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

## TL;DR

This paper introduces a convex variational model for learning convolutional image atoms from incomplete or corrupted data, enabling simultaneous image reconstruction and atom learning with proven stability and numerical efficiency.

## Contribution

It presents a novel convex variational approach for joint image reconstruction and atom learning, including a semi-convex variant for improved numerical performance.

## Key findings

- Proven well-posedness and stability in inverse problems.
- Numerical computation of globally optimal solutions.
- Effective handling of incomplete, noisy, and blurry data.

## 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.