TY - JOUR A1 - Göppel, Simon A1 - Frikel, Jürgen A1 - Haltmeier, Markus T1 - Data-Proximal Complementary ℓ1-TV Reconstruction for Limited Data Computed Tomography JF - Mathematics N2 - 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 ℓ1 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. KW - image reconstruction KW - limited data KW - artifact reduction KW - sparse regularization KW - total variation KW - wedge-adapted curvelets Y1 - 2024 U6 - https://doi.org/10.3390/math12101606 SN - 2227-7390 N1 - Corresponding author der OTH Regensburg: Jürgen Frikel (und Markus Haltmeier) VL - 12 IS - 10 PB - MDPI ER - TY - INPR A1 - Göppel, Simon A1 - Frikel, Jürgen A1 - Haltmeier, Markus T1 - A complementary ℓ1 -TV reconstruction algorithm for limited data CT N2 - In a variety of tomographic applications, data cannot be fully acquired, leading to severely underdetermined image reconstruction. Conventional methods result in reconstructions with significant artifacts. In order to remove these artifacts, regularization methods have to be applied that incorporate additional information. An important example is TV reconstruction which is well known to efficiently compensate for missing data and well reduces 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 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 over different scales, in which case ℓ1 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 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 demonstrate the advantages of our approach in a series of numerical experiments in this context. KW - Image reconstruction KW - limited data KW - artifact reduction KW - sparse regularization KW - complementary image reconstruction KW - wedge-adapted curvelets Y1 - 2023 ER -