In recent years, limited angle CT has become a challenging testing ground for several theoretical and numerical studies, where both variational regularisation and data-driven techniques have been investigated extensively. In this talk, I will present a hybrid reconstruction framework where the proximal operator of an accelerated unrolled scheme is learned to ensure suitable theoretical guarantees. The recipe relays on the interplay between sparse regularization theory, harmonic analysis, microlocal analysis and Plug and Play methods. The numerical results show that these approaches significantly surpasses both pure model- and more data-based reconstruction methods.

It is well known that image reconstructions from limited tomographic data often suffer from significant artifacts and missing features. To remove these artifacts and to compensate for the missing information, reconstruction methods have to incorporate additional information about the objects of interest. An important example of such methods is TV reconstruction. It is well known that this technique can efficiently compensate for missing information and reduce reconstruction artifacts. At the same time, however, tomographic data are also contaminated by noise, which poses an additional challenge. The use of a single penalty term (regularizer) within a variational regularization framework must therefore account for both the missing data and the noise. However, it is known that a single regularizer does not work perfectly for both tasks. In this talk, we introduce a new variational formulation that combines the advantages of two different regularizers, one aimed at accurate reconstruction in the presence of noise and the other aimed at selecting a solution with reduced artifacts. Both reconstructions are linked by a data consistency condition that makes them close to each other in the data domain. We demonstrate the proposed method for the limited angle CT problem using a combined curvelet and TV approach.

Spectral computed tomography is an evolving technique which exploits the property of materials to attenuate X-rays in different ways depending on the specific energy. Compared to conventional CT, spectral CT employs a photon-counting detector that records the energy of individual photons and produce a fine grid of discrete energy-dependent data. In this way it is easier to distinguish materials that have similar attenuation coefficients in an energy range, but different in others. The material decomposition process allows to not only reconstruct the object, but also to estimate the concentration of the materials that compose it.

Different strategies to reconstruct material-specific images have been developed in the last years, but many improvements have yet to be made especially for low-dose cases and few projections. This setup is justified by the slowness and flux limit of the high energy resolution photon counting detectors, but leads to noisier data, especially across the energy channels, and less spatial information. The study of the noise distribution, together with the usage of suitable regularizers and the selection of their parameters become crucial to obtain a good quality reconstruction and material decomposition. The talk will address all these issues by focusing on the case study of materials that have high atomic number with similar attenuation coefficients and K-edges in the considered energy range.

In subsea pipe inspection using X-ray computed tomography (CT), obtaining data is time-consuming and costly due to the challenging underwater conditions. We propose an efficient Bayesian CT reconstruction method with a new class of structural Gaussian priors incorporating known material properties to enhance quality from limited data. Experiments with real and synthetic data demonstrate artifact reduction, increased contrast, and enhanced reconstruction certainty compared to conventional reconstruction methods.

Authhors: Silja L. Christensen, Nicolai A. B. Riis, Felipe Uribe and Jakob S. Jørgensen