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

Accelerating MRI Data Acquisition Using Parallel Imaging and Compressed Sensing

Magnetic Resonance Imaging (MRI) scanners are one of important medical instruments, which can achieve more information of soft issues in human body than other medical instruments, such as Ultrasound, Computed Tomography (CT), Single Photon Emission Computed Tomography (SPECT), Positron Emission Tomography (PET), etc. But MRI\u27s scanning is slow for patience of doctors and patients. In this dissertation, the author proposes some methods of parallel imaging and compressed sensing to accelerate MRI data acquisition. Firstly, a method is proposed to improve the conventional GRAPPA using cross-sampled auto-calibration data. This method use cross-sampled auto-calibration data instead of the conventional parallel-sampled auto-calibration data to estimate the linear kernel model of the conventional GRAPPA. The simulations and experiments show that the cross-sampled GRAPPA can decrease the quantity of ACS lines and reduce the aliasing artifacts comparing to the conventional GRAPPA under same reduction factors. Secondly, a Hybrid encoding method is proposed to accelerate the MRI data acquisition using compressed sensing. This method completely changes the conventional Fourier encoding into Hybrid encoding, which combines the benefits of Fourier and Circulant random encoding, under 2D and 3D situation, through the proposed special hybrid encoding pulse sequences. The simulations and experiments illustrate that the images can be reconstructed by the proposed Hybrid encoding method to reserve more details and resolutions than the conventional Fourier encoding method. Thirdly, a pseudo 2D random sampling method is proposed by dynamically swapping the gradients of x and y axes on pulse sequences, which can be implemented physically as the convention 1D random sampling method. The simulations show that the proposed method can reserve more details than the convention 1D random sampling method. These methods can recover images to achieve better qualities under same situations than the conventional methods. Using these methods, the MRI data acquisitions can be accelerated comparing to the conventional methods

Advanced parallel magnetic resonance imaging methods with applications to MR spectroscopic imaging

Parallel magnetic resonance imaging offers a framework for acceleration of conventional MRI encoding using an array of receiver coils with spatially-varying sensitivities. Novel encoding and reconstruction techniques for parallel MRI are investigated in this dissertation. The main goal is to improve the actual reconstruction methods and to develop new approaches for massively parallel MRI systems that take advantage of the higher information content provided by the large number of small receivers. A generalized forward model and inverse reconstruction with regularization for parallel MRI with arbitrary k-space sub-sampling is developed. Regularization methods using the singular value decomposition of the encoding matrix and pre-conditioning of the forward model are proposed to desensitize the solution from data noise and model errors. Variable density k-space sub-sampling is presented to improve the reconstruction with the common uniform sub-sampling. A novel method for massively parallel MRI systems named Superresolution Sensitivity Encoding (SURE-SENSE) is proposed where acceleration is performed by acquiring the low spatial resolution representation of the object being imaged and the stronger sensitivity variation from small receiver coils is used to perform intra-pixel reconstruction. SURE-SENSE compares favorably the performance of standard SENSE reconstruction for low spatial resolution imaging such as spectroscopic imaging. The methods developed in this dissertation are applied to Proton Echo Planar Spectroscopic Imaging (PEPSI) for metabolic imaging in human brain with high spatial and spectral resolution in clinically feasible acquisition times. The contributions presented in this dissertation are expected to provide methods that substantially enhance the utility of parallel MRI for clinical research and to offer a framework for fast MRSI of human brain with high spatial and spectral resolution

Doctor of Philosophy

dissertationMagnetic Resonance Imaging (MRI) is one of the most important medical imaging technologies in use today. Unlike other imaging tools, such as X-ray imaging or computed tomography (CT), MRI is noninvasive and without ionizing radiation. A major limitation of MRI, however, is its relatively low imaging speed and low spatial-temporal resolution, as in the case of dynamic contrast enhanced magnetic resonance imaging (DCE-MRI). These hinder the clinical use of MRI. In this thesis, we aim to develop novel signal processing techniques to improve the imaging quality and reduce the imaging time of MRI. This thesis consists of two parts, corresponding to our work on parallel MRI and dynamic MRI, respectively. In the first part, we address an important problem in parallel MRI that the coil sensitivities functions are not known exactly and the estimation error often leads to artifacts in the reconstructed image. First, we develop a new framework based on multichannel blind deconvolution (MBD) to jointly estimate the image and the sensitivity functions. For fully sampled MRI, the proposed approach yields more uniform image reconstructions than that of the sum-of-squares (SOS) and other existing methods. Second, we extend this framework to undersampled parallel MRI and develop a new algorithm, termed Sparse BLIP, for blind iterative parallel image reconstruction using compressed sensing (CS). Sparse BLIP reconstructs both the sensitivity functions and the image simultaneously from the undersampled data, while enforcing the sparseness constraint in the image and sensitivities. Superior image constructions can be obtained by Sparse BLIP when compared to other state-of-the-art methods. In the second part of the thesis, we study highly accelerated DCE-MRI and provide a comparative study of the temporal constraint reconstruction (TCR) versus model-based reconstruction. We find that, at high reduction factors, the choice of baseline image greatly affects the convergence of TCR and the improved TCR algorithm with the proposed baseline initialization can achieve good performance without much loss of temporal or spatial resolution for a high reduction factor of 30. The model-based approach, on the other hand, performs inferior to TCR with even the best phase initialization