  The Shinnar-Le-Roux (SLR) algorithm is widely used to design frequency
selective pulses with large flip angles. We improve its design process to
generate pulses with lower energy (by as much as 26%) and more accurate phase
profiles.
  Concretely, the SLR algorithm consists of two steps: (1) an invertible
transform between frequency selective pulses and polynomial pairs that
represent Cayley-Klein (CK) parameters and (2) the design of the CK polynomial
pair to match the desired magnetization profiles. Because the CK polynomial
pair is bi-linearly coupled, the original algorithm sequentially solves for
each polynomial instead of jointly. This results in sub-optimal pulses.
  Instead, we leverage a convex relaxation technique, commonly used for low
rank matrix recovery, to address the bi-linearity. Our numerical experiments
show that the resulting pulses are almost always globally optimal in practice.
For slice excitation, the proposed algorithm results in more accurate linear
phase profiles. And in general the improved pulses have lower energy than the
original SLR pulses.
