FIG. 1.
Illustration of sparse matrices: (a) an example of the sparse design matrix G resulting from an echo-planar imaging experiment with an image size of 32 × 32, an eight-channel coil, and a reduction factor of 4; (b) a schematic illustration of the construction of sparse matrix M, and (c) its pseudoinverse M+. The solid straight lines in (b) and (c) indicate the Cartesian grids in k-space with the small white circles indicating the grid points. The small dark circles indicate random sampling locations. The larger solid circles show the range of the coil sensitivity, while the larger dotted circle in (c) shows the reconstruction kernel width. (b) Matrix M is computed as the autocorrelation of the sampled coil sensitivity. The shaded common area indicates the area over which the summation should be taken to compute Mρρ′. (c) The ρ′-th entry in the ρ-th row of M+ is only nonzero when kρ′ is within a distance of w of kρ′, as indicated by the dotted circle.