  Fourier phasing is the problem of retrieving Fourier phase information from
Fourier intensity data. The standard Fourier phase retrieval (without a mask)
is known to have many solutions which cause the standard phasing algorithms to
stagnate and produce wrong or inaccurate solutions. In this paper Fourier phase
retrieval is carried out with the introduction of a randomly fabricated mask in
measurement and reconstruction. Highly probable uniqueness of solution, up to a
global phase, was previously proved with exact knowledge of the mask. Here the
uniqueness result is extended to the case where only rough information about
the mask's phases is assumed. The exponential probability bound for uniqueness
is given in terms of the uncertainty-to-diversity ratio (UDR) of the unknown
mask. New phasing algorithms alternating between the object update and the mask
update are systematically tested and demonstrated to have the capability of
recovering both the object and the mask (within the object support)
simultaneously, consistent with the uniqueness result. Phasing with a
phase-uncertain mask is shown to be robust with respect to the correlation in
the mask as well as the Gaussian and Poisson noises.
