  The development of fast and accurate image reconstruction algorithms is a
central aspect of computed tomography. In this paper, we investigate this issue
for the sparse data problem in photoacoustic tomography (PAT). We develop a
direct and highly efficient reconstruction algorithm based on deep learning. In
our approach image reconstruction is performed with a deep convolutional neural
network (CNN), whose weights are adjusted prior to the actual image
reconstruction based on a set of training data. The proposed reconstruction
approach can be interpreted as a network that uses the PAT filtered
backprojection algorithm for the first layer, followed by the U-net
architecture for the remaining layers. Actual image reconstruction with deep
learning consists in one evaluation of the trained CNN, which does not require
time consuming solution of the forward and adjoint problems. At the same time,
our numerical results demonstrate that the proposed deep learning approach
reconstructs images with a quality comparable to state of the art iterative
approaches for PAT from sparse data.
