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In the area of magnetic resonance imaging (MRI), an extensive range of non-linear reconstruction algorithms has been proposed which can be used with general Fourier subsampling patterns. However, the design of these subsampling patterns has typically been considered in isolation from the reconstruction rule and the anatomy under consideration. In this paper, we propose a learning-based framework for optimizing MRI subsampling patterns for a specific reconstruction rule and anatomy, considering both the noiseless and noisy settings. Our learning algorithm has access to a representative set of training signals, and searches for a sampling pattern that performs well on average for the signals in this set. We present a novel parameter-free greedy mask selection method and show it to be effective for a variety of reconstruction rules and performance metrics. Moreover, we also support our numerical findings by providing a rigorous justification of our framework via statistical learning theory. [ABSTRACT FROM AUTHOR]

The problem of recovering a structured signal \textx \in \mathbb C^p from a set of dimensionality-reduced linear measurements \textb = \textAx arises in a variety of applications, such as medical imaging, spectroscopy, Fourier optics, and computerized tomography. Due to computational and storage complexity or physical constraints imposed by the problem, the measurement matrix \textA \in \mathbb C^n \times p is often of the form \textA = \textP\Omega{{\Psi}} for some orthonormal basis matrix ${{\Psi}}\in {\mathbb C}^{p \times p} and subsampling operator \textP\Omega: {\mathbb C}^{p} \rightarrow {\mathbb C}^{n} that selects the rows indexed by \Omega. This raises the fundamental question of how best to choose the index set \Omega in order to optimize the recovery performance. Previous approaches to addressing this question rely on nonuniform random subsampling using application-specific knowledge of the structure of \text{x}. In this paper, we instead take a principled learning-based approach in which a fixed index set is chosen based on a set of training signals \text{x}_1,\ldots,\text{x}_m$. We formulate combinatorial optimization problems seeking to maximize the energy captured in these signals in an average-case or worst-case sense, and we show that these can be efficiently solved either exactly or approximately via the identification of modularity and submodularity structures. We provide both deterministic and statistical theoretical guarantees showing how the resulting measurement matrices perform on signals differing from the training signals, and we provide numerical examples showing our approach to be effective on a variety of data sets. [ABSTRACT FROM PUBLISHER]