DEEP EXPECTATION-MAXIMIZATION FOR IMAGE RECONSTRUCTION FROMUNDER-SAMPLED POISSON DATA

International audienceWe consider the reconstruction of an image from a sequence of a few linear measurements corrupted by Poisson noise. This generic problem has many biomedical applications, such as computerized tomography, positron emission tomogra-phy, and optical microscopy. Here, we focus on a computational optics problem where the setup acquires some coefficients of the Hadamard transform of the image of the scene. We formalize this problem in a Bayesian setting where we estimate the missing Hadamard coefficients from those acquired. Then, we propose a deep-learning network that consists of two fully connected layers (FCLs) that map data from the measurement domain to the image domain, followed by convolutional layers that act in the image domain. On the one hand, we set the FCLs so that they compute the best linear solution of the problem. While the first FCL denoises the raw measurements, the second FCL completes the missing measurements from the denoised measurements. The convolutional layers undergo learning through a training phase. We also describe a framework for training the network in the presence of Poisson noise. In particular, our approach includes an estimation of the image intensity, together with a normalization scheme that allows varying noise levels to be handled during training. We compare our network to linear reconstructors and to network variants that do not address the noise issue at all, or that address it implicitly. Finally, we present results from simulated and experimental acquisitions, considering varying noise levels. Our network yields higher reconstruction peak signal-to-noise ratios in scenarios where the actual noise level is higher than that expected and used during the training phase