  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.
