A novel k-space annihilating filter method for unification between compressed sensing and parallel MRI

In this paper, we propose a novel k-space method called ALOHA (Annihilating filter based LOw-rank Hankel matrix Approach) that unifies parallel imaging and compressed sensing as a k-space data interpolation problem. Specifically, ALOHA employs annihilating filter relationships originated from the intrinsic image property originated from the finite rate of innovation model, as well as the multi-coil acquisition physics. By interchanging the annihilating filter with the k-space measurement, a rank-deficient block Hankel structured matrix can be obtained, whose missing elements can be restored by a low rank matrix completion algorithm. To exploit the low rank Hankel structure, we develop an alternating direction method of multiplier (ADMM) method with initialisation from low rank matrix fitting (LMaFit) algorithm. Additionally, we develop a novel pyramidal representation of the Hankel structured matrix to reduce the computational complexity of the algorithm. ALOHA can be universally applied to compressed sensing MRI as well as parallel imaging for both static and dynamic applications. Experimental results with real in vivo data confirmed that ALOHA outperforms the existing state-of-the-art parallel and compressed sensing MRI.