Fourier phase retrieval(PR) is a severely ill-posed inverse problem that
arises in various applications. To guarantee a unique solution and relieve the
dependence on the initialization, background information can be exploited as a
structural priors. However, the requirement for the background information may
be challenging when moving to the high-resolution imaging. At the same time,
the previously proposed projected gradient descent(PGD) method also demands
much background information.
In this paper, we present an improved theoretical result about the demand for
the background information, along with two Douglas Rachford(DR) based methods.
Analytically, we demonstrate that the background required to ensure a unique
solution can be decreased by nearly 1/2 for the 2-D signals compared to the
1-D signals. By generalizing the results into d-dimension, we show that the
length of the background information more than (2dd+1ββ1) folds of
the signal is sufficient to ensure the uniqueness. At the same time, we also
analyze the stability and robustness of the model when measurements and
background information are corrupted by the noise. Furthermore, two methods
called Background Douglas-Rachford (BDR) and Convex Background Douglas-Rachford
(CBDR) are proposed. BDR which is a kind of non-convex method is proven to have
the local R-linear convergence rate under mild assumptions. Instead, CBDR
method uses the techniques of convexification and can be proven to own a global
convergence guarantee as long as the background information is sufficient. To
support this, a new property called F-RIP is established. We test the
performance of the proposed methods through simulations as well as real
experimental measurements, and demonstrate that they achieve a higher recovery
rate with less background information compared to the PGD method