Data-Proximal Complementary ℓ 1 -TV Reconstruction for Limited Data Computed Tomography

In a number of tomographic applications, data cannot be fully acquired, resulting in severely underdetermined image reconstruction. Conventional methods in such cases 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 in compensating for missing data and reducing reconstruction artifacts. On the other hand, tomographic data are 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. A particular regularizer may not be ideal for both tasks. For example, the TV regularizer is a poor choice for noise reduction over multiple scales, in which case ℓ 1 curvelet regularization methods are well suited. To address this issue, in this paper, we present 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. The first stage is mainly aimed at accurate reconstruction in the presence of noise, and the second stage is aimed 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.