Modeled as Least square problem. The gradient method (GM), fast GM (FGM) and optimized GM (OGM) were compared.

Figure 2: Reconstructed cardiac cine images (only systolic frame shown here), the error maps and the convergent speeds using different optimization methods (GM, FGM, OGM) and acceleration rates (R = 2, 4, 6) for parallel imaging (PI). The stepsize α= 1/L and the total number of iteration N = 150. The convergence speed of FGM and OGM are much faster than GM, and OGM has the best convergence performance. For R= 4,6, GM didn’t converge within 150 iterations. For R= 6, FGM and OGM diverged due to the high acceleration rate (optimization problem is ill posed).

Modeled as Lasso problem (l1 penalty, sparsifying transform: Temporal Fourier Transform). ISTA/PGM, FISTA/FPGM and POGM were compared.

Figure 4: Reconstructed cardiac cine images (only systolic frame shown here), the error maps and the convergent speeds using different optimization methods (ISTA, FISTA, POGM) and acceleration rates (R = 4, 6, 8) for compressive sensing (CS). The stepsize α= 1/L and the total number of iteration N = 150. The convergence speed of FISTA and POGM are much faster than ISTA, and POGM has the best convergence performance. For R= 8, ISTA didn’t converge within 150 iterations.