Deep Expectation-Maximization For Image Reconstruction From Under-Sampled Poisson Data

Many biomedical imaging techniques, such as computerized tomography, positron emission tomography, and optical microscopy, involve reconstruction of an image from a sequence of a few linear measurements that are corrupted by Poisson noise. In this study, we focus on computational optics, and more precisely single-pixel imaging, where the set-up acquires some of the coefficients of the Hadamard transform of the image of the scene.Recently, this problem has benefited from the advent of deep learning. Although deep methods were initially considered as black boxes, they are now understood as learnable optimisation schemes. Here, we propose a network architecture based on the expectation-maximization algorithm that optimizes the maximum a posteriori of the unknown image for measurements corrupted by Poisson noise. This leads to an interpretable network that generalizes several existing approaches.Finally, we present some reconstruction results from simulated data and from experimental acquisitions from a single-pixel camera. Our network yields higher reconstruction peak signal-to-noise ratios than other similar approaches.