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We study an auto-calibration problem in which a transform-sparse signal iscompressive-sensed by multiple sensors in parallel with unknown sensingparameters. The problem has an important application in pMRI reconstruction,where explicit coil calibrations are often difficult and costly to achieve inpractice, but nevertheless a fundamental requirement for high-precisionreconstructions. Most auto-calibrated strategies result in reconstruction thatcorresponds to solving a challenging biconvex optimization problem. Wetransform the auto-calibrated parallel sensing as a convex optimization problemusing the idea of `lifting'. By exploiting sparsity structures in the signaland the redundancy introduced by multiple sensors, we solve a mixed-normminimization problem to recover the underlying signal and the sensingparameters simultaneously. Robust and stable recovery guarantees are derived inthe presence of noise and sparsity deficiencies in the signals. For the pMRIapplication, our method provides a theoretically guaranteed approach toself-calibrated parallel imaging to accelerate MRI acquisitions underappropriate assumptions. Developments in MRI are discussed, and numericalsimulations using the analytical phantom and simulated coil sensitives arepresented to support our theoretical results.

Auto-Calibration and Biconvex Compressive Sensing with Applications to Parallel MRI