Reducing acquisition time is a crucial challenge for many imaging techniques. Compressed sensing (CS) theory offers an appealing framework to address this issue since it provides theoretical guarantees on the reconstruction of sparse signals by projection on a low-dimensional linear subspace. In this paper, we focus on a setting where the imaging device allows us to sense a fixed set of measurements. We first discuss the choice of an optimal sampling subspace allowing perfect reconstruction of sparse signals. Its design relies on the random drawing of independent measurements. We discuss how to select the drawing distribution and show that a mixed strategy involving partial deterministic sampling and independent drawings can help in breaking the so-called coherence barrier. Unfortunately, independent random sampling is irrelevant for many acquisition devices owing to acquisition constraints. To overcome this limitation, the notion of a variable density sampler (VDS) is introduced and defined as a stochastic process with a prescribed limit empirical measure. It encompasses samplers based on independent measurements or continuous curves. The latter are crucial to extend CS results to actual applications. We propose two original approaches to designing a continuous VDS, one based on random walks over the acquisition space and one based on the travelling salesman problem. Following theoretical considerations and retrospective CS simulations in magnetic resonance imaging, we intend to highlight the key properties of a VDS to ensure accurate sparse reconstructions, namely its limit empirical measure and its mixing time.

Purpose To present a new optimition‐driven design of optimal k‐space trajectories in the context of compressed sensing: Spreading Projection Algorithm for Rapid K‐space sampLING (SPARKLING). TheoryThe SPARKLING algorithm is a versatile method inspired from stippling techniques that automatically generates optimized sampling patterns compatible with MR hardware constraints on maximum gradient amplitude and slew rate. These non‐Cartesian sampling curves are designed to comply with key criteria for optimal sampling: a controlled distribution of samples (e.g., variable density) and a locally uniform k‐space coverage. MethodsEx vivo and in vivo prospective T2*‐weighted acquisitions were performed on a 7‐Tesla scanner using the SPARKLING trajectories for various setups and target densities. Our method was compared to radial and variable‐density spiral trajectories for high‐resolution imaging. ResultsCombining sampling efficiency with compressed sensing, the proposed sampling patterns allowed up to 20‐fold reductions in MR scan time (compared to fully sampled Cartesian acquisitions) for two‐dimensional T2*‐weighted imaging without deterioration of image quality, as demonstrated by our experimental results at 7 Tesla on in vivo human brains for a high in‐plane resolution of 390 μm. In comparison to existing non‐Cartesian sampling strategies, the proposed technique also yielded superior image quality. ConclusionsThe proposed optimization‐driven design of k‐space trajectories is a versatile framework that is able to enhance MR sampling performance in the context of compressed sensing.