In a variety of tomographic applications, data cannot be fully acquired, leading to a severely underdetermined image reconstruction problem. In such cases, conventional methods generate reconstructions with significant artifacts. In order to remove these artifacts, regularization methods must be applied that beneficially incorporate additional information. An important example of such methods is TV reconstruction. It is well-known that this technique can efficiently compensate for the missing data and reduce reconstruction artifacts. At the same time, however, tomographic data is also contaminated by noise, which poses an additional challenge. The use of a single penalty term (regularizer) within a variational regularization framework must therefore account for both, the missing data and the noise. However, a single regularizer may not be ideal for both tasks. For example, the TV regularizer is a poor choice for noise reduction across different scales, in which case $\ell^1$-curvelet regularization methods work well. To address this issue, in this paper we introduce a novel variational regularization framework that combines the advantages of two different regularizers. The basic idea of our framework is to perform reconstruction in two stages, where the first stage mainly aims at accurate reconstruction in the presence of noise, and the second stage aims at artifact reduction. Both reconstruction stages are connected by a data consistency condition, which makes them close to each other in the data domain. The proposed method is implemented and tested for limited view CT using a combined curvelet-TV-approach. To this end, we define and implement a curvelet transform adapted to the limited view problem and demonstrate the advantages of our approach in a series of numerical experiments in this context.

翻译：在多种断层成像应用中，数据无法完全采集，导致重建问题严重欠定。此时，传统方法会产生带有显著伪影的重建结果。为消除这些伪影，必须采用能够有效引入额外信息的正则化方法，其中TV重建是重要范例。众所周知，该技术能有效补偿缺失数据并减少重建伪影。然而，断层数据同时受到噪声污染，构成额外挑战。在变分正则化框架中使用单一惩罚项（正则子）必须兼顾缺失数据与噪声，但单一正则子可能难以同时胜任这两项任务。例如，TV正则子在跨尺度降噪方面表现不佳，而$\ell^1$-curvelet正则化方法在此类场景中效果显著。为解决此问题，本文提出一种新型变分正则化框架，融合两种不同正则子的优势。该框架的基本思想是采用两阶段重建：第一阶段主要针对含噪数据实现精确重建，第二阶段致力于伪影抑制。两个重建阶段通过数据保真度条件相关联，确保其在数据域内保持接近。我们针对有限角度CT问题，采用curvelet-TV联合方法实现并测试了所提算法。为此，我们定义并实现了适应有限角度问题的curvelet变换，并通过系列数值实验展示了该方法的优势。