We develop for the first time a mathematical framework in which the class of projection algorithms can be applied to high numerical aperture (NA) phase retrieval. Within this framework, we first analyze the basic steps of solving the high-NA phase retrieval problem by projection algorithms and establish the closed forms of all the relevant projection operators. 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 point-spread-function (PSF) is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which deals with the scalar PSF. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval. Importantly, the improved performance of projection methods over the other classes of phase retrieval algorithms in the low-NA setting now also becomes applicable to the high-NA case. This is demonstrated by the accompanying numerical results which show that available solution approaches for high-NA phase retrieval are outperformed by projection methods.

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.

We present an efficient phase retrieval approach for imaging systems with high numerical aperture based on the vectorial model of the point spread function. The algorithm is in the class of alternating minimization methods and can be adjusted for applications with either known or unknown amplitude of the field in the pupil. The algorithm outperforms existing solutions for high-numerical-aperture phase retrieval: (1) the generalization of the method of Hanser et al., based on extension of the scalar diffraction theory by representing the out-of-focus diversity applied to the image by a spherical cap, and (2) the method of Braat et al., which assumes through the use of extended Nijboer–Zernike expansion the phase to be smooth. The former is limited in terms of accuracy due to model deviations, while the latter is of high computational complexity and excludes phase retrieval problems where the phase is discontinuous or sparse. Extensive numerical results demonstrate the efficiency, robustness, and practicability of the proposed algorithm in various practically relevant simulations.

We consider the extension of the traditional projection-based phase retrieval algorithms by increasing the problem dimensionality and introducing novel projection operators. The approach is demonstrated on an example of phase retrieval for the high-NA case.

We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged mappings to expansive set-valued mappings that characterizes a type of strong calmness of the fixed point mapping. The second component to this analysis is an extension of the well-established notion of metric subregularity - or inverse calmness - of the mapping at fixed points. Convergence of expansive fixed point iterations is proved using these two properties, and quantitative estimates are a natural by-product of the framework. To demonstrate the application of the theory, we prove, for the first time, a number of results showing local linear convergence of nonconvex cyclic projections for inconsistent (and consistent) feasibility problems, local linear convergence of the forward-backward algorithm for structured optimization without convexity, strong or otherwise, and local linear convergence of the Douglas-Rachford algorithm for structured nonconvex minimization. This theory includes earlier approaches for known results, convex and nonconvex, as special cases.

This paper proposes an algorithm for solving structured optimization problems, which covers both the backward–backward and the Douglas–Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the corresponding operator is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point iterations are established. When applied to nonconvex feasibility including potentially inconsistent problems, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets. In this special case, we refine known linear convergence criteria for the Douglas–Rachford (DR) algorithm. As a consequence, for feasibility problem with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems.

We present necessary conditions for monotonicity of fixed point iterations of mappings that may violate the usual nonexpansive property. Notions of linear-type monotonicity of fixed point sequences—weaker than Fejér monotonicity—are shown to imply metric subregularity. This, together with the almost averaging property recently introduced by Luke et al. (Math Oper Res, 2018. https://doi.org/10.1287/moor.2017.0898), guarantees linear convergence of the sequence to a fixed point. We specialize these results to the alternating projections iteration where the metric subregularity property takes on a distinct geometric characterization of sets at points of intersection called subtransversality. Subtransversality is shown to be necessary for linear convergence of alternating projections for consistent feasibility.

We investigate the role of error bounds, or metric subregularity, in the convergence of Picard iterations of nonexpansive maps in Hilbert spaces. Our main results show, on one hand, that the existence of an error bound is sufficient for strong convergence and, on the other hand, that an error bound exists on bounded sets for nonexpansive mappings possessing a fixed point whenever the space is finite dimensional. In the Hilbert space setting, we show that a monotonicity property of the distances of the Picard iterations is all that is needed to guarantee the existence of an error bound. The same monotonicity assumption turns out also to guarantee that the distance of Picard iterates to the fixed point set converges to zero. Our results provide a quantitative characterization of strong convergence as well as new criteria for when strong, as opposed to just weak, convergence holds.

We present a convex relaxation-based algorithm for large-scale general phase retrieval problems. General phase retrieval problems include, e.g., the estimation of the phase of the optical field in the pupil plane based on intensity measurements of a point source recorded in the image (focal) plane. The non-convex problem of finding the complex field that generates the correct intensity is reformulated into a rank constraint problem. The nuclear norm is used to obtain the convex relaxation of the phase retrieval problem. A new iterative method referred to as convex optimization-based phase retrieval (COPR) is presented, with each iteration consisting of solving a convex problem. In the noise-free case and for a class of phase retrieval problems, the solutions of the minimization problems converge linearly or faster towards a correct solution. Since the solutions to nuclear norm minimization problems can be computed using semidefinite programming, and this tends to be an expensive optimization in terms of scalability, we provide a fast algorithm called alternating direction method of multipliers (ADMM) that exploits the problem structure. The performance of the COPR algorithm is demonstrated in a realistic numerical simulation study, demonstrating its improvements in reliability and speed with respect to state-of-the-art methods.