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

TL;DR

This work tackles auto-calibration in parallel compressive sensing by lifting the bilinear measurement model to a linear operator acting on a lifted matrix, then promoting block-sparsity across coils via an $ ext{1,2}$-norm. Under mild incoherence and subspace assumptions for coil gains and transform-sparse signals, the authors prove robust recovery guarantees with high probability when the per-coil sample count $L$ and the number of coils $C$ satisfy logarithmic, sparsity, and coherence-dependent bounds. Theoretical results are complemented by simulations and real pMRI data demonstrating improved performance over standard $ ext{l}_1$ methods and favorable comparisons to ENLIVE, particularly in higher acceleration regimes. The approach provides a convex, provably reliable route to auto-calibrated parallel MRI reconstructions with practical implications for faster, higher-quality imaging. Overall, the paper advances the understanding of when and how auto-calibrated pMRI can achieve stable recovery using convex optimization and lifting under realistic measurement ensembles.

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.

Figures (15)

• Figure 2: Phase transition plot of performance by solving \ref{['main0']} directly. The figures show empirical rate of success for a fixed sampling rate $L$ and different pairs of $(C,k,n)$. The numbers 1.0 means 100% rate of success and 0.0 means 0% rate of success. Observe that the transitional curve is improved for more $C$. However, when $C$ reaches a certain level the improvements in the empirical rate of success saturates.
• Figure 3: Repeated the same experimental setup as in Fig:\ref{['fig:2']}, with 1% of noise added.
• Figure 4: Phase transition plot of performance by solving \ref{['main0']} and using db4 wavelet transform as the sparsifying transformation matrix.
• Figure 5: Phase transition plot of performance by minimizing $\sum\limits_{i \in [C]}\|\textbf{X}(:,i)\|_2$ instead. Observe that compared to the proposed method, the improvement in the success rate is not obvious for including larger $C$.
• Figure 6: The empirical rate of success for a fixed k and varying $(L,n,C)$.

Theorems & Definitions (12)

• Lemma 1
• Lemma 2
• Theorem 1
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9