Convex combination of alternating projection and Douglas–Rachford operators for phase retrieval

Abstract

We present the convergence analysis of convex combination of the alternating projection and Douglas–Rachford operators for solving the phase retrieval problem. New convergence criteria for iterations generated by the algorithm are established by applying various schemes of numerical analysis and exploring both physical and mathematical characteristics of the phase retrieval problem. Numerical results demonstrate the advantages of the algorithm over the other widely known projection methods in practically relevant simulations.

Some New Characterizations of Intrinsic Transversality in Hilbert Spaces

“…Transversality was also used to prove linear convergence of the Douglas-Rachford algorithm [16,49] and its relaxations [52]. A practical application of these results is to the phase retrieval problem where transversality is sufficient for linear convergence of alternating projections, the Douglas-Rachford algorithm and actually any convex combinations of the two algorithms [53].…”

Projection methods for high numerical aperture phase retrieval

“…This algorithm covers both T DR (by setting β = 1) and T AP (by setting β = 0). When A is affine, T DRAP is convex combination of these two operators [57]. The latter also explains its name DRAP which stands for Doughlas-Rachford and AP.…”

“…Recently, however, a framework has been established that accommodates fixed point iterations built from compositions and averages of set-valued, expansive mappings [41]. The extended analysis scheme is based on the theory of pointwise almost averaged mappings and has been applied to prove, for the first time, local (linear) convergence of fundamental algorithms like cyclic projections and relaxed DR for solving inconsistent, nonconvex feasibility problems; see, for example, [15,38,54,55,57]. It is worth mentioning that (low-NA) phase retrieval has been a main motivation of the mathematical development in [41].…”

“…We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms in the presence of noise. Making use of the vectorial PSF is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which mostly deals with the scalar PSF, see, for example, [5,18,20,21,36,37,52,53,55,57]. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval.…”