Inverse problems are at the heart of many practical problems such as image reconstruction or nondestructive testing. A characteristic feature is their instability with respect to data perturbations. To stabilize the inversion process, regularization methods must be developed and applied. In this paper, we introduce the concept of filtered diagonal frame decomposition, which extends the classical filtered SVD to the case of frames. The use of frames as generalized singular systems allows a better match to a given class of potential solutions and is also beneficial for problems where the SVD is not analytically available. We show that filtered diagonal frame decompositions yield convergent regularization methods, derive convergence rates under source conditions and prove order optimality. Our analysis applies to bounded and unbounded forward operators. As a practical application of our tools, we study filtered diagonal frame decompositions for inverting the Radon transform as an unbounded operator on $L^2({\mathbb{R}}^2)$.

逆问题是许多实际问题的核心，例如图像重建或无损检测。一个特征是它们在数据扰动方面的不稳定性。为了稳定反演过程，必须开发和应用正则化方法。在本文中，我们引入了滤波对角帧分解的概念，将经典的滤波 SVD 扩展到帧的情况。使用框架作为广义奇异系统可以更好地匹配给定类别的潜在解决方案，并且对于 SVD 在分析上不可用的问题也是有益的。我们展示了过滤的对角帧分解产生收敛的正则化方法，在源条件下推导出收敛速度并证明顺序最优性。我们的分析适用于有界和无界前向运算符。作为我们工具的实际应用，我们研究了过滤对角帧分解，以将 Radon 变换作为无界算子在${\mathrm{大号}}^2({\mathbb{R}}^2)$.