k-t Group Sparse Reconstruction Method for Dynamic Compressed MRI

Introduction: Up to now, besides sparsity, the standard compressed sensing methods used in MR do not exploit any other prior information about the underlying signal. In general, the MR data in its sparse representation always exhibits some structure. As an example, for dynamic cardiac MR data, the signal support in its sparse representation (x-f space) is always in compact form [1]. In this work, exploiting the structural properties of sparse representation, we propose a new formulation titled ‘k-t group sparse compressed sensing’. This formulation introduces a constraint that forces a group structure in sparse representation of the reconstructed signal. The k-t group sparse reconstruction achieves much higher temporal and spatial resolution than the standard l1 method at high acceleration factors (9-fold acceleration). Method: Our proposed method consists of two steps; group assignment and signal recovery. The signal support in x-f space is assumed to be known a priori. In group assignment step, based on the prior knowledge of signal support we assign elements in x-f space to distinct groups. A simple illustration of group assignment step is shown in Fig.1. The group assignment is done by running the modified run-length encoding (RLE) scheme [2] on signal support elements. This scheme assigns those support elements to a single group that are adjacent to each other in x-f space [Fig.1(c)]. Each of the elements in x-f space that are not part of signal support is assigned as a single element group. Once the groups are assigned to all elements in x-f space, we recover the signal in x-f space by ‘k-t group sparse’ reconstruction. The proposed reconstruction method recovers the signal by minimizing the number of non-sparse groups in x-f space, subject to the data constraints [3]. The formulation is as follows: Let X be the signal in x-f space whose elements Xi {i=1, 2,.., N} after the group assignment step are assigned to K distinct groups {g1,g2,..,gK} which are nonoverlapping and whose union gives the signal X. The k-t group sparse formulation is given as: minX||X||1,2 subject to AX=b, where ||X||1,2 is the mixed l1-l2 norm given as ||X||1,2 =||X1||2+||X2||2+.......+||XK||2, ||Xj||2 being the l2 norm of the vector containing all elements in x-f space assigned to the group {gj} , A is the measurement matrix, b is the set of measurements in k-t space. The summation over l2 norms