Inverting spectrogram measurements via aliased Wigner distribution deconvolution and angular synchronization

Abstract

We propose a two-step approach for reconstructing a signal $\textbf x\in \mathbb{C}^d$ from subsampled discrete short-time Fourier transform magnitude (spectogram) measurements: first, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix $\widehat{\textbf{x}}\widehat{\textbf{x}}^{*}.$ Secondly, we use angular synchronization to solve for $\widehat{\textbf{x}}$ (and then for $\textbf{x}$ by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems; one which guarantees the recovery of discrete, bandlimited signals $\textbf{x}\in \mathbb{C}^{d}$ from fewer than $d$ short-time Fourier transform magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.

On Recovery Guarantees for Angular Synchronization

“…We remind the reader that the goal of ptychography is to estimate the a signal y ∈ C d from phaseless measurements through localized masks. A recent method for recovering the signal y from such observation is the BlockPR algorithm by Iwen et al [14], see [9,[22][23][24] for follow-up works developing this algorithm further that also rely on weighted angular synchronization. The BlockPR algorithm proceeds by combining neighboring masks to obtain estimates for the products of entries located close to each other.…”

“…Combined, they will grant us inequality (18). The first one is required to transit from the signs of z back to z using Inequality (23). The scaling factor guarantees that vector…”

“…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, .…”

Toward Fast and Provably Accurate Near-field Ptychographic Phase Retrieval

“…where x ∈ C d is the sample being imaged, m ∈ C d is a mask which represents the probe's incident illumination on (a portion of) the sample, • denotes the Hadamard (pointwise) product, S k is a shift operator, and D : C d → C d is a function that describes the diffraction of the probe radiation from the sample to the plane of the detector after possibly passing though, e.g, a lens. Prior work in the computational mathematics community related to ptychographic imaging has primarily focused on far-field 1 ptychography (FFP) in which D is the action of a discrete Fourier transform matrix (see, e.g., [10,11,12,13,14,15]) in (1). Here, in contrast, we consider the less well studied setting of near-field ptychography (NFP) which describes situations where the masked sample is too close to the detector to be well described by the FFP model.…”