This work is based on the concepts introduced in [4,5], mainly in order to achieve better reconstruction, both the inter-echo correlation and the intra-image spatial redundancy need to be exploited. In [4,5], the inter image correlation information assumed that the similarity amongst the images result in the locations of the images edges being the same over all the acquired echo images. Based on this assumption, the previous work showed that when the transform coefficients of the different echoes are stacked as columns of an MMV matrix or concatenated as a long vector, the matrix or vector thus formed is row-sparse or group-sparse respectively. Thus it required solving an optimization problem which promotes the signal’s row/group sparsity.Our work differs from its predecessors in the optimization problem used for reconstruction.

As mentioned above, when the transform coefficients of the echo images are stacked in MMV form, the resulting matrix is row-sparse. Such a row-sparse matrix is low-rank as well (the rank is less than or equal to the number of non-zero rows). The key difference between this work and [4,5] is that it uses this extra information regarding rank deficiency of the MMV matrix along with row/group sparsity. Compared to [4,5] we use more information regarding the structure of the unknown signal (row/group sparsity and low-rank property) compared to [4,5] (only row/group sparsity).As mentioned above, owing to the partial sampling of the K-space, the reconstruction problem is under-determined and prior information regarding the solution is required.

Intuitively, the greater the information we have regarding the unknown signal (solution), the better is the reconstruction. In the context of row-sparse MMV recovery, it has been theoretically proven in [6,7] that using the extra information that the MMV matrix has low rank (and not only the row-sparsity information), better reconstruction results can indeed be obtained (a row-sparse matrix will be obviously low-rank as well, and its rank will be less than or equal to the number of non-zero rows). Motivated by these studies, we propose to solve the multi-echo MRI reconstruction problem by formulating an optimization problem that exploits both the row/group sparsity and the low-rank properties Cilengitide of the unknown signals (to be reconstructed).The problem of rank-deficient row-sparse MMV recovery has been studied before [6,7].

However multi-echo MRI reconstruction can only be formulated as a MMV recovery when the same sampling mask is used for collecting the K-space samples for all echoes. This is however a restrictive scenario. In general, we should be able to solve the problem even when different sampling masks are used for sampling the K-space data for every echo. This would require formulating the reconstruction as group-sparse vector recovery problem [5].