Dalca, and Mert R. Sabuncu In compressed sensing MRI, k-space measurements are under-sampled to achieve accelerated scan times. There are two fundamental problems in compressed sensing MRI: (1) where to sample and (2) how to reconstruct. In this paper, we tackle both problems simultaneously, using a novel unsupervised, end-to-end learning framework, called LOUPE. Our method trains a neural network model on a set of full-resolution MRI scans, which are retrospectively under-sampled and forwarded to an antialiasing model that computes a reconstruction, which is in turn compared with the input. In our experiments, we demonstrate that LOUPEoptimized under-sampling masks are data-dependent, varying significantly with the imaged anatomy, and perform well with different reconstruction methods. We present empirical results obtained with a large-scale, publicly available knee MRI dataset, where LOUPE offered the most superior reconstruction quality across different conditions. Even with an aggressive 8-fold acceleration rate, LOUPE's reconstructions contained much of the anatomical detail that was missed by alternative masks and reconstruction methods. Our experiments also show how LOUPE yielded optimal under-sampling patterns that were significantly different for brain vs knee MRI scans. I NTRODUCTION M AGNETIC Resonance Imaging (MRI) is a ubiquitous, noninvasive, and versatile biomedical imaging technology. A central challenge in MRI is long scan times, which constrains accessibility and increases costs. One remedy is to accelerate MRI via compressed sensing [1], [2]. In compressed sensing MRI, k-space data (i.e., the Fourier transform of the image) is sampled below the Nyquist-Shannon rate [1], which is often referred to as "under-sampling." Given an under-sampled set of measurements, the objective is to "reconstruct" the full-resolution MRI.

Acquisition of Magnetic Resonance Imaging (MRI) scans can be accelerated by under-sampling in k-space (i.e., the Fourier domain). In this paper, we consider the problem of optimizing the sub-sampling pattern in a data-driven fashion. Since the reconstruction model's performance depends on the sub-sampling pattern, we combine the two problems. For a given sparsity constraint, our method optimizes the sub-sampling pattern and reconstruction model, using an end-to-end learning strategy. Our algorithm learns from full-resolution data that are under-sampled retrospectively, yielding a sub-sampling pattern and reconstruction model that are customized to the type of images represented in the training data. The proposed method, which we call LOUPE (Learning-based Optimization of the Under-sampling PattErn), was implemented by modifying a U-Net, a widely-used convolutional neural network architecture, that we append with the forward model that encodes the under-sampling process. Our experiments with T1-weighted structural brain MRI scans show that the optimized sub-sampling pattern can yield significantly more accurate reconstructions compared to standard random uniform, variable density or equispaced under-sampling schemes.