In a number of tomographic applications, data cannot be fully acquired, resulting in a severely underdetermined image reconstruction. In such cases, conventional methods lead to reconstructions with significant artifacts. To overcome these artifacts, regularization methods are applied that incorporate additional information. An important example is TV reconstruction, which is known to be efficient at compensating for missing data and reducing reconstruction artifacts. At the same time, however, tomographic data is also contaminated by noise, which poses an additional challenge. The use of a single regularizer must therefore account for both the missing data and the noise. However, a particular regularizer may not be ideal for both tasks. For example, the TV regularizer is a poor choice for noise reduction across multiple scales, in which case $\ell^1$ curvelet regularization methods are well suited. To address this issue, in this paper we introduce a novel variational regularization framework that combines the advantages of 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 proximity condition. The proposed method is implemented and tested for limited-view CT using a combined curvelet-TV approach. We define and implement a curvelet transform adapted to the limited-view problem and illustrate the advantages of our approach in numerical experiments.

翻译：在众多层析成像应用中，数据无法完全采集，导致图像重建面临严重的欠定性问题。在此类情况下，传统方法会产生具有显著伪影的重建结果。为克服这些伪影，需采用融合额外信息的正则化方法。其中TV重建是重要范例，其能有效补偿缺失数据并减少重建伪影。然而，层析数据同时受噪声污染，这构成额外挑战。使用单一正则化器必须兼顾缺失数据与噪声处理，但特定正则化器可能无法同时胜任这两项任务。例如，TV正则化器在多尺度降噪方面表现欠佳，而$\ell^1$曲波正则化方法在此领域具有优势。针对该问题，本文提出一种新颖的变分正则化框架，可融合不同正则化器的优势。该框架的核心思想为采用两阶段重建：第一阶段主要针对噪声环境下的精确重建，第二阶段致力于伪影抑制。两重建阶段通过数据近端条件相互关联。我们以曲波-TV联合方法实现并验证了该方案在有限视角CT中的应用。通过定义并实现适配有限视角问题的曲波变换，数值实验展示了本方法的优越性。