We consider a class of regularization methods for inverse problems where a coupled regularization is employed for the simultaneous reconstruction of data from multiple sources. Applications for such a setting can be found in multi-spectral or multimodality inverse problems, but also in inverse problems with dynamic data. We consider this setting in a rather general framework and derive stability and convergence results, including convergence rates. In particular, we show how parameter choice strategies adapted to the interplay of different data channels allow to improve upon convergence rates that would be obtained by treating all channels equally. Motivated by concrete applications, our results are obtained under rather general assumptions that allow to include the Kullback-Leibler divergence as data discrepancy term. To simplify their application to concrete settings, we further elaborate several practically relevant special cases in detail. To complement the analytical results, we also provide an algorithmic framework and source code that allows to solve a class of jointly regularized inverse problems with any number of data discrepancies. As concrete applications, we show numerical results for multi-contrast MR and joint MR-PET reconstruction.

中文翻译：

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与多个数据差异耦合的正则化

我们考虑一类用于逆问题的正则化方法，其中采用耦合正则化来同时重建来自多个源的数据。这种设置的应用可以在多光谱或多模态反问题中找到，也可以在动态数据的反问题中找到。我们在一个相当一般的框架中考虑这个设置，并得出稳定性和收敛结果，包括收敛速度。特别是，我们展示了适应不同数据通道相互作用的参数选择策略如何提高通过平等对待所有通道获得的收敛速度。受具体应用的启发，我们的结果是在相当普遍的假设下获得的，这些假设允许将 Kullback-Leibler 散度作为数据差异项。为了简化它们在具体设置中的应用，我们进一步详细阐述了几个实际相关的特殊情况。为了补充分析结果，我们还提供了一个算法框架和源代码，可以解决一类具有任意数量数据差异的联合正则化逆问题。作为具体应用，我们展示了多对比 MR 和联合 MR-PET 重建的数值结果。