  Applying standard algorithms to sparse data problems in photoacoustic
tomography (PAT) yields low-quality images containing severe under-sampling
artifacts. To some extent, these artifacts can be reduced by iterative image
reconstruction algorithms which allow to include prior knowledge such as
smoothness, total variation (TV) or sparsity constraints. These algorithms tend
to be time consuming as the forward and adjoint problems have to be solved
repeatedly. Further, iterative algorithms have additional drawbacks. For
example, the reconstruction quality strongly depends on a-priori model
assumptions about the objects to be recovered, which are often not strictly
satisfied in practical applications. To overcome these issues, in this paper,
we develop direct and efficient reconstruction algorithms based on deep
learning. As opposed to iterative algorithms, we apply a convolutional neural
network, whose parameters are trained before the reconstruction process based
on a set of training data. For actual image reconstruction, a single evaluation
of the trained network yields the desired result. Our presented numerical
results (using two different network architectures) demonstrate that the
proposed deep learning approach reconstructs images with a quality comparable
to state of the art iterative reconstruction methods.
