TY - JOUR A1 - Göppel, Simon A1 - Frikel, Jürgen A1 - Haltmeier, Markus T1 - Feature Reconstruction from Incomplete Tomographic Data without Detour JF - Mathematics N2 - In this paper, we consider the problem of feature reconstruction from incomplete X-ray CT data. Such incomplete data problems occur when the number of measured X-rays is restricted either due to limit radiation exposure or due to practical constraints, making the detection of certain rays challenging. Since image reconstruction from incomplete data is a severely ill-posed (unstable) problem, the reconstructed images may suffer from characteristic artefacts or missing features, thus significantly complicating subsequent image processing tasks (e.g., edge detection or segmentation). In this paper, we introduce a framework for the robust reconstruction of convolutional image features directly from CT data without the need of computing a reconstructed image first. Within our framework, we use non-linear variational regularization methods that can be adapted to a variety of feature reconstruction tasks and to several limited data situations. The proposed variational regularization method minimizes an energy functional being the sum of a feature dependent datafitting term and an additional penalty accounting for specific properties of the features. In our numerical experiments, we consider instances of edge reconstructions from angular under-sampled data and show that our approach is able to reliably reconstruct feature maps in this case. KW - Computed tomography KW - Radon transform KW - reconstruction KW - limited data KW - sparse data KW - feature reconstruction, KW - edge detection Y1 - 2022 U6 - https://doi.org/10.3390/math10081318 VL - 10 IS - 8 PB - MDPI ER - TY - INPR A1 - Göppel, Simon A1 - Frikel, Jürgen A1 - Haltmeier, Markus T1 - Translation invariant diagonal frame decomposition for the Radon transform N2 - In this article, we address the challenge of solving the ill-posed reconstruction problem in computed tomography using a translation invariant diagonal frame decomposition (TIDFD). First, we review the concept of a TI-DFD for general linear operators and the corresponding filter-based regularization concept. We then introduce the TI-DFD for the Radon transform on L 2 (R 2) and provide an exemplary construction using the TI wavelet transform. Presented numerical results clearly demonstrate the benefits of our approach over non-translation invariant counterparts. KW - DFD KW - translation invariance KW - Radon transform KW - inverse problems KW - regularization Y1 - 2023 ER -