As this problem has not been encountered before there is no algorithm to solve it. This is another significant contribution of this paper. In the appendix, we derive an efficient solution to this rank-deficient group-sparse recovery problem.The next section describes the theory behind the proposed method. Section 3 describes the experimental results. The conclusions of the work are discussed in Section 4. The derivation of the algorithm is relegated to the appendix since it may not be of interest to the majority of readers.2.?Theory2.1. Literature ReviewThe previous work that is directly relevant to us is found in [4,5]. Multi-echo images pertain to the same cross section and only vary in tissue contrasts; the positions of edges (tissue boundaries) remain the same in all images.

The wavelet transform encodes the discontinuities in images. The wavelet coefficients have high values along edges and are zeroes or near zeroes in smooth areas. Since the edges of the multi-echo images are aligned, the wavelet transform of the images will have high values at similar positions (along edges) in the different images. When the wavelet transform coefficients of different images are stacked as column vectors of a Multiple Measurement Vector (MMV) matrix, the resulting matrix is row-sparse (such a matrix is of size n �� N, assuming that the wavelet coefficient vector is of length n and there are N such echo images). The MMV matrix is row-sparse since Carfilzomib only those rows that correspond to edge positions have high values and the rest of the rows are zeroes or near zeroes [4].

Alternately the transform coefficients of the N echoes can be concatenated into a vector of length nN. The thus formed long vector can be grouped according to positions, i.e., the jth group is formed by collecting the jth coefficient of the transform coefficients of each one of the N echoes. There will be n such groups of size N and the concatenated wavelet coefficient vector will be group-sparse [5]. Thus one can see that row-sparsity and group-sparsity are the same; these two forms arise out of the difference in arranging the vectors.The aforesaid studies use two prior pieces of information��that the images are spatially redundant and are correlated with each other. The wavelet transform decorrelates the spatial redundancies in the individual images leading to a sparse representation; the fact that concatenated wavelet transform coefficients form a row-sparse matrix [4] or a group-sparse vector [5] arises from the inter-image correlation.In this work, we follow the same assumptions as [4,5] but improve upon their results by incorporating more information regarding the solution.