  Deep neural network approaches to inverse imaging problems have produced
impressive results in the last few years. In this paper, we consider the use of
generative models in a variational regularisation approach to inverse problems.
The considered regularisers penalise images that are far from the range of a
generative model that has learned to produce images similar to a training
dataset. We name this family \textit{generative regularisers}. The success of
generative regularisers depends on the quality of the generative model and so
we propose a set of desired criteria to assess generative models and guide
future research. In our numerical experiments, we evaluate three common
generative models, autoencoders, variational autoencoders and generative
adversarial networks, against our desired criteria. We also test three
different generative regularisers on the inverse problems of deblurring,
deconvolution, and tomography. We show that restricting solutions of the
inverse problem to lie exactly in the range of a generative model can give good
results but that allowing small deviations from the range of the generator
produces more consistent results.
