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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. Toovercome these artifacts, regularization methods are applied that incorporateadditional information. An important example is TV reconstruction, which isknown to be efficient at compensating for missing data and reducingreconstruction artifacts. At the same time, however, tomographic data is alsocontaminated by noise, which poses an additional challenge. The use of a singleregularizer 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. Toaddress this issue, in this paper we introduce a novel variationalregularization framework that combines the advantages of differentregularizers. The basic idea of our framework is to perform reconstruction intwo stages, where the first stage mainly aims at accurate reconstruction in thepresence of noise, and the second stage aims at artifact reduction. Bothreconstruction stages are connected by a data proximity condition. The proposedmethod is implemented and tested for limited-view CT using a combinedcurvelet-TV approach. We define and implement a curvelet transform adapted tothe limited-view problem and illustrate the advantages of our approach innumerical experiments.

Data-proximal complementary $\ell^1$-TV reconstruction for limited data CT