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A variational model for learning convolutional image atoms from corruptedand/or incomplete data is introduced and analyzed both in function space andnumerically. Building on lifting and relaxation strategies, the proposedapproach is convex and allows for simultaneous image reconstruction andatom-learning in a general, inverse problems context. Further, motivated by animproved numerical performance, also a semi-convex variant is included in theanalysis and the experiments of the paper. For both settings, fundamentalanalytical properties allowing in particular to ensure well-posedness andstability results for inverse problems are proven in a continuous setting.Exploiting convexity, globally optimal solutions are further computednumerically for applications with incomplete, noisy and blurry data andnumerical results are shown.

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