  We study an auto-calibration problem in which a transform-sparse signal is
compressive-sensed by multiple sensors in parallel with unknown sensing
parameters. The problem has an important application in pMRI reconstruction,
where explicit coil calibrations are often difficult and costly to achieve in
practice, but nevertheless a fundamental requirement for high-precision
reconstructions. Most auto-calibrated strategies result in reconstruction that
corresponds to solving a challenging biconvex optimization problem. We
transform the auto-calibrated parallel sensing as a convex optimization problem
using the idea of `lifting'. By exploiting sparsity structures in the signal
and the redundancy introduced by multiple sensors, we solve a mixed-norm
minimization problem to recover the underlying signal and the sensing
parameters simultaneously. Robust and stable recovery guarantees are derived in
the presence of noise and sparsity deficiencies in the signals. For the pMRI
application, our method provides a theoretically guaranteed approach to
self-calibrated parallel imaging to accelerate MRI acquisitions under
appropriate assumptions. Developments in MRI are discussed, and numerical
simulations using the analytical phantom and simulated coil sensitives are
presented to support our theoretical results.
