Frame Coherence and Sparse Signal Processing

The sparse signal processing literature often uses random sensing matrices to obtain performance guarantees. Unfortunately, in the real world, sensing matrices do not always come from random processes. It is therefore desirable to evaluate whether an arbitrary matrix, or frame, is suitable for sensing sparse signals. To this end, the present paper investigates two parameters that measure the coherence of a frame: worst-case and average coherence. We first provide several examples of frames that have small spectral norm, worst-case coherence, and average coherence. Next, we present a new lower bound on worst-case coherence and compare it to the Welch bound. Later, we propose an algorithm that decreases the average coherence of a frame without changing its spectral norm or worst-case coherence. Finally, we use worst-case and average coherence, as opposed to the Restricted Isometry Property, to garner near-optimal probabilistic guarantees on both sparse signal detection and reconstruction in the presence of noise. This contrasts with recent results that only guarantee noiseless signal recovery from arbitrary frames, and which further assume independence across the nonzero entries of the signal---in a sense, requiring small average coherence replaces the need for such an assumption

Support Recovery with Sparsely Sampled Free Random Matrices

Consider a Bernoulli-Gaussian complex $n$-vector whose components are $V_i = X_i B_i$, with X_i \sim \Cc\Nc(0,\Pc_x) and binary $B_i$ mutually independent and iid across $i$. This random $q$-sparse vector is multiplied by a square random matrix \Um, and a randomly chosen subset, of average size $n p$, $p \in [0,1]$, of the resulting vector components is then observed in additive Gaussian noise. We extend the scope of conventional noisy compressive sampling models where \Um is typically %A16 the identity or a matrix with iid components, to allow \Um satisfying a certain freeness condition. This class of matrices encompasses Haar matrices and other unitarily invariant matrices. We use the replica method and the decoupling principle of Guo and Verd\'u, as well as a number of information theoretic bounds, to study the input-output mutual information and the support recovery error rate in the limit of $n \to \infty$. We also extend the scope of the large deviation approach of Rangan, Fletcher and Goyal and characterize the performance of a class of estimators encompassing thresholded linear MMSE and $\ell_1$ relaxation

Optimized and accelerated 19F‐MRI of inhaled perfluoropropane to assess regional pulmonary ventilation

Purpose To accelerate 19F‐MR imaging of inhaled perfluoropropane using compressed sensing methods, and to optimize critical scan acquisition parameters for assessment of lung ventilation properties. Methods Simulations were performed to determine optimal acquisition parameters for maximal perfluoropropane signal‐to‐noise ratio (SNR) in human lungs for a spoiled gradient echo sequence. Optimized parameters were subsequently employed for 19F‐MRI of inhaled perfluoropropane in a cohort of 11 healthy participants using a 3.0 T scanner. The impact of 1.8×, 2.4×, and 3.0× undersampling ratios on 19F‐MRI acquisitions was evaluated, using both retrospective and prospective compressed sensing methods. Results 3D spoiled gradient echo 19F‐MR ventilation images were acquired at 1‐cm isotropic resolution within a single breath hold. Mean SNR was 11.7 ± 4.1 for scans acquired within a single breath hold (duration = 18 s). Acquisition of 19F‐MRI scans at shorter scan durations (4.5 s) was also demonstrated as feasible. Application of both retrospective (n = 8) and prospective (n = 3) compressed sensing methods demonstrated that 1.8× acceleration had negligible impact on qualitative image appearance, with no statistically significant change in measured lung ventilated volume. Acceleration factors of 2.4× and 3.0× resulted in increasing differences between fully sampled and undersampled datasets. Conclusion This study demonstrates methods for determining optimal acquisition parameters for 19F‐MRI of inhaled perfluoropropane and shows significant reduction in scan acquisition times (and thus participant breath hold duration) by use of compressed sensing