  We develop a data-driven regularization method for the severely ill-posed
problem of photoacoustic image reconstruction from limited view data. Our
approach is based on the regularizing networks that have been recently
introduced and analyzed in [J. Schwab, S. Antholzer, and M. Haltmeier. Big in
Japan: Regularizing networks for solving inverse problems (2018),
arXiv:1812.00965] and consists of two steps. In the first step, an intermediate
reconstruction is performed by applying truncated singular value decomposition
(SVD). In order to prevent noise amplification, only coefficients corresponding
to sufficiently large singular values are used, whereas the remaining
coefficients are set zero. In a second step, a trained deep neural network is
applied to recover the truncated SVD coefficients. Numerical results are
presented demonstrating that the proposed data driven estimation of the
truncated singular values significantly improves the pure truncated SVD
reconstruction. We point out that proposed reconstruction framework can
straightforwardly be applied to other inverse problems, where the SVD is either
known analytically or can be computed numerically.
