Solving inverse problems is central to a variety of important applications, such as biomedical image reconstruction and non-destructive testing. These problems are characterized by the sensitivity of direct solution methods with respect to data perturbations. To stabilize the reconstruction process, regularization methods have to be employed. Well-known regularization methods are based on frame expansions, such as the wavelet-vaguelette (WVD) decomposition, which are well adapted to the underlying signal class and the forward model and furthermore allow efficient implementation. However, it is well known that the lack of translational invariance of wavelets and related systems leads to specific artifacts in the reconstruction. To overcome this problem, in this paper we introduce and analyze the translation invariant diagonal frame decomposition (TI-DFD) of linear operators as a novel concept generalizing the SVD. We characterize ill-posedness via the TI-DFD and prove that a TI-DFD combined with a regularizing filter leads to a convergent regularization method with optimal convergence rates. As illustrative example, we construct a wavelet-based TI-DFD for one-dimensional integration, where we also investigate our approach numerically. The results indicate that filtered TI-DFDs eliminate the typical wavelet artifacts when using standard wavelets and provide a fast, accurate, and stable solution scheme for inverse problems.

翻译：求解逆问题是生物医学图像重建、无损检测等众多重要应用的核心。这类问题的特点在于直接求解法对数据扰动极其敏感。为稳定重构过程，必须采用正则化方法。经典的正则化方法基于框架展开，如小波-小波变换（WVD）分解，这类方法既能适应底层信号类别和正演模型，又能实现高效运算。然而众所周知，小波及其相关系统缺乏平移不变性会导致重构中出现特定伪影。为克服这一难题，本文提出并分析线性算子的平移不变对角框架分解（TI-DFD）这一新概念，将其作为奇异值分解（SVD）的推广形式。我们通过TI-DFD刻画逆问题的不适定性，并证明将TI-DFD与正则化滤波器相结合可得到具有最优收敛速度的收敛性正则化方法。以一维积分为例，我们构建了基于小波的TI-DFD，并对其进行了数值研究。结果表明：在使用标准小波时，经滤波的TI-DFD能消除典型的小波伪影，为逆问题提供快速、精确且稳定的求解方案。