  We investigate compressed sensing (CS) techniques for reducing the number of
measurements in photoacoustic tomography (PAT). High resolution imaging from CS
data requires particular image reconstruction algorithms. The most established
reconstruction techniques for that purpose use sparsity and
$\ell^1$-minimization. Recently, deep learning appeared as a new paradigm for
CS and other inverse problems. In this paper, we compare a recently invented
joint $\ell^1$-minimization algorithm with two deep learning methods, namely a
residual network and an approximate nullspace network. We present numerical
results showing that all developed techniques perform well for deterministic
sparse measurements as well as for random Bernoulli measurements. For the
deterministic sampling, deep learning shows more accurate results, whereas for
Bernoulli measurements the $\ell^1$-minimization algorithm performs best.
Comparing the implemented deep learning approaches, we show that the nullspace
network uniformly outperforms the residual network in terms of the mean squared
error (MSE).
