Sparsity in MRI RF excitation pulse design

Magnetic resonance imaging (MRI) may be viewed as a two-stage experiment that yields a non-invasive spatial mapping of hydrogen nuclei in living subjects. Nuclear spins within a subject are first excited using a radio-frequency (RF) excitation pulse and proportions of excited spins are then detected using a resonant coil; images are then reconstructed from this data. Excitation pulses need to be tailored to a user\´s specific needs and in most applications need to be as short as possible, due to spin relaxation, tissue heating, signal-to-noise ratio (SNR), and data readout limitations. The design of short-duration excitation pulses is an important topic and the focus of our work. One may show that RF excitation pulse design, under certain conditions, involves choosing to deposit energy in a continuous, 3-D, Fourier-like domain ("excitation k-space") in order to form some desired excitation in the spatial domain. Energy may only be deposited along a 1-D contour, and there are limitations on where and how it may be placed; the most important fact is that excitation pulse duration directly corresponds to the length of the chosen contour and the rate it is traversed. The problem then is to find a sparse "trajectory" (and corresponding energy deposition) within this k-space such that a high-fidelity version of the desired excitation is formed in the spatial domain. We show how sparsity and simultaneous sparsity are applicable to 2-D and 3-D excitation pulse design and present a novel instance where simultaneous sparsity is desirable. We then discuss how to apply sparse approximation concepts to produce RF pulses. These "sparsity-enforced" designs, generated via convex relaxation techniques, significantly outperform conventional pulses: for fixed pulse duration, sparsity-enforced pulses always produce higher-fidelity excitations.