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

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.Comment: Keywords: Self-calibration, Compressive sensing, Convex optimization, Random matrices, Parallel MR

Compressive PCA for Low-Rank Matrices on Graphs

We introduce a novel framework for an approxi- mate recovery of data matrices which are low-rank on graphs, from sampled measurements. The rows and columns of such matrices belong to the span of the first few eigenvectors of the graphs constructed between their rows and columns. We leverage this property to recover the non-linear low-rank structures efficiently from sampled data measurements, with a low cost (linear in n). First, a Resrtricted Isometry Property (RIP) condition is introduced for efficient uniform sampling of the rows and columns of such matrices based on the cumulative coherence of graph eigenvectors. Secondly, a state-of-the-art fast low-rank recovery method is suggested for the sampled data. Finally, several efficient, parallel and parameter-free decoders are presented along with their theoretical analysis for decoding the low-rank and cluster indicators for the full data matrix. Thus, we overcome the computational limitations of the standard linear low-rank recovery methods for big datasets. Our method can also be seen as a major step towards efficient recovery of non- linear low-rank structures. For a matrix of size n X p, on a single core machine, our method gains a speed up of $p^2/k$ over Robust Principal Component Analysis (RPCA), where k << p is the subspace dimension. Numerically, we can recover a low-rank matrix of size 10304 X 1000, 100 times faster than Robust PCA