On relaxed averaged alternating reflections (RAAR) algorithm for phase retrieval with structured illumination

In this paper, we consider the phase retrieval problem with structured illumination, which leads to a pixel-dependent deterministic phase shift term in the physical model. The existence of the phase shift term can ease the numerical algorithm for phase retrieval. The relaxed averaged alternating reflections (RAAR) algorithm is modified to adapt to two or more diffraction patterns, and the modified RAAR algorithm operates in the Fourier domain rather than the space domain. The RAAR is motivated by a linear combination of the reflection projection and projection onto the measurement space, with parameter β trading off the two projections. Although the local convergence of the RAAR algorithm with an initialization within the basin of attraction is proved for 0 < β ⩽ 1 , the numerical performance with a random initialization varies with different values of β . Numerical simulations are presented to demonstrate the effectiveness and stability of the algorithm with 0.5 < β < 1 , compared to the ER ( β = 0.5 ) method and Douglas–Rachford ( β = 1 ) method. The numerical global convergence of the RAAR with 0.5 < β < 1 is also illustrated in our tests.