  Phase retrieval is in general a non-convex and non-linear task and the
corresponding algorithms struggle with the issue of local minima. We consider
the case where the measurement samples within typically very small and
disconnected subsets are coherently linked to each other - which is a
reasonable assumption for our objective of antenna measurements. Two classes of
measurement setups are discussed which can provide this kind of extra
information: multi-probe systems and holographic measurements with multiple
reference signals. We propose several formulations of the corresponding phase
retrieval problem. The simplest of these formulations poses a linear system of
equations similar to an eigenvalue problem where a unique non-trivial
null-space vector needs to be found. Accurate phase reconstruction for
partially coherent observations is, thus, possible by a reliable solution
process and with judgment of the solution quality. Under ideal, noise-free
conditions, the required sampling density is less than two times the number of
unknowns. Noise and other observation errors increase this value slightly.
Simulations for Gaussian random matrices and for antenna measurement scenarios
demonstrate that reliable phase reconstruction is possible with the presented
approach.
