Conjugate phase retrieval in shift-invariant spaces generated by a Gaussian

TL;DR

This paper investigates the problem of conjugate phase retrieval within Gaussian shift-invariant spaces, demonstrating unique determination from phaseless Hermite samples and providing explicit reconstruction methods for finite coefficient functions.

Contribution

It establishes the unique recoverability of functions in Gaussian shift-invariant spaces from phaseless Hermite samples and offers explicit reconstruction procedures for finite coefficient cases.

Findings

Modulus function determined from phaseless Hermite samples.

Unique function recovery up to unimodular constant and conjugation.

Explicit reconstruction method for finite coefficient functions.

Abstract

Conjugate phase retrieval considers the recovery of a function, up to a unimodular constant and conjugation, from its phaseless measurements. In this paper, we explore the conjugate phase retrieval in a shift-invariant space generated by a Gaussian funciton. First, we show that the modulus function in the Gaussian shift-invariant space can be determined from the phaseless Hermite samples taken on a discrete sampling set. We then show that a function in the shift-invariant space generated by a Gaussian can be uniquely determined, up to a unimodular constant and conjugation, from its phaseless Hermite samples on a discrete set. For the functions with finite coefficient sequences, we provide an explicit reconstruction procedure.