  The boom of non-uniform sampling and compressed sensing techniques
dramatically alleviates the lengthy data acquisition problem of magnetic
resonance imaging. Sparse reconstruction, thanks to its fast computation and
promising performance, has attracted researchers to put numerous efforts on it
and has been adopted in commercial scanners. To perform sparse reconstruction,
choosing a proper algorithm is essential in providing satisfying results and
saving time in tuning parameters. The pFISTA, a simple and efficient algorithm
for sparse reconstruction, has been successfully extended to parallel imaging.
However, its convergence criterion is still an open question. And the existing
convergence criterion of single-coil pFISTA cannot be applied to the parallel
imaging pFISTA, which, therefore, imposes confusions and difficulties on users
about determining the only parameter - step size. In this work, we provide the
guaranteed convergence analysis of the parallel imaging version pFISTA to solve
the two well-known parallel imaging reconstruction models, SENSE and SPIRiT.
Along with the convergence analysis, we provide recommended step size values
for SENSE and SPIRiT reconstructions to obtain fast and promising
reconstructions. Experiments on in vivo brain images demonstrate the validity
of the convergence criterion. Besides, experimental results show that compared
to using backtracking and power iteration to determine the step size, our
recommended step size achieves more than five times acceleration in
reconstruction time in most tested cases.
