Fast imaging applications in magnetic resonance imaging (MRI) frequently involve undersampling of k-space data to achieve the desired temporal resolution. However, high temporal resolution images generated from undersampled data suffer from aliasing artifacts. In radial k-space sampling, this manifests as undesirable streaks that obscure image detail. Compressed sensing reconstruction has been shown to reduce such streak artifacts, based on the assumption of image sparsity. Here, compressed sensing is implemented with three different radial sampling schemes (golden-angle, bit-reversed, and random sampling), which are compared over a range of spatiotemporal resolutions. The sampling methods are implemented in static scenarios where different undersampling patterns could be compared. Results from point spread function studies, simulations, phantom and in vivo experiments show that the choice of radial sampling pattern influences the quality of the final image reconstructed by the compressed sensing algorithm. While evenly undersampled radial trajectories are best for specific temporal resolutions, golden-angle radial sampling results in the least overall error when various temporal resolutions are considered. Reduced temporal fluctuations from aliasing artifacts in golden-angle sampling translates to improved compressed sensing reconstructions overall. Magn Reson Med 67:363-377,

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The North Atlantic Treaty Organization (NATO) Stockpile Planning Committee (SPC) periodically determines if NATO member nations have the necessary munitions for a full range of mission types, accomplished through the use of a model that minimizes the cost of the required stockpile. We were tasked to examine how the methodology of this model could be modified to allow individual nations to better determine their requirements for Precision-Guided Munitions (PGMs). The approach we undertook involves augmenting the methodology of the model with a multi-objective optimization approach using a genetic algorithm, in which the solution is optimized along two competing objectives: total cost (which is minimized), and the usage of PGMs (which is maximized). We recommended that the SPC consider including this change in all future versions of ACROSS.

The Frank-Wolfe (FW) algorithm has been widely used in solving nuclear norm constrained problems, since it does not require projections. However, FW often yields high rank intermediate iterates, which can be very expensive in time and space costs for large problems. To address this issue, we propose a rank-drop method for nuclear norm constrained problems. The goal is to generate descent steps that lead to rank decreases, maintaining low-rank solutions throughout the algorithm. Moreover, the optimization problems are constrained to ensure that the rank-drop step is also feasible and can be readily incorporated into a projection-free minimization method, e.g., FW. We demonstrate that by incorporating rank-drop steps into the FW algorithm, the rank of the solution is greatly reduced compared to the original FW or its common variants.

Frank-Wolfe methods (FW) have gained significant interest in the machine learning community due to its ability to efficiently solve large problems that admit a sparse structure (e.g. sparse vectors and low-rank matrices). However the performance of the existing FW method hinges on the quality of the linear approximation. This typically restricts FW to smooth functions for which the approximation quality, indicated by a global curvature measure, is reasonably good. In this paper, we propose a modified FW algorithm amenable to nonsmooth functions by optimizing for approximation quality over all affine functions given a neighborhood of interest. We analyze theoretical properties of the proposed algorithm and demonstrate that it overcomes many issues associated with existing methods in the context of nonsmooth low-rank matrix estimation.