Fourier phase retrieval with a single mask by Douglas–Rachford algorithms

Abstract

The Fourier-domain Douglas-Rachford (FDR) algorithm is analyzed for phase retrieval with a single random mask. Since the uniqueness of phase retrieval solution requires more than a single oversampled coded diffraction pattern, the extra information is imposed in either of the following forms: 1) the sector condition on the object; 2) another oversampled diffraction pattern, coded or uncoded. For both settings, the uniqueness of projected fixed point is proved and for setting 2) the local, geometric convergence is derived with a rate given by a spectral gap condition. Numerical experiments demonstrate global, power-law convergence of FDR from arbitrary initialization for both settings as well as for 3 or more coded diffraction patterns oversampling. In practice, the geometric convergence can be recovered from the power-law regime by a simple projection trick, resulting in highly accurate reconstruction from generic initialization.

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

“…The proof, which is a mimic of the same result in literature [24], is given in next section. Theorem 4.…”

“…When β " 1, the unrelaxed RAAR algorithm becomes HIO algorithm in Fourier domain [24]. Since each of the algorithms involve the same basic operations at each iteration, the rate of convergence of one algorithm relative to another boils down to iteration counts.…”

“…In the regime of Fresnel, the phase shift is automatically introduced in the original signal/image. In this paper, we consider the more stable RAAR algorithm instead of the HIO method used in [24]. A common and vexing problem for the HIO method is that iterates will oscillation around the solution in the noiseless case and even wander away from the neighborhood of the solution in the presence of noise (see our numerical simulations in Section 5).…”

“…In this setting, even the ER algorithm behaves well for nonnegative image. And recently the local geometrical convergence to a solution for HIO is proved [24,25]. An important difference between the optimization approaches in [20][21][22] and the standard iterative projection methods is that their coded diffraction patterns are not oversampled.…”

“…An important difference between the optimization approaches in [20][21][22] and the standard iterative projection methods is that their coded diffraction patterns are not oversampled. We emphasize that reducing the number of coded diffraction patterns is crucial for the diffract-before-destruct approach and oversampling is a small price to pay with current sensor technology [24].…”