The inherent slow imaging speed of Magnetic Resonance Image (MRI) has spurred the development of various acceleration methods, typically through heuristically undersampling the MRI measurement domain known as k-space. Recently, deep neural networks have been applied to reconstruct undersampled k-space data and have shown improved reconstruction performance. While most of these methods focus on designing novel reconstruction networks or new training strategies for a given undersampling pattern, e.g., Cartesian undersampling or Non-Cartesian sampling, to date, there is limited research aiming to learn and optimize k-space sampling strategies using deep neural networks. This work proposes a novel optimization framework to learn k-space sampling trajectories by considering it as an Ordinary Differential Equation (ODE) problem that can be solved using neural ODE. In particular, the sampling of k-space data is framed as a dynamic system, in which neural ODE is formulated to approximate the system with additional constraints on MRI physics. In addition, we have also demonstrated that trajectory optimization and image reconstruction can be learned collaboratively for improved imaging efficiency and reconstruction performance. Experiments were conducted on different in-vivo datasets (e.g., brain and knee images) acquired with different sequences. Initial results have shown that our proposed method can generate better image quality in accelerated MRI than conventional undersampling schemes in Cartesian and Non-Cartesian acquisitions.
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