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

2024-01-18

arXiv.org

AI Summary

This work addresses the challenge of sparse signal reconstruction in parallel magnetic resonance imaging (pMRI) when receiver coil sensitivity profiles are unknown. We propose a lifting-based self-calibrating reconstruction method that reformulates the nonconvex joint estimation problem as a convex optimization task, integrating mixed-norm minimization with transform-domain sparsity modeling. For the first time, our approach establishes theoretical robust recovery guarantees against both noise and insufficient sparsity—without requiring pre-calibration. By fully exploiting redundancy across multi-coil acquisitions, the method achieves high-fidelity, fully automatic calibration and reconstruction on both simulated and in vivo MRI data. It significantly outperforms existing bi-convex or heuristic self-calibration techniques, overcoming the fundamental limitation of prior methods: the absence of rigorous theoretical recovery guarantees.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Non-convex Optimization
• Computer Vision: Medical and Biological Imaging

Application Category

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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.

Problem

• Solving auto-calibration for parallel MRI with unknown sensor parameters
• Transforming biconvex compressive sensing into convex optimization via lifting
• Providing guaranteed signal recovery despite noise and sparsity limitations

Innovation

• Lifting transforms biconvex problem to convex optimization
• Mixed-norm minimization recovers signal and parameters simultaneously
• Method provides robust recovery guarantees with noise consideration