922 research outputs found

    Signal Recovery in Perturbed Fourier Compressed Sensing

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    In many applications in compressed sensing, the measurement matrix is a Fourier matrix, i.e., it measures the Fourier transform of the underlying signal at some specified `base' frequencies {ui}i=1M\{u_i\}_{i=1}^M, where MM is the number of measurements. However due to system calibration errors, the system may measure the Fourier transform at frequencies {ui+δi}i=1M\{u_i + \delta_i\}_{i=1}^M that are different from the base frequencies and where {δi}i=1M\{\delta_i\}_{i=1}^M are unknown. Ignoring perturbations of this nature can lead to major errors in signal recovery. In this paper, we present a simple but effective alternating minimization algorithm to recover the perturbations in the frequencies \emph{in situ} with the signal, which we assume is sparse or compressible in some known basis. In many cases, the perturbations {δi}i=1M\{\delta_i\}_{i=1}^M can be expressed in terms of a small number of unique parameters PMP \ll M. We demonstrate that in such cases, the method leads to excellent quality results that are several times better than baseline algorithms (which are based on existing off-grid methods in the recent literature on direction of arrival (DOA) estimation, modified to suit the computational problem in this paper). Our results are also robust to noise in the measurement values. We also provide theoretical results for (1) the convergence of our algorithm, and (2) the uniqueness of its solution under some restrictions.Comment: New theortical results about uniqueness and convergence now included. More challenging experiments now include

    Doctor of Philosophy

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    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
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