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.

翻译：我们研究了一个自校准问题，其中变换稀疏信号通过多个传感器并行压缩感知，且传感参数未知。该问题在并行磁共振成像（pMRI）重建中具有重要应用，实践中线圈的显式校准往往困难且成本高昂，但却是高精度重建的基本要求。大多数自校准策略得到的重建结果对应于求解一个具有挑战性的双凸优化问题。我们利用“提升”的概念将自校准并行传感转化为凸优化问题。通过利用信号的稀疏结构以及多个传感器引入的冗余性，我们求解一个混合范数最小化问题来同时恢复底层信号和传感参数。在存在噪声和信号稀疏性不足的情况下，推导出了稳健且稳定的恢复保证。针对pMRI应用，我们的方法为在适当假设下加速MRI采集提供了具有理论保证的自校准并行成像途径。文中讨论了MRI的发展，并通过使用解析模型和模拟线圈灵敏度的数值仿真来支持我们的理论结果。