A Convex Variational Model for Learning Convolutional Image Atoms from Incomplete Data

A Chambolle; M Holler; T Pock

doi:10.1007/s10851-019-00919-7

A Convex Variational Model for Learning Convolutional Image Atoms from Incomplete Data

J Math Imaging Vis. 2020;62(3):417-444. doi: 10.1007/s10851-019-00919-7. Epub 2019 Nov 18.

Authors

A Chambolle¹, M Holler², T Pock³

Affiliations

¹ 1Centre de Mathématiques Appliquées, École Polytechnique, Paris, France.
² 2Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria.
³ 3Institute of Computer Graphics and Vision, Graz University of Technology, Graz, Austria.

Abstract

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.

Keywords: Convex relaxation; Convolutional Lasso; Functional lifting; Inverse problems; Learning approaches; Machine learning; Texture reconstruction; Variational methods.