Figure 2:

General concept used in CS. The original signal x should be sparse, and the system matrix E should be incoherent. The effect of this is that E^Ty, where y is the k-space data and E^T is the adjoint of E, should be basically sparse peaks mixed with noise-like artifacts, which are easily recovered by a nonlinear CS reconstruction. Most significant peaks are usually recovered at the initial iteration ${\widehat{\mathit{x}}}_1$. However, remaining sparse elements can be seen when the residual is mapped back to image domain, via ${\mathit{E}}^T(\mathit{y}-\mathit{E}{\widehat{\mathit{x}}}_1)$, and they are recovered in the following iterations, such as ${\widehat{\mathit{x}}}_2$.