Total Deep Variation: A Stable Regularization Method for Inverse Problems

Erich Kobler, Alexander Effland, Karl Kunisch, Thomas Pock

Research output: Contribution to journalArticlepeer-review


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 several imaging tasks.

Original languageEnglish
Pages (from-to)9163 - 9180
Number of pages18
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Issue number12
Publication statusPublished - 2022


  • Convolutional neural networks
  • gradient flow
  • image restoration
  • Inverse problems
  • inverse problems
  • mean-field optimal control problem
  • medical imaging
  • Noise reduction
  • Optimal control
  • Stability analysis
  • Task analysis
  • Training
  • Trajectory
  • variational methods

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition
  • Computational Theory and Mathematics
  • Artificial Intelligence
  • Applied Mathematics


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