Learned imaging with constraints and uncertainty quantification

We outline new approaches to incorporate ideas from deep learning into wave-based least-squares imaging. The aim, and main contribution of this work, is the combination of handcrafted constraints with deep convolutional neural networks, as a way to harness their remarkable ease of generating natural images. The mathematical basis underlying our method is the expectation-maximization framework, where data are divided in batches and coupled to additional "latent" unknowns. These unknowns are pairs of elements from the original unknown space (but now coupled to a specific data batch) and network inputs. In this setting, the neural network controls the similarity between these additional parameters, acting as a "center" variable. The resulting problem amounts to a maximum-likelihood estimation of the network parameters when the augmented data model is marginalized over the latent variables.