A convex variational model for learning convolutional image atoms from incomplete data

Antonin Chambolle, Thomas Pock, Martin Holler*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A variational model for learning convolutional image atoms from corrupted and/or incomplete data is introduced and analyzed both in function space and numerically. Building on lifting and relaxation strategies, the proposed approach is convex and allows for simultaneous image reconstruction and atom learning in a general, inverse problems context. Further, motivated by an improved numerical performance, also a semi-convex variant is included in the analysis and the experiments of the paper. For both settings, fundamental analytical properties allowing in particular to ensure well-posedness and stability results for inverse problems are proven in a continuous setting. Exploiting convexity, globally optimal solutions are further computed numerically for applications with incomplete, noisy and blurry data and numerical results are shown.
Original languageEnglish
Pages (from-to)417-444
Number of pages28
JournalJournal of Mathematical Imaging and Vision
Issue number3
Publication statusPublished - 1 Apr 2020


  • Convex relaxation
  • Convolutional Lasso
  • Functional lifting
  • Inverse problems
  • Learning approaches
  • Machine learning
  • Texture reconstruction
  • Variational methods

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Applied Mathematics
  • Geometry and Topology
  • Computer Vision and Pattern Recognition
  • Statistics and Probability
  • Modelling and Simulation

Cite this