study, we classify images from the MNIST dataset after severe blurring due to diffraction. Our numerical experiments demonstrate the ability to classify highly blurred images that would be otherwise indistinguishable by direct imaging without the physical pre-processing of the optical field.