Recovery of Damped Exponentials Using Structured Low Rank Matrix Completion

POSE: POSition Encoding for accelerated quantitative MRI

Quantitative MRI utilizes multiple acquisitions with varying sequence parameters to sufficiently characterize a biophysical model of interest, resulting in undesirable scan times. Here we propose, validate and demonstrate a new general strategy for accelerating MRI using subvoxel shifting as a source of encoding called POSition Encoding (POSE). The POSE framework applies unique subvoxel shifts along the acquisition parameter dimension, thereby creating an extra source of encoding. Combining with a biophysical signal model of interest, accelerated and enhanced resolution maps of biophysical parameters are obtained. This has been validated and demonstrated through numerical Bloch equation simulations, phantom experiments and in vivo experiments using the variable flip angle signal model in 3D acquisitions as an application example. Monte Carlo simulations were performed using in vivo data to investigate our method's noise performance. POSE quantification results from numerical Bloch equation simulations of both a numerical phantom and realistic digital brain phantom concur well with the reference method, validating our method both theoretically and for realistic situations. NIST phantom experiment results show excellent overall agreement with the reference method, confirming our method's applicability for a wide range of T1 values. In vivo results not only exhibit good agreement with the reference method, but also show g-factors that significantly outperforms conventional parallel imaging methods with identical acceleration. Furthermore, our results show that POSE can be combined with parallel imaging to further accelerate while maintaining superior noise performance over parallel imaging that uses lower acceleration factors.

Reducing the Effects of Motion Artifacts in fMRI: A Structured Matrix Completion Approach

Functional MRI (fMRI) is widely used to study the functional organization of normal and pathological brains. However, the fMRI signal may be contaminated by subject motion artifacts that are only partially mitigated by motion correction strategies. These artifacts lead to distance-dependent biases in the inferred signal correlations. To mitigate these spurious effects, motion-corrupted volumes are censored from fMRI time series. Censoring can result in discontinuities in the fMRI signal, which may lead to substantial alterations in functional connectivity analysis. We propose a new approach to recover the missing entries from censoring based on structured low rank matrix completion. We formulated the artifact-reduction problem as the recovery of a super-resolved matrix from unprocessed fMRI measurements. We enforced a low rank prior on a large structured matrix, formed from the samples of the time series, to recover the missing entries. The recovered time series, in addition to being motion compensated, are also slice-time corrected at a fine temporal resolution. To achieve a fast and memory-efficient solution for our proposed optimization problem, we employed a variable splitting strategy. We validated the algorithm with simulations, data acquired under different motion conditions, and datasets from the ABCD study. Functional connectivity analysis showed that the proposed reconstruction resulted in connectivity matrices with lower errors in pair-wise correlation than non-censored and censored time series based on a standard processing pipeline. In addition, seed-based correlation analyses showed improved delineation of the default mode network. These demonstrate that the method can effectively reduce the adverse effects of motion in fMRI analysis.

Improved MUSSELS reconstruction for high-resolution multi-shot diffusion weighted imaging

PURPOSE

MUSSELS is a one-step iterative reconstruction method for multishot diffusion weighted (msDW) imaging. The current work presents an efficient implementation, termed IRLS MUSSELS, that enables faster reconstruction to enhance its utility for high-resolution diffusion MRI studies.

METHODS

The recently proposed MUSSELS reconstruction belongs to a new class of parallel imaging-based methods that recover artifact-free DWIs from msDW data without needing phase compensation. The reconstruction is achieved via structured low-rank matrix completion algorithms, which are computationally demanding due to the large size of the Hankel matrices and their associated computations involving singular value decompositions. Because of this, computational demands of the MUSSELS reconstruction scales as the matrix size and the number of shots increases, which hinders its practical utility for high-resolution applications. In this work, we derive a computationally efficient MUSSELS formulation by modifying the iterative reweighted least squares (IRLS) method that were proposed earlier to solve such problems. Using whole-brain in vivo data, we show the utility of the IRLS MUSSELS for routine high-resolution studies with reduced computational burden.

RESULTS

IRLS MUSSELS provides about five times faster reconstruction for matrix sizes 192 × 192 and 256 × 256 compared to the earlier MUSSELS implementation. The widely employed conjugate symmetry priors can also be incorporated into IRLS MUSSELS to reduce blurring of the partial Fourier acquisitions, without incurring much computational burden.

CONCLUSIONS

The proposed method is observed to be computationally efficient to enable routine high-resolution studies. The computational complexity matches the traditional msDWI reconstruction methods and provides improved reconstruction results with the additional constraints.

MULTI-ECHO RECOVERY WITH FIELD INHOMOGENEITY COMPENSATION USING STRUCTURED LOW-RANK MATRIX COMPLETION

Echo-planar imaging (EPI), which is the main workhorse of functional MRI, suffers from field inhomogeneity-induced geometric distortions. The amount of distortion is proportional to the readout duration, which restricts the maximum achievable spatial resolution. The spatially varying nature of the T 2 * decay makes it challenging for EPI schemes with a single echo time to obtain good sensitivity to functional activations in different brain regions. Despite the use of parallel MRI and multislice acceleration, the number of different echo times that can be acquired in a reasonable TR is limited. The main focus of this work is to introduce a rosette-based acquisition scheme and a structured low-rank reconstruction algorithm to overcome the above challenges. The proposed scheme exploits the exponential structure of the time series to recover distortion-free images from several echoes simultaneously.

Reconstruction of Binary Shapes from Blurred Images via Hankel-structured Low-rank Matrix Recovery

With the dominance of digital imaging systems, we are often dealing with discrete-domain samples of an analog image. Due to physical limitations, all imaging devices apply a blurring kernel on the input image before taking samples to form the output pixels. In this paper, we focus on the reconstruction of binary shape images from few blurred samples. This problem has applications in medical imaging, shape processing, and image segmentation. Our method relies on representing the analog shape image in a discrete grid much finer than the sampling grid. We formulate the problem as the recovery of a rank r matrix that is formed by a Hankel structure on the pixels. We further propose efficient ADMM-based algorithms to recover the low-rank matrix in both noiseless and noisy settings. We also analytically investigate the number of required samples for successful recovery in the noiseless case. For this purpose, we study the problem in the random sampling framework, and show that with O(r log4(n1n2)) random samples (where the size of the image is assumed to be n1 x n2) we can guarantee the perfect reconstruction with high probability under mild conditions. We further prove the robustness of the proposed recovery in the noisy setting by showing that the reconstruction error in the noisy case is bounded when the input noise is bounded. Simulation results confirm that our proposed method outperform the conventional total variation minimization in the noiseless settings.

Calibration-Free B0 Correction of EPI Data Using Structured Low Rank Matrix Recovery

We introduce a structured low rank algorithm for the calibration-free compensation of field inhomogeneity artifacts in echo planar imaging (EPI) MRI data. We acquire the data using two EPI readouts that differ in echo-time. Using time segmentation, we reformulate the field inhomogeneity compensation problem as the recovery of an image time series from highly undersampled Fourier measurements. The temporal profile at each pixel is modeled as a single exponential, which is exploited to fill in the missing entries. We show that the exponential behavior at each pixel, along with the spatial smoothness of the exponential parameters, can be exploited to derive a 3-D annihilation relation in the Fourier domain. This relation translates to a low rank property on a structured multi-fold Toeplitz matrix, whose entries correspond to the measured k-space samples. We introduce a fast two-step algorithm for the completion of the Toeplitz matrix from the available samples. In the first step, we estimate the null space vectors of the Toeplitz matrix using only its fully sampled rows. The null space is then used to estimate the signal subspace, which facilitates the efficient recovery of the time series of images. We finally demonstrate the proposed approach on spherical MR phantom data and human data and show that the artifacts are significantly reduced.

RECOVERY OF NOISY POINTS ON BANDLIMITED SURFACES: KERNEL METHODS RE-EXPLAINED

We introduce a continuous domain framework for the recovery of points on a surface in high dimensional space, represented as the zero-level set of a bandlimited function. We show that the exponential maps of the points on the surface satisfy annihilation relations, implying that they lie in a finite dimensional subspace. The subspace properties are used to derive sampling conditions, which will guarantee the perfect recovery of the surface from finite number of points. We rely on nuclear norm minimization to exploit the low-rank structure of the maps to recover the points from noisy measurements. Since the direct estimation of the surface is computationally prohibitive in very high dimensions, we propose an iterative reweighted algorithm using the "kernel trick". The iterative algorithm reveals deep links to Laplacian based algorithms widely used in graph signal processing; the theory and the sampling conditions can serve as a basis for discrete-continuous domain processing of signals on a graph.

DENOISING AND DEINTERLEAVING OF EPSI DATA USING STRUCTURED LOW-RANK MATRIX RECOVERY

Echo-planar spectroscopic imaging (EPSI) sequence with spectrally interleaving is often used to rapidly collect metabolic MRI data. The main problem in using it on high field scanners is the presence of spurious peaks resulting from phase distortions between interleaves as well as the low signal to noise ratio. We introduce a novel structured low-rank framework for the simultaneous denoising and deinterleaving of spectrally interleaved EPSI data. The proposed algorithm exploits annihilation relations resulting from the linear predicability of exponential signals as well as due to uncorrected phase relations between interleaves. The algorithm is formulated as a structured nuclear norm minimization of a block Hankel matrix, derived from the interleaves. Experiments using hyperpolarized 13 C mouse kidney EPSI data demonstrate the ability of the algorithm to remove ghost peaks from EPSI data collected using bipolar readout gradients.