• Compressed Sensing
• Magnetic resonance imaging (MRI)
• Matrix Completion

ACCELERATED MAGNETIC RESONANCE IMAGING (MRI) is among the most important topics in technological research of the MRI scientific community. Fast acquisitions are in fact needed to improve the image quality and the patient comfort during the MRI exam.
A MRI exam is composed by the acquisition of several images able to represent a given volume of interest with several image contrast. This gives the possibility to characterize the biological tissues by increasing the amount of information available for the diagnosis. The acquisition time of each image depends mainly on its dimensions (in terms of acquisition volume and spatial resolution), and it may be long for some applications. In dynamic MRI, the need for cardiac and respiratory synchronization further increases the acquisition time. For this reason some advanced techniques, like 4D Flow MRI, have not been widely used in the clinical routine.
To reduce the scan time in MRI it is possible to use several techniques as well as their combination. Non-cartesian acquisition trajectories reduce the scan time by sampling more efficiently the k-space. Parallel Imaging and Compressed Sensing (CS), instead, reduce the acquisition time by undersampling the k-space. Artefacts resulting from the undersampled k-space are then removed exploiting the knowledge of coil sensitivity maps (in parallel MRI) or the compressibility of MRI images (in CS-MRI). During the years, CS and related techniques like Matrix Completion, have gained an increasing interest among the scientific community.
A faithful reconstruction using CS, requires incoherent k-space sampling and a nonlinear reconstruction that enforces compressibility of the target image. Therefore, in fast MRI, it is important to design incoherent acquisition strategies and reconstruction methods that can improve the reconstruction accuracy.
In this doctoral thesis I describe novel acquisitions strategies and reconstruction methods for accelerated dynamic MRI. The main contributions of this thesis are:

- A novel non-cartesian trajectory based on a stack of variable density spirals is proposed for accelerated 3D CS-MRI and 2D dynamic CS-MRI.

- A novel reconstruction method, based on Low Rank plus Sparse Matrix Completion, is proposed for accelerated 4D Flow MRI.