Well-conditioned ptychographic imaging via lost subspace completion

Abstract

Ptychography, a special case of the phase retrieval problem, is a popular method in modern imaging. Its measurements are based on the shifts of a locally supported window function. In general, direct recovery of an object from such measurements is known to be an ill-posed problem. Although for some windows the conditioning can be controlled, for a number of important cases it is not possible, for instance for Gaussian windows. In this paper we develop a subspace completion algorithm, which enables stable reconstruction for a much wider choice of windows, including Gaussian windows. The combination with a regularization technique leads to improved conditioning and better noise robustness.

On Recovery Guarantees for Angular Synchronization

“…in [14] for the case of unweighted graphs. Their proof is based on the Cheeger inequality that is only available for the normalized Laplacian, which is why the minimization problem in their theorem has a different normalization than (9). In the special case that deg( ) is a constant for all (as in [14]), the two normalizations agree up to a constant.…”

“…Suppose that G = (V , E, W ) is a weighted graph with τ G > 0. Letx ∈ T d be the minimizer of the LSP (8) and z be the minimizer of the ER (9). Set R ∈ C d×d as R , j = W…”

“…Corollary 1 Suppose G = (V , E) is an unweighted graph with τ G > 0. Let z ∈ C d be the minimizer of the ER (9). Then…”

“…This problem is commonly referred to as angular synchronization or phase synchronization. It frequently arises in various applications such as recovery from phaseless measurements [1,9,14,17,20,[22][23][24], ordering of data from relative ranks [7], digital communications [15], jigsaw puzzle solving [12] and distributed systems [10]. The problem of angular synchronization is also closely related to the broader problem of pose graph optimization [5], which appears in robotics and computer vision and group synchronization problems [19,21].…”

“…In many applications such as certain algorithms for ptychography [9,14,[22][23][24], noisy observations of only a strict subset of the set of differences are available. To mathematically describe this restriction we will work with the quantity E := {( , j) ∈ {1, .…”