Deep Expectation-Maximization for Single-Pixel Image Reconstruction With Signal-Dependent Noise

Image reconstruction from a sequence of a few linear measurements that are corrupted by signal-dependent normally distributed noise is an inverse problem with many biomedical imaging applications, such as computerized tomography and optical microscopy. In this study, we focus on single-pixel imaging, where the set-up acquires a down-sampled Hadamard transform of an image of the scene. Deep learning is a computationally efficient framework to solve inverse problems in imaging. Several neural-network architectures provide a link between deep and optimization-based image reconstruction methods. These deep-learning methods rely on a forward operator and lead to more interpretable networks. Here, we propose a novel network architecture obtained by unrolling an heuristic expectation-maximization algorithm. In particular, we compute the maximum a posteriori estimate of the unknown image given measurements corrupted by normally distributed signal-dependent noise. We show that the so-called expectation-maximization reconstruction network (EM-Net) applies to mixed Skellam-Gaussian noise models that are common in single-pixel imaging. We present reconstruction results from simulated and experimental single-pixel acquisitions. We show that EM-Net generalizes very well to noise levels not seen during training, despite having fewer learned parameters than alternative methods. The proposed EM-Net generally reconstructs images with fewer artifacts and higher signal-to-noise ratios, in particular in high-noise situations compared to other state of the art reconstruction algorithms that do not estimate the noise covariance.