  Various problems in computer vision and medical imaging can be cast as
inverse problems. A frequent method for solving inverse problems is the
variational approach, which amounts to minimizing an energy composed of a data
fidelity term and a regularizer. Classically, handcrafted regularizers are
used, which are commonly outperformed by state-of-the-art deep learning
approaches. In this work, we combine the variational formulation of inverse
problems with deep learning by introducing the data-driven general-purpose
total deep variation regularizer. In its core, a convolutional neural network
extracts local features on multiple scales and in successive blocks. This
combination allows for a rigorous mathematical analysis including an optimal
control formulation of the training problem in a mean-field setting and a
stability analysis with respect to the initial values and the parameters of the
regularizer. In addition, we experimentally verify the robustness against
adversarial attacks and numerically derive upper bounds for the generalization
error. Finally, we achieve state-of-the-art results for numerous imaging tasks.
