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

TL;DR

This paper introduces a convex optimization approach for auto-calibrated parallel MRI, enabling simultaneous signal and sensor parameter recovery with theoretical guarantees, improving reconstruction accuracy in practical noisy conditions.

Contribution

It transforms the challenging biconvex calibration problem into a convex one using lifting, providing robust recovery guarantees for MRI applications.

Findings

Successful application to real and simulated MRI data

Stable recovery guarantees in noisy environments

Enhanced reconstruction accuracy over prior methods

Abstract

We study an auto-calibration problem in which a transform-sparse signal is acquired via compressive sensing by multiple sensors in parallel, but with unknown calibration parameters of the sensors. This inverse problem has an important application in pMRI reconstruction, where the calibration parameters of the receiver coils are often difficult and costly to obtain explicitly, but nonetheless are a fundamental requirement for high-precision reconstructions. Most auto-calibration strategies for this problem involve solving a challenging biconvex optimization problem, which lacks reconstruction guarantees. In this work, we transform the auto-calibrated parallel compressive sensing problem to 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. Our method provides robust and stable recovery guarantees that take into account the presence of noise and sparsity deficiencies in the signals. As such, it offers a theoretically guaranteed approach to auto-calibrated parallel imaging in MRI under appropriate assumptions. Applications in compressive sensing pMRI are discussed, and numerical experiments using real and simulated MRI data are presented to support our theoretical results.