A novel data-driven reconstruction algorithm for quantum image sensors is proposed. Binary observations are efficiently decoded by modeling the reconstruction structure as a two-layer neural network, where optimal coefficients are obtained via error backpropagation. Such a model encapsulates the structure of state-of-the-art algorithms, yet it presents a considerably faster alternative which adapts to input examples without a priori statistical information. Simulations on natural and synthetic datasets show accurate reconstructions with structural similarities consistent with the state of the art, while requiring approximately 5 times less computational cost.