Multilevel Gauss–Newton methods for phase retrieval problems

Abstract

The phase retrieval problem is of wide interest because it appears in a number of interesting application areas in physics. Several kinds of phase retrieval problems appeared in laser optics over the past decade. In this paper we consider the numerical solution of two phase retrieval problems for an unknown smooth function f with compact support. We approximate f by a linear spline. The corresponding spline coefficients are iteratively determined by local Gauss-Newton methods, where convenient initial guesses are constructed by a multilevel strategy. We close with some numerical tests which illustrate our method.

Sparse Phase Retrieval of One-Dimensional Signals by Prony's Method

“…Applications of this problem occur in electron microscopy, wave front sensing, laser optics [6,7] as well as in X-ray crystallography and speckle imaging [8]. For the posed problem, we will show that for sparse signals the given Fourier intensities are already sufficient for an almost sure unique recovery, and we will give a construction algorithm to recover f .…”

Sparse phase retrieval of structured signals by Prony's method

“…Using the assumption that the differences T j − T k differ pairwise for j = k, we have rearranged the terms of the sum (2) such that the distinct non-zero frequencies τ := T j − T k together with the zero frequency τ 0 := 0 are ordered by size. The N (N − 1) + 1 frequency differences τ for = − N (N −1) /2, .…”

“…Recovery problems of this kind appear in a wide range of applications such as in electron microscopy, wave front sensing, laser optics [1,2] as well as in X-ray crystallography and speckle imaging [3]. Due to the loss of the phase, in general, the problem is ill-posed and cannot be solved uniquely.…”

Ambiguities in one‐dimensional phase retrieval of structured functions

“…If we choose the centred linear B-spline φ(t) := (1 − |t |) χ [−1,1] (t) as generator function, then the unknown function f is a linear spline function. This specific phase retrieval problem was introduced in [1,2]. Obviously, the considered phase retrieval problem is not uniquely solvable.…”