Phase retrieval by iterated projections

TL;DR

The paper introduces the difference map, a unifying iterative framework for phase retrieval built from two elementary projections and a real parameter $β$, unifying Fourier modulus, support, histogram, and atomicity constraints. For the standard optics case (Fourier modulus and support), the special choice $β = 1$ reproduces Fienup's hybrid input-output map, while other $β$ values also yield effective maps without traditional input-output counterparts. The authors analyze local convergence, characterize fixed points as solutions, and show that stagnation is avoided by the difference-map dynamics; they extend the approach to crystallography via Sayre's equation and atomicity projections, with numerical experiments indicating feasibility for structures comprising hundreds of atoms. The work bridges imaging-phase retrieval and crystallographic phase problems, offering a versatile, geometry-based framework with practical success in synthetic data experiments. Overall, the difference map provides a flexible tool that can incorporate diverse a priori constraints and offers insights into convergence behavior and algorithmic design for complex phase-retrieval tasks.

Abstract

Several strategies in phase retrieval are unified by an iterative "difference map" constructed from a pair of elementary projections and a single real parameter $β$. For the standard application in optics, where the two projections implement Fourier modulus and object support constraints respectively, the difference map reproduces the "hybrid" form of Fienup's input-output map for $β= 1$. Other values of $β$ are equally effective in retrieving phases but have no input-output counterparts. The geometric construction of the difference map illuminates the distinction between its fixed points and the recovered object, as well as the mechanism whereby stagnation is avoided. When support constraints are replaced by object histogram or atomicity constraints, the difference map lends itself to crystallographic phase retrieval. Numerical experiments with synthetic data suggest that structures with hundreds of atoms can be solved.

Figures (13)

• Figure 1: Examples of elementary projections. (a) Random positive density; (b) Fourier modulus projection of (a); (c) histogram projection of (b); (d) atomicity projection of (b).
• Figure 2: Constraint subspaces (perpendicular rods) in the neighborhood of a solution (a) and a trap (b). The two circular disks in (a) represent points which project to the intersection of the constraint subspaces, $\rho_{1\cap 2}$, under action of $\pi_1 \circ f_2$ and $\pi_2 \circ f_1$ respectively; their intersection is the set of fixed points of the difference map. When the constraint subspaces do not intersect, as in (b), the action of the difference map is to move the iterates along the axis of minimum separation.
• Figure 3: Plot of the map $f_{\alpha}$, Eq. (\ref{['falpha']}), composed with itself three times for $\alpha=0.37$ and $c=8/3$. Points of intersection with the straight line give the three fixed points $\lambda=0,c^{-1},1$; the divergent behavior near $\lambda=-1$ and $\lambda=2$ corresponds to the "uranium" instability.
• Figure 4: Object (a) and corresponding Fourier modulus (b) used in subsequent phase retrieval experiments.
• Figure 6: Same as Figure 5 but with histogram constraints. Far fewer iterations are required to recover the same object.