  Phase unwrapping is the process of recovering a continuous phase signal from
an original signal wrapped in the ($-\pi$,$\pi$] interval. It is a critical
step of coherent signal processing, with applications such as synthetic
aperture radar, acoustic imaging, magnetic resonance, X-ray crystallography,
and seismic processing. In the field of computational optics, this problem is
classically treated as a norm-minimization problem, in which one seeks to
minimize the differences between the gradients of the original wrapped signal
and those of the continuous unwrapped signal. When the L0-norm is considered,
the number of differences should be minimized, leading to a difficult
combinatorial optimization problem. We propose an approximate model for the
L0-norm phase unwrapping problem in 2D, in which the singularities of the
wrapped phase image are associated with a graph where the vertices have $-1$ or
$+1$ polarities. The objective is to find a minimum-cost balanced spanning
forest where the sum of the polarities is equal to zero in each tree. We
introduce a set of primal and dual heuristics, a branch-and-cut algorithm, and
a hybrid metaheuristic to efficiently find exact or heuristic solutions. These
approaches move us one step closer to optimal solutions for 2D L0-norm phase
unwrapping; such solutions were previously viewed, in the signal processing
literature, as highly desirable but not achievable.
