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

arXiv: https://arxiv.org/abs/1812.03077 · HTML: https://mlsanity.com/paper/1812.03077/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2018.

## Built on by (most cited first, 3 of 3)

- [Convergent regularization in inverse problems and linear plug-and-play denoisers](https://mlsanity.com/paper/2307.09441/) (2023)
- [A Note on the Regularity of Images Generated by Convolutional Neural Networks](https://mlsanity.com/paper/2204.10588/) (2022)
- [Deep learning based dictionary learning and tomographic image reconstruction](https://mlsanity.com/paper/2108.11730/) (2021)

## Abstract (arXiv)

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.