### Temporal Regularization Use in Dynamic Contrast-Enhanced MRI

, 2011

"... I am incredibly grateful to so many people for all of the love, support, and guidance I have received over the years. Of course, I’d like to thank my advisor, Jeff Fessler, who, for better or for worse, convinced me to go to grad school in the first place. He inspired me as an undergrad, and he insp ..."

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I am incredibly grateful to so many people for all of the love, support, and guidance I have received over the years. Of course, I’d like to thank my advisor, Jeff Fessler, who, for better or for worse, convinced me to go to grad school in the first place. He inspired me as an undergrad, and he inspires me still today. I think it’s safe to say that this dissertation wouldn’t exist without Jeff’s endless guidance, patience, and support. Thank you. I also want to thank the other members of my committee: Doug Noll, who first taught me the nuts and bolts of MRI, and whose group meetings and group members were an important part of my graduate education; Tom Chenevert, whose insights and knowledge regarding “real life ” DCE-MRI were invaluable to this research project; and

### Submission to: Magnetic Resonance in Medicine Correspondence to:

"... Running head: Auto-calibration parallel imaging reconstruction method without calibration data A calibrationless parallel imaging reconstruction method, termed simultaneous autocalibrating and k-space estimation (SAKÉ), is presented. It is a data-driven, coil-by-coil reconstruction method that does ..."

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Running head: Auto-calibration parallel imaging reconstruction method without calibration data A calibrationless parallel imaging reconstruction method, termed simultaneous autocalibrating and k-space estimation (SAKÉ), is presented. It is a data-driven, coil-by-coil reconstruction method that does not require fully sampled calibrating signals. In SAKÉ, an under-sampled multi-channel dataset is structured into a single matrix and data reconstruction is formulated as a structured low-rank matrix completion problem. An iterative solution that implements a projection-onto-sets algorithm with singular value hard-thresholding is described. Reconstruction results are demonstrated for undersampled, multi-channel Cartesian and non-Cartesian data with no calibration data. These exhibit excellent image quality comparable to those obtained with calibration data. Finally, improved image quality is demonstrated by combining SAKÉ with waveletbased compressed sensing. This method could benefit MR applications where acquiring accurate calibration data is limiting or not possible at all.

### A Rapid and Robust Numerical Algorithm for Sensitivity Encoding with Sparsity Constraints: Self-Feeding Sparse SENSE

"... The method of enforcing sparsity during magnetic resonance imaging reconstruction has been successfully applied to partially parallel imaging (PPI) techniques to reduce noise and artifact levels and hence to achieve even higher acceleration factors. However, there are two major problems in the exist ..."

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The method of enforcing sparsity during magnetic resonance imaging reconstruction has been successfully applied to partially parallel imaging (PPI) techniques to reduce noise and artifact levels and hence to achieve even higher acceleration factors. However, there are two major problems in the existing sparsity-constrained PPI techniques: speed and robustness. By introducing an auxiliary variable and decomposing the original minimization problem into two subproblems that are much easier to solve, a fast and robust numerical algorithm for sparsity-constrained PPI technique is developed in this work. The specific implementation for a conventional Cartesian trajectory data set is named self-feeding Sparse Sensitivity Encoding (SENSE). The computational cost for the proposed method is two conventional SENSE reconstructions plus one spatially adaptive image denoising procedure. With reconstruction time approximately doubled, images with a much lower root mean square error (RMSE) can be achieved at high acceleration factors. Using a standard eight-channel head coil, a net acceleration factor of 5 along one dimension can be achieved with low RMSE. Furthermore, the algorithm is insensitive to the choice of parameters. This work improves the clinical applicability of SENSE at high acceleration factors. Key words: partially parallel imaging; g-factor; sparsity constraint; prior information; compressed sensing; numerical algorithm 1

### On Theoretical Limits in Parallel Magnetic Resonance Imaging

, 2008

"... Abstract—Based on a Fourier series expression of true image, receiver sensitivities, and measurements, it is possible to give theoretical limits for the perfect reconstructability of image and sensitivities in parallel magnetic resonance imaging. These limits depend on the smoothness of the sensitiv ..."

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Abstract—Based on a Fourier series expression of true image, receiver sensitivities, and measurements, it is possible to give theoretical limits for the perfect reconstructability of image and sensitivities in parallel magnetic resonance imaging. These limits depend on the smoothness of the sensitivities, number of receiver coils, and size of the acquired k-space measurement window. Different types of a priori information can be incorporated in the determination of these limits. Furthermore, the method employed is constructive and can serve as the basis for a nonlinear reconstruction scheme, as is shown using data from a simulated phantom. Index Terms—Magnetic resonance imaging, Image reconstruction, Fourier series, Newton method I.

### Magnetic Resonance in Medicine 61:145–152 (2009) Regularized Sensitivity Encoding (SENSE) Reconstruction Using Bregman Iterations

"... In parallel imaging, the signal-to-noise ratio (SNR) of sensitivity encoding (SENSE) reconstruction is usually degraded by the ill-conditioning problem, which becomes especially serious at large acceleration factors. Existing regularization methods have been shown to alleviate the problem. However, ..."

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In parallel imaging, the signal-to-noise ratio (SNR) of sensitivity encoding (SENSE) reconstruction is usually degraded by the ill-conditioning problem, which becomes especially serious at large acceleration factors. Existing regularization methods have been shown to alleviate the problem. However, they usually suffer from image artifacts at high acceleration factors due to the large data inconsistency resulting from heavy regularization. In this paper, we propose Bregman iteration for SENSE regularization. Unlike the existing regularization methods where the regularization function is fixed, the method adaptively updates the regularization function using the Bregman distance at different iterations, such that the iteration gradually removes the aliasing artifacts and recovers fine structures before the noise finally comes back. With a discrepancy principle as the stopping criterion, our results demonstrate that the reconstructed image using Bregman iteration preserves both sharp edges lost in Tikhonov regularization and fines structures missed in total variation (TV) regularization, while reducing more noise and aliasing artifacts. Magn Reson Med 61:

### Parallel Magnetic Resonance Imaging as Approximation in a Reproducing Kernel Hilbert Space

, 2014

"... In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more ..."

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In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. To understand and design k-space sampling patterns, a theoretical framework is needed to analyze how well arbitrary sam-pling patterns reconstruct unsampled k-space using receive coil information. As shown here, reconstruction from samples at arbitrary locations can be un-derstood as approximation of vector-valued functions from the acquired sam-ples and formulated using a Reproducing Kernel Hilbert Space (RKHS) with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between approximation theory and paral-lel imaging. Theoretical tools from approximation theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional image-domain g-factor noise anal-ysis to both noise amplification and approximation errors in k-space. This is demonstrated with numerical examples.

### 1On Theoretical Limits in Parallel Magnetic Resonance Imaging

"... Abstract—Based on a Fourier series expression of true image, receiver sensitivities, and measurements, it is possible to give theoretical limits for the perfect reconstructability of image and sensitivities in parallel magnetic resonance imaging. These limits depend on the smoothness of the sensitiv ..."

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Abstract—Based on a Fourier series expression of true image, receiver sensitivities, and measurements, it is possible to give theoretical limits for the perfect reconstructability of image and sensitivities in parallel magnetic resonance imaging. These limits depend on the smoothness of the sensitivities, number of receiver coils, and size of the acquired k-space measurement window. Different types of a priori information can be incorporated in the determination of these limits. Fur-thermore, the method employed is constructive and can serve as the basis for a nonlinear reconstruction scheme, as is shown using data from a simulated phantom. Index Terms—Magnetic resonance imaging, Image reconstruction, Fourier series, Newton method

### IMAGE RECONSTRUCTION FROM PHASED-ARRAY MRI DATA BASED ON MULTICHANNEL BLIND DECONVOLUTION

"... In this paper we consider image reconstruction from multichannel phased array MRI data without prior knowledge of the coil sensitivity functions. A new framework based on multichannel blind deconvolution (MBD) is developed for joint estimation of the image function and the sensitivity functions in k ..."

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In this paper we consider image reconstruction from multichannel phased array MRI data without prior knowledge of the coil sensitivity functions. A new framework based on multichannel blind deconvolution (MBD) is developed for joint estimation of the image function and the sensitivity functions in k-space. By exploiting the smoothness of the estimated functions in the spatial domain, we develop a regularization approach in conjunction with MBD to obtain good reconstruction of the image function. Experimental results using simulated and real data demonstrate that the proposed reconstruction algorithm can better removes the sensitivity weighting in the reconstructed images compared to the sum-of-squares (SoS) approach. Index Terms — Phased array MRI, multichannel deconvolution, regularization, image restoration

### Automatic High-Bandwidth Calibration and Reconstruction of Arbitrarily Sampled Parallel MRI

, 2014

"... Today, many MRI reconstruction techniques exist for undersampled MRI data. Regularization-based techniques inspired by compressed sensing allow for the reconstruction of undersampled data that would lead to an ill-posed reconstruction problem. Parallel imaging enables the reconstruction of MRI image ..."

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Today, many MRI reconstruction techniques exist for undersampled MRI data. Regularization-based techniques inspired by compressed sensing allow for the reconstruction of undersampled data that would lead to an ill-posed reconstruction problem. Parallel imaging enables the reconstruction of MRI images from undersampled multi-coil data that leads to a well-posed reconstruction problem. Autocalibrating pMRI techniques encompass pMRI techniques where no explicit knowledge of the coil sensivities is required. A first purpose of this paper is to derive a novel autocalibration approach for pMRI that allows for the estimation and use of smooth, but high-bandwidth coil profiles instead of a compactly supported kernel. These high-bandwidth models adhere more accurately to the physics of an antenna system. The second purpose of this paper is to demonstrate the feasibility of a parameter-free reconstruction algorithm that combines autocalibrating pMRI and compressed sensing. Therefore, we present several techniques for automatic parameter estimation in MRI reconstruction. Experiments show that a higher reconstruction accuracy can be had using high-bandwidth coil models and that the automatic parameter choices yield an acceptable result.