(A) compressed sensing framework for high resolution dynamic MRI

Abstract

In dynamic MRI, spatio-temporal resolution is a very important issue. Recently, compressed sensing approaches have gained recognition as a highly attractive imaging technique since they enable accelerated acquisition by breaking Nyquist sampling requirements. According to compressed sensing, by solving an $l_1$ minimization problem, accurate reconstruction can be achieved from severely under-sampled measurements when an unknown signal can be sparsely represented and the sampling basis is incoherent with respect to the modeling dictionary. These conditions can be well satisfied in dynamic MRI. First, by exploiting temporal redundancies of dynamic MRI, it can be easily sparsified. For example, when an object has periodic motion such as in the case of the heart, a Fourier transform along the temporal direction effectively sparsifies the corresponding dynamic images. For non-periodic dynamic images, we can use a Karhunen Loeve Transform (KLT), which is well known as an optimal energy compaction transform. Second, an incoherent sampling basis can be constructed by arbitrarily choosing sampling patterns in k-t space such as random or radial sampling patterns. Based on this concept, we developed the k-t FOCUSS algorithm, which reconstructs aliasing-free dynamic MR images from down-sampled measurements in cartesian or radial trajectories.

More recently, researchers have been interested in solving an $l_p$ minimization problem for $0 \leq p < 1$ based on the observation that the number of required measurements for an exact sparse reconstruction is more greatly reduced than in solving an $l_1$ minimization. However, when $0 \leq p<1$, the problem is no longer a convex optimization, and thus a combinatorial number of local minima exist. To deal with this problem, we adopted an empirical Bayesian approach, so-called sparse Bayesian learning (SBL), in dynamic MRI. Theoretically, SBL achieves an $l_0$ minimization solution at the global minimum and filters out a large amou...