On systems with quasi-discrete spectrum

In this paper we re-examine the theory of systems with quasi-discrete spectrum initiated in the 1960's by Abramov, Hahn, and Parry. In the first part, we give a simpler proof of the Hahn--Parry theorem stating that each minimal topological system with quasi-discrete spectrum is isomorphic to a certain affine automorphism system on some compact Abelian group. Next, we show that a suitable application of Gelfand's theorem renders Abramov's theorem --- the analogue of the Hahn-Parry theorem for measure-preserving systems --- a straightforward corollary of the Hahn-Parry result. In the second part, independent of the first, we present a shortened proof of the fact that each factor of a totally ergodic system with quasi-discrete spectrum (a "QDS-system") has again quasi-discrete spectrum and that such systems have zero entropy. Moreover, we obtain a complete algebraic classification of the factors of a QDS-system. In the third part, we apply the results of the second to the (still open) question whether a Markov quasi-factor of a QDS-system is already a factor of it. We show that this is true when the system satisfies some algebraic constraint on the group of quasi-eigenvalues, which is satisfied, e.g., in the case of the skew shift.Comment: 25 pages. Accepted for publication in Studia Mathematic

End-to-end Memory-Efficient Reconstruction for Cone Beam CT

Cone Beam CT plays an important role in many medical fields nowadays, but the potential of this imaging modality is hampered by lower image quality compared to the conventional CT. A lot of recent research has been directed towards reconstruction methods relying on deep learning. However, practical application of deep learning to CBCT reconstruction is complicated by several issues, such as exceedingly high memory costs of deep learning methods for fully 3D data. In this work, we address these limitations and propose LIRE: a learned invertible primal-dual iterative scheme for Cone Beam CT reconstruction. Memory requirements of the network are substantially reduced while preserving its expressive power, enabling us to train on data with isotropic 2mm voxel spacing, clinically-relevant projection count and detector panel resolution on current hardware with 24 GB VRAM. Two LIRE models for small and for large Field-of-View setting were trained and validated on a set of 260 + 22 thorax CT scans and tested using a set of 142 thorax CT scans plus an out-of-distribution dataset of 79 head \& neck CT scans. For both settings, our method surpasses the classical methods and the deep learning baselines on both test sets. On the thorax CT set, our method achieves PSNR of 33.84 $\pm$ 2.28 for the small FoV setting and 35.14 $\pm$ 2.69 for the large FoV setting; U-Net baseline achieves PSNR of 33.08 $\pm$ 1.75 and 34.29 $\pm$ 2.71 respectively. On the head \& neck CT set, our method achieves PSNR of 39.35 $\pm$ 1.75 for the small FoV setting and 41.21 $\pm$ 1.41 for the large FoV setting; U-Net baseline achieves PSNR of 33.08 $\pm$ 1.75 and 34.29 $\pm$ 2.71 respectively. Additionally, we demonstrate that LIRE can be finetuned to reconstruct high-resolution CBCT data with the same geometry but 1mm voxel spacing and higher detector panel resolution, where it outperforms the U-Net baseline as well

Deep Cardiac MRI Reconstruction with ADMM

Cardiac magnetic resonance imaging is a valuable non-invasive tool for identifying cardiovascular diseases. For instance, Cine MRI is the benchmark modality for assessing the cardiac function and anatomy. On the other hand, multi-contrast (T1 and T2) mapping has the potential to assess pathologies and abnormalities in the myocardium and interstitium. However, voluntary breath-holding and often arrhythmia, in combination with MRI's slow imaging speed, can lead to motion artifacts, hindering real-time acquisition image quality. Although performing accelerated acquisitions can facilitate dynamic imaging, it induces aliasing, causing low reconstructed image quality in Cine MRI and inaccurate T1 and T2 mapping estimation. In this work, inspired by related work in accelerated MRI reconstruction, we present a deep learning (DL)-based method for accelerated cine and multi-contrast reconstruction in the context of dynamic cardiac imaging. We formulate the reconstruction problem as a least squares regularized optimization task, and employ vSHARP, a state-of-the-art DL-based inverse problem solver, which incorporates half-quadratic variable splitting and the alternating direction method of multipliers with neural networks. We treat the problem in two setups; a 2D reconstruction and a 2D dynamic reconstruction task, and employ 2D and 3D deep learning networks, respectively. Our method optimizes in both the image and k-space domains, allowing for high reconstruction fidelity. Although the target data is undersampled with a Cartesian equispaced scheme, we train our model using both Cartesian and simulated non-Cartesian undersampling schemes to enhance generalization of the model to unseen data. Furthermore, our model adopts a deep neural network to learn and refine the sensitivity maps of multi-coil k-space data. Lastly, our method is jointly trained on both, undersampled cine and multi-contrast data.Comment: 12 pages, 3 figures, 2 tables. CMRxRecon Challenge, MICCAI 202