  We introduce a new algorithm for complex image reconstruction with separate
regularization of the image magnitude and phase. This optimization problem is
interesting in many different image reconstruction contexts, although is
nonconvex and can be difficult to solve. In this work, we first describe a
novel implementation of the previous proximal alternating linearized minization
(PALM) algorithm to solve this optimization problem. We then make enhancements
to PALM, leading to a new algorithm named PALMNUT that combines the PALM
together with Nesterov's momentum and a novel approach that relies on uncoupled
coordinatewise step sizes derived from coordinatewise Lipschitz-like bounds.
Theoretically, we establish that a version of PALMNUT (without Nesterov's
momentum) monotonically decreases the objective function, leading to guaranteed
convergence in many cases of interest. Empirical results obtained in the
context of magnetic resonance imaging demonstrate that PALMNUT has
computational advantages over common existing approaches like alternating
minimization. Although our focus is on the application to separate magnitude
and phase regularization, we expect that the same approach may also be useful
in other nonconvex optimization problems with similar objective function
structure.
