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.

中文翻译：

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用于从不完整数据中学习卷积图像原子的凸变分模型。


介绍了一种从损坏和/或不完整的数据中学习卷积图像原子的变分模型，并在函数空间和数值上进行了分析。基于提升和松弛策略，所提出的方法是凸的，并且允许在一般的逆问题环境中同时进行图像重建和原子学习。此外，在改进数值性能的推动下，本文的分析和实验中还包含了半凸变体。对于这两种设置，基本分析属性特别允许确保逆问题的适定性和稳定性结果在连续设置中得到证明。利用凸性，针对具有不完整、噪声和模糊数据的应用进一步以数值方式计算全局最优解，并显示数值结果。