  Since the 1970s, much of traditional interferometric imaging has been built
around variations of the CLEAN algorithm, in both terminology, methodology, and
algorithm development. Recent developments in applying new algorithms from
convex optimization to interferometry has allowed old concepts to be viewed
from a new perspective, ranging from image restoration to the development of
computationally distributed algorithms. We present how this has ultimately led
the authors to new perspectives in wide-field imaging, allowing for the first
full individual non-coplanar corrections applied during imaging over extremely
wide-fields of view for the Murchison Widefield Array (MWA) telescope.
Furthermore, this same mathematical framework has provided a novel
understanding of wide-band polarimetry at low frequencies, where instrumental
channel depolarization can be corrected through the new
$\delta\lambda^2$-projection algorithm. This is a demonstration that new
algorithm development outside of traditional radio astronomy is valuable for
the new theoretical and practical perspectives gained. These perspectives are
timely with the next generation of radio telescopes coming online.
